Document [original]

Non-equilibrium self-assembly

processes of complex dipolar

colloids

vorgelegt von:

Diplomphysiker

Florian Kogler

aus

Ulm, Baden-W¨

urttemberg

Von der Fakult¨

at II – Mathematik und Naturwissenschaften

der Technische Universit¨

at Berlin

zur Erlangung des akademischen Grades

Doktor der Naturwissenschaften

Dr. rer. nat.

genehmigte Dissertation

Promotionsausschuss:

Vorsitzender: Prof. Dr. Martin Schoen

1. Gutachterin: Prof. Dr. Sabine H. L. Klapp

2. Gutachter: Prof. Dr. Joachim Dzubiella

3. Gutachterin: Prof. Dr. Carol K. Hall

Tag der wissenschaftlichen Aussprache: 26.2.2016

Berlin, Deutschland.

2016

D-83

Declaration of Authorship

Ich versichere hiermit eidesstattlich, dass die vorliegende Dissertation

selbst¨

andig verfasst wurde, die benutzten Hilfsmittel und Quellen kor-

rekt aufgef ¨

uhrt sind und im Falle von Co-Autorenschaft die Darstellung

des Eigenanteils zutreffend ist.

Acknowledgments

At this point I want to take the chance to say ’thank you’ to all the peo-

ple who supported me in writing this thesis, be it by scientific or personal

means.

Special thanks go to Prof. Dr. Klapp for her endless efforts and all the very

helpful advices, hints and discussions. Her commitment and support was

crucial for the success of this work.

I also want to thank Prof. Dr. Hall for her hospitality during my stay in

her group and her encouraging words. I always felt welcome and learned

a lot. During this stay in Raleigh I spent a lot of time with Dr. Bharti,

talking and discussing on science and everything else. Without him, ev-

erything would have been much more difficult. Also, I want to thank Prof.

Dr. Velev for giving me the chance to relate my work to what happens in

experiments.

My work was financed by the IRTG 1524, and I appreciate very much all

the effort of Petra, Beatrix, Daniela and Prof. Dr. Schoen to keeping every-

thing together. Additionally, I want to thank Prof. Dr. Schoen and Prof.

Dzubiella who are willing to be part of my PhD committee.

Finally, special thanks go to the members of the AG Klapp for making

daily life a nice thing and for being supportive and for helping out, when

work becomes difficult.

Zusammenfassung

Diese Arbeit besch¨

aftigt sich mit dem Einfluss induzierter dipolarer

Wechselwirkungen auf die nichtgleichgewichts Strukturbildung in quasi

zweidimensionalen kolloidalen Systemen. Einerseits werden konkrete

experimentell realisierte Systeme theoretisch modelliert und mit Hilfe

von Computersimulationen der Brownschen Dynamik untersucht. Zur

Absch¨

atzung von Phasen ¨

uberg¨

angen werden weiterhin Methoden der

Dichtefunktionaltheorie angewandt. Andererseits werden generische

Modelle entwickelt um grundlegende Mechanismen der Strukturbildung

zu indentifizieren. Die zugrundeliegenden experimentellen Systeme

entsprechen dabei zwei prototypischen nichgleichgewichts Prozessen:

einmal der Bahnenbildung (lane formation) und einmal der diffusions-

beschr¨

ankten Aggregation (diffusion limited aggregation).

Im Falle der Bahnenbildung werden Januspartikel in einem elektrischen

Wechselfeld betrachtet, die, je nach Feldfrequenz, entweder kettenartige

Strukturen oder einen Schwimmmechanismus ausbilden. Die theoretis-

che Untersuchung einer explizit dipolaren Modellierung zeigt sowohl die

Kettenbildung als auch neuartige Bahnenbildung im Schwimmerregime.

Zus¨

atzlich wird ein abstrakteres Referenzsystem behandelt, welches es er-

stmalig erlaubt eine Verbindung zwischen spinodialer Entmischung und

Bahnenbildung herzustellen. Weiterhin wird hier zum ersten mal Bah-

nenbildung in Systemen mit attraktiven Teilchenwechselwirkungen un-

tersucht.

Der zweite Teil der Arbeit behandelt die feldgesteuerte Aggregation kol-

loidaler Teilchen im diffusionsbeschr¨

ankten Regime. Hier werden eben-

falls zwei Modellsysteme untersucht. Zun¨

achst werden experimentell

beobachtete Strukturen analysiert und mit Ergebnissen aufwendiger

Computersimulationen verglichen. Zu diesem Zweck wird eine neue

Methode der Strukturanalyse eingef ¨

uhrt und angewandt. Die Experi-

mente wurden von Dr. Bhuvnesh Bharti an der North Carolina State

University durchgef ¨

uhrt und von mir analysiert. Die theoretischen

Betrachtungen erlauben hierbei, im Gegensatz zu den Experimenten,

das Aggregationsverhalten in Abh¨

angigkeit der Partikeldichte zu un-

tersuchen. Erg¨

anzend dazu wird ein zweites Modellsystem betrachtet,

welches auf generische Eigenschaften der feldgesteuerten Aggregation

ausgerichtet ist. Hier liegt besonderes Augenmerk auf dem Verhalten im

diffusionsbeschr¨

ankten Aggregationsregime. Letzteres ist ein ausgiebig

untersuchter nichtgleichgewichts Prozess, der hier um neue Facetten, also

feldinduzierete dipolare Wechselwirkungen, erg¨

anzt wird.

iii

Abstract

This work investigates the influence of induced dipolar interactions on the

non-equilibrium structure formation in quasi-two dimensional colloidal

systems. On the one hand, theoretical models for certain experimentally

realized systems are developed and investigated by means of Brownian

Dynamics Simulations. To estimate phase transitions, basic density func-

tional theory methods are applied. On the other hand, also generic models

are developed with the aim to unveil the underlying mechanisms of struc-

ture formation and particle aggregation. The experimental systems moti-

vating this work correspond to two prototype non-equilibrium processes,

which are lane formation and diffusion limited aggregation.

In the case of lane formation it is Janus particles in AC electric fields

which are considered. These particles form either chain-like aggregates or

yield a self-propulsion mechanism, depending on the field frequency. The

investigation of an explicitly dipolar model shows chain formation as well

as lane formation in the self-propulsion regime. Additionally, a generic

system is considered, which allows for the first time to make a connec-

tion between lane formation and spinodal decomposition. Furthermore,

this is the first study on lane formation in systems with attractive particle

interactions.

The second part of this work investigates the field directed assembly

of colloidal particles in the diffusion limited aggregation regime. Again,

two model systems are developed and investigated. First, experimentally

observed structures are analyzed and compared to results from extensive

computer simulations. To this end, a new method for structure analysis is

introduced and applied. Experiments were performed by Dr. B. Bharti at

North Carolina State University and analyzed by myself. The theoretical

considerations allow, in contrast to experiments, to investigate the depen-

dency of the aggregation behavior on the particle density. Additionally, a

second model system, aiming on the generic features of particle aggrega-

tion is investigated. Here, special attention is paid on the diffusion limited

aggregation. The latter is an extensively studied non-equilibrium process,

which is extended by new features, namely the influence of field induced

dipolar interactions.

Publications

Most of this work is based on the publications listed below. Texts are

partially rewritten and some extensions are introduced. Contributions

from Co-Authors presented in this work are indicated in the text.

Lane formation in a system of dipolar microswimmers Florian

Kogler and Sabine H. L. Klapp, Europhys. Lett. 110,10004, (2015).

Using Brownian Dynamics (BD) simulations we investigate the non-

equilibrium structure formation of a two-dimensional (2D) binary sys-

tem of dipolar colloids propelling in opposite directions. Despite of a

pronounced tendency for chain formation, the system displays a transi-

tion towards a laned state reminiscent of lane formation in systems with

isotropic repulsive interactions. However, the anisotropic dipolar inter-

actions induce novel features: First, the lanes have themselves a complex

internal structure characterized by chains or clusters. Second, laning oc-

curs only in a window of interaction strengths. We interprete our findings

by a phase separation process and simple force balance arguments.

Generic model for tunable colloidal aggregation in multidirec-

tional fields Florian Kogler, Orlin D. Velev, Carol K. Hall and Sabine

H. L. Klapp, Soft Matter,11,7356 -7366, (2015)

Based on Brownian Dynamics computer simulations in two dimensions

we investigate aggregation scenarios of colloidal particles with directional

interactions induced by multiple external fields. To this end we propose

a model which allows continuous change in the particle interactions from

point-dipole-like to patchy-like (with four patches). We show that, as a re-

sult of this change, the non-equilibrium aggregation occurring at low den-

sities and temperatures transforms from conventional diffusion-limited

cluster aggregation (DLCA) to slippery DLCA involving rotating bonds;

this is accompanied by a pronounced change of the underlying lattice

structure of the aggregates from square-like to hexagonal ordering. In-

creasing the temperature we find a transformation to a fluid phase, con-

sistent with results of a simple mean-field density functional theory.

Contents

1 Introduction 1

1.1Complex Colloids . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2Active Matter . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.2.1Colloidal self-propulsion . . . . . . . . . . . . . . . . 5

1.2.2Collective behavior of active colloids . . . . . . . . . 6

1.2.3Lane formation . . . . . . . . . . . . . . . . . . . . . 7

1.2.4Self-propelling metallo-dielectric Janus particles . . 8

1.3Non-Equilibrium Aggregation of Colloids . . . . . . . . . . 10

1.3.1Disordered solids - glasses and gels . . . . . . . . . . 10

1.3.2Diffusion limited aggregation . . . . . . . . . . . . . 12

1.3.3Aggregated networks of superparamagnetic colloids 14

1.4Outline of this Thesis . . . . . . . . . . . . . . . . . . . . . . 15

2 Theoretical Concepts and Methods 17

2.1Langevin Equation . . . . . . . . . . . . . . . . . . . . . . . 18

2.1.1Ovderdamped limit . . . . . . . . . . . . . . . . . . . 19

2.2Phase Transitions . . . . . . . . . . . . . . . . . . . . . . . . 20

2.3Phase Coexistence . . . . . . . . . . . . . . . . . . . . . . . . 20

2.4Pair Correlation Function . . . . . . . . . . . . . . . . . . . . 22

2.5Direct Correlation Function . . . . . . . . . . . . . . . . . . 23

2.5.1Random phase approximation . . . . . . . . . . . . . 24

2.6Ornstein-Zernike Equation . . . . . . . . . . . . . . . . . . 25

2.7Kirkwood-BuffEquation . . . . . . . . . . . . . . . . . . . . 26

3 Theoretical Models of Complex Dipolar Colloids 29

3.1Colloidal Interaction Potentials . . . . . . . . . . . . . . . . 30

3.1.1Representations of anisotropic pair interactions in

two-dimensions . . . . . . . . . . . . . . . . . . . . . 32

3.2The DDP Model for Dipolar Microswimmers in Experiment 34

3.2.1The driven Lennard Jones fluid - A generic model

for attractive microswimmers . . . . . . . . . . . . . 37

3.2.2Target quantities . . . . . . . . . . . . . . . . . . . . 38

3.3Colloids in Crossed Fields . . . . . . . . . . . . . . . . . . . 39

3.4The CDP Model for Superparamagnetic Particles in Exper-

iment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

3.5Characterization of Network Structures - Energy Histograms 46

Contents

3.6The EDP Model - A Generic Approach towards Mutidirec-

tional Field Induced Dipolar Interactions . . . . . . . . . . 50

4 Lane Formation of Dipolar Microswimmers 55

4.1Background . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

4.2Lane Formation of Dipolar Microswimmers . . . . . . . . . 57

4.2.1Complex lanes - Internal structure . . . . . . . . . . 59

4.2.2Complex lanes - Characteristic lane widths . . . . . 60

4.3Mechanisms of Lane Formation . . . . . . . . . . . . . . . . 63

4.3.1Lane formation in the isotropic model . . . . . . . . 64

4.3.2The laning state diagram for dipolar swimmers . . . 67

4.4Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

5 Mutidirectional Colloidal Assembly 71

5.1Field Controlled Assembly of Colloidal Networks - Experi-

ment and Simulation . . . . . . . . . . . . . . . . . . . . . . 73

5.1.1Background . . . . . . . . . . . . . . . . . . . . . . . 73

5.1.2Structure formation in experiment . . . . . . . . . . 74

5.1.3Comparing simulation and experiment . . . . . . . . 74

5.1.4State diagram . . . . . . . . . . . . . . . . . . . . . . 78

5.2Diffusion Limited Aggregation in Mutidirectional Fields . . 83

5.2.1Background . . . . . . . . . . . . . . . . . . . . . . . 83

5.2.2The interaction anisotropy - Effect on local order . . 86

5.2.3Transient character of aggregates . . . . . . . . . . . 90

5.2.4The interaction anisotropy - Effect on large-scale

fractal structure . . . . . . . . . . . . . . . . . . . . . 92

5.2.5Beyond DLCA - Higher temperatures . . . . . . . . . 94

5.2.6Spotlight on higher densities . . . . . . . . . . . . . 99

5.3Small and Large-Scale Structures . . . . . . . . . . . . . . . 102

5.3.1Strange compactifaction - A hybrid of anisotropic

and slippery DLA? . . . . . . . . . . . . . . . . . . . 104

5.4Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

6 Conclusion and Outlook 107

Appendix 117

Bibliography 127

viii

List of Figures

1.1Examples of complex colloids and their assembly. . . . . . . 2

1.2Examples of colloidal microswimmers and their collective

behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.3Experimental image of metallo-dielectric Janus-particles. . 8

1.4Self-propulsion of Metallo-dielectric Janus-particles. . . . . 9

1.5Examples of non-equilibrium colloidal aggregates - Fractal

structures and gels. . . . . . . . . . . . . . . . . . . . . . . . 12

1.6Field directed aggregation of superparamagnetic polymer

particles under the presence of crossed external fields. . . . 15

2.1Phase diagram of simple fluid - Coexistence region . . . . . 21

3.1Energy ’landscape’ of a single dipole particles. . . . . . . . . 32

3.2Potential energy ’landscape’ for two DDPs with opposite

orientation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

3.3Angle-averaged DDP potential . . . . . . . . . . . . . . . . . 37

3.4Energy ’landscape’ of the complex model for superparam-

agnetic particles in crossed external fields. . . . . . . . . . . 44

3.5Basic building blocks for colloidal networks. . . . . . . . . . 46

3.6Comparison of colloidal networks in experiment and simu-

lation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

3.7Sketch of the generic model for colloids in crossed fields. . 51

3.8Energy ’landscapes’ for the generic model in crossed exter-

nal fields. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

4.1Laning order parameter as function of driving force for

DDP and SS model . . . . . . . . . . . . . . . . . . . . . . . 57

4.2Snapshots of the DDP system - equilibrium aggregates and

laned structures. . . . . . . . . . . . . . . . . . . . . . . . . . 58

4.3Internal structures of lanes via pair correlation functions. . 59

4.4Large scale structure of lanes in the DDP model via pair

correlation functions. . . . . . . . . . . . . . . . . . . . . . . 61

4.5Lane width in the DDP model - dependency on the driving

force. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

4.6Large scale structure of lanes in the SS model via pair cor-

relation functions. . . . . . . . . . . . . . . . . . . . . . . . . 63

4.7The laning transition in the driven Lennard-Jones fluid and

the laning state diagram. . . . . . . . . . . . . . . . . . . . . 65

List of Figures

4.8Schematic sketch for competition between attraction and

driving force . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

4.9Effective interaction potential for the LJ-system. . . . . . . . 67

4.10 State diagram for lane formation in the DDP system. . . . . 69

5.1Comparison of field directed string fluids and network

structures of superparamagnetic particles in experiment

and simulation. . . . . . . . . . . . . . . . . . . . . . . . . . 76

5.2Non-equilibrium state diagram for the aggregation of su-

perparamagnetic particles in crossed fields at density ρ∗=

0.35. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

5.3Collapse of field assembled network structure in simulation. 81

5.4Simulation snapshots at ρ∗=0.3and T∗=0.05 for (a) δ=

0.1σ, (b) δ=0.21σand (c) δ=0.3σ. Particles are colored

according to their value of φ4

i. . . . . . . . . . . . . . . . . . 87

5.5Results for simulations with N=1800 at temperature

T∗=0.05 and density ρ∗=0.3. (a) Orientational bond

order parameters Φ4for square (black) and Φ6(yellow)

for hexagonal particle arrangements. (b) Mean coordina-

tion number ¯

zas function of charge separation δat times

t=100,200,300τb. . . . . . . . . . . . . . . . . . . . . . . . 88

5.6Minimum energy of a particle with six neighbors in hexag-

onal arrangement as function of δ(black) and energy for a

particle in rectangular arrangement with 4neighbors (red). 89

5.7Time correlation functions obtained from simulations with

N=1800 at temperature T∗=0.05 and density ρ∗=0.3.

(a) [(b)] Time evolution of the bond [angle] auto-correlation

function cb(t) [ca(t)] for three different charge separations

δ=0.1σ , 0.21σand 0.3σcolored in yellow, purple and black

respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

5.8Fractal dimension Dfas a function of charge separation at

ρ∗=0.3and T∗=0.05. At δ=0.21σwe find a bimodal

distribution of fractal dimension with peaks at Df=1.48

(solid line) and 1.6(dashed line). . . . . . . . . . . . . . . . 93

5.9Temperature dependence of the system properties at den-

sity ρ∗=0.3for charge separations δ=0.1,0.21,0.3σcol-

ored in yellow, purple and black, respectively. (a) Fractal

dimension Dfevaluated at t≈250τb, (b) Mean coordina-

tion number, (c) Orientational order parameters Φ4and Φ6.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

5.10 Bond auto correlation function cb(t) for different tempera-

tures T∗at charge separation δ=0.3σand density ρ∗=0.3.96

5.11 Simulation snapshots at ρ∗=0.3with δ=0.21σat (a) T∗=

0.15, (b) T∗=0.3, and (c) T∗=0.45. Particles are colored

according to their value φ4

i. . . . . . . . . . . . . . . . . . . 97

List of Figures

5.12 Numerical solutions to Eq.5.10 as function of T∗for density

ρ∗=0.3and charge separations δ=0.1,0.21,0.3σcolored in

yellow, purple and black, respectively. . . . . . . . . . . . . 98

5.13 System at T∗=0.05, density ρ∗=0.7and δ=0.21σ. The

color-code gives the orientational bond-order parameter φ4

of each particle i. . . . . . . . . . . . . . . . . . . . . . . . . 100

5.14 (a) Orientational bond order parameter Φ4and Φ6as func-

tion of density ρ∗for δ=0.21σat T∗=0.05. (b) Mean coor-

dination number ¯

zas function of ρ∗at T∗=0.05 for differ-

ent δ=0.1σ , 0.21σand 0.3σcolored in yellow, purple and

black, respectively. . . . . . . . . . . . . . . . . . . . . . . . 101

5.15 Small- and large-scale images of network structures in ex-

periment and Simulation . . . . . . . . . . . . . . . . . . . . 103

6.1Simulation snapshots at ρ∗=0.3,T∗=1and µ∗=3.5for

driving forces (a) f∗

d=1, (b) f∗

d=3and (c) f∗

d=5. Brighter

dots in particles indicate the particle orientation. . . . . . . 111

6.2Mean square displacement in the active dipole fluid at ρ∗=

0.6,T∗=1and f∗=2for different dipole strengths. . . . . . 112

Introduction

Controlling colloidal self-organization processes is nowadays a

promising route towards the design of new functional materials

and research aiming on unveiling the underlying mechanisms is a

rapidly developing discipline. The scientific understanding of col-

loidal self-assembly processes is mostly based on statistical physics,

thermodynamics and chemistry. Together, these fields of science pro-

vide great knowledge about the nature of particle interactions and

fundamental theories for the equilibrium phase behavior of particle

ensembles. However, the interplay between more complex particle

interactions and the out-of-equilibrium collective behavior lacks a

general understanding. In this introductory chapter, we review cer-

tain aspects of the present status of the scientific understanding of

complex collective colloidal behavior.

1.1 Complex Colloids

Colloidal suspensions are soft matter systems, consisting of nano to mi-

crometer sized solid particles, so called colloids, which are suspended in

a carrier liquid. A single colloid in a fluid of temperature Tis subject to

thermal fluctuations and undergoes Brownian motion. Furthermore, the

thermal energy (≈kBTwith kBbeing Boltzmann’s constant) also results in

a threshold for the influence of interactions between colloids. It is only

colloidal interactions associated with energies ϵ≳kBTwhich are able to

dominate over thermal fluctuations.

An important strategy in exploiting colloidal self-assembly is based on

manipulating the way colloids interact and the variety of possible col-

loidal interactions is enormous. A simple categorization is given by their

1.1Complex Colloids

Figure 1.1: (a) Microscopic photographs of rod-, pyramid and cube-like

colloids [1]. (b) Aggregation of permanent dipole particles in

an external field [2]. (c) Particles with an off-centered mag-

netic inlay forming reversible doublet and triplet aggregates

[3]. (d) Multivalent colloidal ’super-atoms’ with various num-

ber of patches [4]. (e) Experimental image of a magnetic Janus-

particles. Differences in the brightness of the hemispheres cor-

respond to different surface materials [5]. (f) and (g) tube-

like aggregates of magnetic Janus-particles in precessing fields

field conditions [6].

origins. There exist forces exerted on the colloids by the solvent, e.g.,

depletion forces, by pair interactions between the colloids, e.g., electro-

static interactions, and by external fields e.g., gravitation or electromag-

netic fields. An important example is the van der Waals interaction, which

results from the instantaneous microscopic polarization of colloids. Im-

portantly, van der Waals forces are present in every colloidal suspension

and the presence of these short-ranged attractive forces can lead to parti-

cle aggregation. However, many colloidal particles are negatively charged,

often by purpose, and repel each other due to electrostatic interactions.

This counter acts the attraction due to the van der Waals interaction and

can prevent aggregation. Thus, the stability of the homogeneous phase is

governed by the interplay between attractive and repulsive forces. Control

over this interplay can be achieved by screening the electrostatic interac-

tions via the concentration of electrolytes in the liquid [7].

1Introduction

Anisotropic interactions

Of special interest however are particles with anisotropic interactions as

they allow an even richer phase behavior [8]. These anisotropies can be

introduced in different ways. One major example is changing particle

shapes towards elongated, rod-like objects. At high packing fractions, en-

sembles of such particles are named liquid crystals and form the basis for

many daily life technologies [9]. But also other shapes like stars [10], pyra-

mids or cubes have gathered attention [11,1,12]. Microscopic images of

such colloids are shown in Fig. 1.1(a).

Interaction anisotropies can also be achieved by functionalizing dis-

crete parts of the particle surface. An important approach is here to

build Janus-particles [13,14], where a certain part of the particle sur-

face is coated by a material different from the particle body. Partially

metal-coated polystyrene spheres are one example [15] but also carbon-

coated [16] or amphiphilic [17,18] and hydrophobic [19] Janus-particles

have been studied. In Fig. 1.1(e) a microscopic image of a cobalt coated

magnetic Janus-particle is shown [5]. Also functionalizing the particle

surface via polymers, i. e. via DNA, allows to introduce selective particle

interactions [20,21,8].

Such efforts also follow the idea to mimic the valency of atoms by col-

loids [4]. In Fig. 1.1(d) we show microscopic images of complex colloids

with DNA functionalized patches, which results in different interaction

valencies. Additionally, images of ’colloidal molecules’ assembled from

these particles are shown. Such systems are still accessible by microscopes

and easy to handle at room temperature but may be able to undergo ag-

gregation in analogy to atoms.

Dipolar interactions

Directional or anisotropic interactions can also be realized by introducing

magnetic and/or electric dipole and/or multipole moments which allow

selective particle bonding [22,11,5,23,24,13]. In Fig. 1.1(b), a computer

simulation result showing chain formation in a system of permanent mag-

netic particles is presented [2]. Here, fine particle chains are oriented

along an external magnetic field, which is an example for the general

property of dipolar particles to orient along fields and to form chains [22].

Additionally, colloids with a dislocated permanent magnetic interaction

site are shown in the microscopic image Fig. 1.1(c). These particles are

able to form reversibly self-assembled non-linear aggregates, which addi-

tionally can be tuned by external fields [3].

Such complex dipole particles allow a variety of self-assembled struc-

tures [11], from field directed networks [25,26], staggered chains and

pseudo-crystalline states [15] to compact clusters with strange magnetiza-

tion properties [5,27], or magnetic tubes [6] as shown in Fig. 1.1(f) and (g)

for magnetic Janus particles in experiment and simulation, respectively.

1.2Active Matter

Within this class, particles with field-induced dipolar interactions [10,

24,28,25,15,26] are especially interesting because switching the fields on

and offcan be equivalent to switching the particle interactions on and off.

This means that aggregation mechanisms [29,30] can be ’dialed in’. Fur-

thermore, the orientation of inductive fields may be used to direct particle

aggregation [31,25,28,15,26,32,33]. Exemplary, we show in Fig. 1.3

microscopic images of metallo dielectric Janus particles (a). Under the

presence of an external electric field pointing vertically here the particles

are (b) aggregated into staggered chains at low density [15] and (c) form

percolated networks at higher density [26].

In general, recent experimental progress in the synthesis and directional

binding of nanometer to micrometer sized patchy and anisotropic parti-

cles makes possible the assembly of colloidal structures with multiple di-

rected bonds [14,13,4,34]. In consequence, such directed self-assembly

processes may be exploited for the formation of new functional materi-

als with specific and/or adjustable properties. Hence, understanding the

interplay between externally induced particle properties, external fields

and thermodynamic conditions, e.g., temperature, is of fundamental in-

terest in modern material science, but also from a statistical physics point

of view.

1.2 Active Matter

In recent years the design of active self-propelling colloidal particles has

attracted a lot of attention [35]. Literature partially uses the terms ’self-

propelling’ and ’active’ as synonyms [36]. However, self-propelling par-

ticles are a subcategory of active particles. Generally speaking, active

particles transform and dissipate energy to pertain some sort of (complex)

non-equilibrium behavior, while self-propelling particles generate trans-

lational motion. The energy fueling the self-propulsion can be harvested

from the environment or stored inside the particle. An example for har-

vesting active systems is protein machines [37,38], which consume adeno-

sine triphosphate (ATP) from the solvent and catalyze a chemical reaction.

The energy associated with this reaction can then be used to perform con-

trolled motion.

Of special interest here are self-propelling colloids which perform a run

and tumbling motion [39], where the direction of motion is subject to

(rotational) diffusion. This in contrast to the case of other motile non-

equilibrium systems [40] e.g., field driven particles [29,41], particles with

enhanced diffusion in external fields [2,42] or colloids under shear [43].

Here it is not the colloids which consume energy and transform it into mo-

tion by some internal mechanism; rather the colloids are passive subjects

to external conditions.

1Introduction

Figure 1.2: (a) Magnetically self-assembled single flagella swimmer [44].

(b) Complex ’snake-swimmer’ composed of magnetic chains

attached to a gold bead [45]. Both swimmers (a) and (b) require

an external actuation of the flagella-like parts via alternating

magnetic fields. (c) Phase separation into a clustered and a di-

lute phase in a system of self-propelling repulsive spheres in

two-dimensions. Left computer simulation, right microscopic

image from experiment [16]. (d) Simulated transition towards

a laned state in two-dimensions [46]. Darker particles move

downwards with increasing velocity from left to right.

1.2.1 Colloidal self-propulsion

Several colloidal self-propulsion mechanisms have been proposed and in-

vestigated in experiment [47]. Some of the presented micro swimming

devices are larger aggregates of i.e., magnetic colloids (see Fig. 1.2(a) and

(b)), which propel by means of a periodical mechanical deformation of

flagella like sub structures [48,44,45]. The mechanical deformation is

fueled by external fields and requires explicit control by humans. Also

active liquid crystals have gather attention [49,50].

Other swimmers are essentially Janus particles, where a certain part

1.2Active Matter

of the colloid surface is influencing the solvent and generating some sort

of gradient. This can be done by an asymmetric chemical reaction [51,

16,47] of e.g., a metal coated surface part, which locally catalyzes the

decomposition of a dissolved fuel. This chemical reaction and the energy

associated with it pertains a chemical gradient at the particle surface and

fuels the propulsion.

Also light induced chemical reactions [52] or localized heating (self-

thermophoresis) [53,16] of the coated hemisphere have been used to pro-

pel Janus-particles; In the latter setup, the temperature gradient induces

complex asymmetric solvent flows around the particle which effectively

exert a force on one of its hemispheres. Another microswimming Janus

particle, which will be investigated in detail in Chapter 4, is fueled by an

uni-axial electric field inducing dipole moments of different strength in

its two hemispheres [15]. Again, this results in complex solvent flows ef-

fectively propelling the particle [54]. In the investigated two-dimensional

set up, the external field orientates these particles, thereby restricting the

propulsion to the directions perpendicular to the field vector [55].

Importantly, many of these experimental studies are concerned with the

propulsion mechanism of a single particle rather than with the collective

behavior of large numbers of these particles.

1.2.2 Collective behavior of active colloids

From a theoretical point of view, the collective behavior of active colloids

has been studied extensively [56,57,35,58]. Basic model systems of ac-

tive particles [59,57,60,61] have gathered a lot of attention. The fa-

mous Vicsek-model [59] and its derivatives [62], where the particles have

a mechanism which tries to align their propulsion directions, lead to the

idea of a motility induced non-equilibrium pseudo-phase transition [63]

as an analogue to the ferromagnetic transition in the Ising model. Also

motility induced crystallization phenomena [64,65,66], swarm forma-

tion [60], turbulence and vortex arrays [67] have been reported. Very re-

cently, experimental and theoretical evidence of motility induced phase

separation processes in systems of self-propelled rods, so called active ne-

matics [49], and discs without an alignment mechanism [68,16,69,70,

71,72] and even without attractive forces has been found [73,58]. Im-

portantly, such phase separation is unknown in equilibrium systems of

hard discs. In Fig. 1.2(c) experimental and simulated phase separated

states of solely repulsive spheres in an quasi two-dimensional setup are

shown [16].

Approaches to describe such phenomena often try to map the non-

equilibrium behavior onto the equilibrium behavior of an effective refer-

ence system. This is done by incorporating the self-propulsion mechanism

into effective potentials [69,74] or by extensions of thermodynamic theo-

ries i.e., an effective Cahn-Hillard equation [75] or non-equilibrium equa-

1Introduction

tions of states [76]. Still, a well founded theory for the (pseudo) phase

behavior of active particles has not been established by now [77] and the

effective thermodynamic descriptions and/or mappings onto passive pair

potentials have been challenged recently [78]. For particles with more

complicated interactions i.e., dipole-dipole interactions [79], theoretical

studies are rare. The above mentioned dipolar Janus swimmers [55] are

therefore of special interest; as they are able to change their direction of

motion, although only perpendicular to the external field, they can be seen

as self-propelling particles with complex anisotropic dipolar interactions.

In principle, they form a binary mixture of dipolar self-propelling parti-

cles, whereas particles of different species are moving in opposite direc-

tions. This general set up has previously been investigated for repulsive

spheres and shows another motility induced non-equilibrium phase tran-

sition known as laning [80].

1.2.3 Lane formation

Lane formation is a prototype of a non-equilibrium self-organization pro-

cess, where an originally homogenous mixture of particles (or other types

of ”agents”) moving in opposite directions segregates into macroscopic

lanes composed of different species. This ubiquitous phenomenon occurs,

e.g., in driven binary mixtures of colloidal particles [81,82,80], migrat-

ing macro-ions [83] or in binary plasmas [84,85,86]. The latter have

been investigated under microgravity conditions on board of the Inter-

national Space Station [87,88]. In addition, also self-propelling systems

with aligned velocities such as bacteria in channels [89] and humans in

pedestrian zones [66] or swarms of ants [90] undergo lane-formation. In

Fig. 1.2(d) the transition towards a laned state in large scale computer

simulations of soft discs driven in opposite directions is shown. With in-

creasing propulsion velocities, from left to right, lane formation becomes

more and more pronounced.

In particular, studies of charged colloids have revealed many funda-

mental aspects of laning such as the impact of density [91], the occurrence

of a freezing process [65], the role of hydrodynamics [92], the accompa-

nying microscopic dynamics (particularly, the so-called dynamical lock-

ing) [82], and the impact of anisotropic friction [93].

All of these models involve isotropic and repulsive interactions between

the colloidal particles. Then, the laning transition in two-dimensions is a

smooth crossover and subject to finite-size effects [46].

Also lattice models have been investigated from a solely theoretical

point of view [94]. Furthermore, the laning transition seems to bear some

similarities to equilibrium decomposition processes in binary mixtures of

(attractive) colloidal particles. In Chapter 3, results establishing a connec-

tion between lane formation of dipolar Janus swimmers and equilibrium

phase separation processes are discussed.

1.2Active Matter

Figure 1.3: Experimental image of (a) metallo-dielectric Janus-particles.

Differences in the brightness of the hemispheres correspond

to different surface materials. (b) Microscopic photograph of

staggered chains and (c) percolated networks. Images taken

from [26].

In general, recent progress in colloidal chemistry has generated a va-

riety of complex, anisotropic particles (see, e.g., [14,13,3]), which can

perform controlled translational motion in an external field [55,26], like

simple charged colloids, but display complex self-assembly behavior al-

ready in equilibrium [26,95]. In driven ensembles of such particles one

may therefore expect a wealth of new phenomena induced by the interplay

between self-assembly due to pair-interactions, on the one hand, and dy-

namical self-organization processes such as laning, on the other hand. The

consequences are so far only poorly understood, contrary to the widely

studied case of active (self-propelled) particles [40,16,70] with more

simple isotropic interactions. Hence, dipolar microswimmers are impor-

tant model systems for the interplay between laning and non-trivial self-

assembly.

1.2.4 Self-propelling metallo-dielectric Janus particles

The experimental dipolar Janus swimmers considered here consist of a

polystyrene sphere with radius ≈5µm and one hemisphere partially cov-

ered by a gold patch (see Fig. 1.3(a)). Dispersed in weakly ionized water

and confined between two glass plates a quasi two-dimensional geometry

is realized. Application of an in plane AC-electric field Eext induces effec-

tively multipolar moments in the metallo-dielectric colloids and aligns the

plane between their hemispheres along the external field, which has been

observed in experiment and shown theoretically by electrostatic calcula-

tions of a single particle in an electric field [15,55]. Interestingly, under

reasonable experimental field conditions, the energy difference between

an field aligned orientation and an orientation rotated by 90°around the

out-of-plane axis (perpendicular to Eext) is in the order of ≈100kBT[15].

This large energy difference effectively suppresses any rotations of the par-

ticles around the out-of-plane axis. In a truly two-dimensional setup this

means that particles have a fixed orientation, with the gold patched part

1Introduction

Figure 1.4: Self-propulsion of Metallo-dielectric Janus-particles: (Left)

Sketch of a single particle and the distribution of charges on

its surface. The solvent flows generated by these distributions

and indicated by arrows result in a propulsion force perpen-

dicular to the external field E. (Right) Photograph from exper-

iment visualizing how particles with opposite orientation of

the gold-patched hemisphere move under presence of the ex-

ternal electric AC-field E. Images taken from [55].

pointing either to the left or to the right, as depicted on the left side of

Fig. 1.4. Depending on the frequency fand strength Eext of the exter-

nal field different structures and particle behaviors can be observed. At

high frequencies (f > 10kHz) and low field strengths particles form stag-

gered chains (see Fig. 1.3(b)). Also colloidal network formation has been

reported for the high frequency regime in experiment [26] and theory [95]

(see Fig. 1.3(c)). Here, particles arrange into chains along and perpendicu-

lar to the electric field. Most important to us, increasing the field strength

and keeping f < 50kHz leads to the appearance of a phenomenon called

induced-charge electro-phoresis (ICEP) [55]. There particles become self-

propelled by converting field transmitted energy into kinetic energy. An

explanation for this mechanism is given in [54,55] and assumes that Eext

induces electric dipole moments in the particles and polarizes the electri-

cal double layer around them (see Fig. 1.4). The polarized charge distri-

bution in the electrical double layer becomes inverted in each half cycle

of the alternating external field and forms, for a homogeneous particle,

a quadrupolar flow pattern [96]. This pattern pertains an inflow of fluid

along the field orientation and an outflow perpendicular to the field. How-

ever, here it is Janus-particles which are considered, and as the induced

charge in the gold covered hemisphere is stronger (due to the higher po-

larizability of the gold patch), a larger amount of fluid is flowing around

the metallic hemisphere. Therefore a larger net ejection of fluid is present

at the metallic side of the particle and in consequence, a net force is act-

ing on the gold covered hemisphere. These complex solvent flows effec-

tively generate a propulsion force orthogonal to the external field in the

direction of the dielectric hemisphere. The theoretical derivation of the

1.3Non-Equilibrium Aggregation of Colloids

velocity of such a dielectric Janus particle undergoing ICEP results in a

velocity [55,54]

v=9

ϵσ E2

ext

η(1−δ)

(1.1)

whereas the parameter δrelates to the capacitance of the double layer, Eext

is the amplitude of the external field, σis the particle diameter, and ϵand

ηare the permittivity and the viscosity of the bulk fluid, respectively.

Compared to other self-propelling particles these are high velocity parti-

cles moving with up to 30µm/s quadratically depending on the external

field strength and linear on the particle size, as determined from exper-

iments and in semi-quantitatively agreement with theory [55]. Interest-

ingly, we can see that the dipolar coupling energy is proportional to E2

ext

and thus also v.

Finally, these particles can be seen as active microswimmers because

they transfer externally supplied energy into kinetic energy while being

able to change their direction of motion by rotations around the field ori-

entation Eext.

1.3 Non-Equilibrium Aggregation of Colloids

Besides the equilibrium aggregation of complex colloids and the collec-

tive behavior in active systems, also the non-equilibrium aggregation of

colloidal particles is of major interest. In contrast to active systems, par-

ticles do not consume energy to pertain their motility, but rather un-

dergo aggregation processes which prevent relaxation into the equilib-

rium state [97]. Several non-equilibrium situations are possible, for exam-

ple uni-directional field controlled assembly [26,95,40,98] or particle de-

position of colloids [99] and/or macromolecules [100] on surfaces. Inter-

estingly, permanent magnetic particles in rotating magnetic fields [6,41]

bear some similarities to active systems, as here the particles are able to

synchronize their rotational motions to each other without following the

rotation of the field directly. This synchronization then leads to aggrega-

tion of complex structures like layers [41] or tubes in case of Janus parti-

cles [6].

1.3.1 Disordered solids - glasses and gels

Of special importance to the understanding of non-equilibrium aggre-

gates are disordered solids like glasses and gels. Glasses are high den-

sity arrested states, where an effect called caging traps the particles in,

from equilibrium statistical physics point of view, unfavorable disordered

configurations [97]. The arrested state is a result from a ’historical’ pro-

cess (quench), where a liquid is cooled fast enough to avoid crystalliza-

tion and particles get stuck in liquid-like arrangements. The caging is

reflected by tremendously slow relaxational dynamics. For example, the

1Introduction

time a particle needs to travel a distance equal to its diameter can be in-

creased up to 15 orders of magnitude by only slightly decreasing temper-

ature [97]. Glasses are realizable in systems of solely repulsively interact-

ing discs. However, there exists also a re-entrance effect of the glass phase

by increasing temperature [97] if short-ranged attractive interactions are

present.

Another important subset of the many classes of self-assembled non-

equilibrium structures are gels. Although there is an ongoing debate on

how to define a gel properly [101], it is generally assumed that the forma-

tion of a gel requires some sort of attractive particle interactions. Here, the

term gel is used for colloidal systems in a disordered state at low volume

fractions, with very slow relaxational dynamics. Additionally, stable per-

colated colloidal networks are generally considered to be the underlying

micro-structures of gels. Such system-spanning cross-linked particle clus-

ters may already result from simple isotropic attractive interactions, like

the well known Lennard-Jones interaction. However, many patchy parti-

cle models strongly favor chaining and branching [102,103,104,105] of

aggregates and therefore structures which are occupying larger parts of

the accessible space. This increases the systems capability to form cross-

linked particle networks already at low volume fractions and hence gela-

tion. Importantly, in the basic model for a two-dimensional ferrofluid [22]

(dipole-dipole interaction due to centered point dipole moments in hard-

discs), particles favor chain and ring-like structures rather than branch-

ing. In consequence, it is unclear whether such ferrofluids undergo gela-

tion [106,107,108,109]. Nevertheless, more complex dipolar models [26,

95,105,108] form gels. Interestingly, many generic models for patchy par-

ticles have been proposed [110,8,105,103,111]. Such approaches often

aim on a conceptional understanding of network formation and gelation

of patchy particles and allow a theoretical understanding of the compe-

tition between condensation and chaining/branching e.g., via Wertheim

theory [105,101]. In fact, the formation of a (non-equilibrium) gel1is

intimately related to the equilibrium phase behavior of the system and of-

ten seen as an arrested phase separation process created by a quench into

the coexistence region. Instead of undergoing the full liquid-gas phase

separation or a spinodal decomposition, initially formed (chain-like) con-

densates undergo a freezing process [112]. This is because interactions

become much stronger than kBT, preventing particles from dissociating

due to thermal fluctuations. In Fig. 1.5(b) and (c) experimental images

of polyethylene (PMMA) polymer mixtures in a state of arrested phase

separation are shown [112]. On the right (Fig. 1.5(c)), discrete frozen

clusters are not able to percolate, while on the left (Fig. 1.5(b)) a single

percolated cluster exists at slightly higher volume fraction and forms a

gel. Still, single PMMA-particles are present, corresponding to the par-

1This holds only for non-equilibrium gels. For a discussion of ’equilibrium routes’ to-

wards gelation see [101].

1.3Non-Equilibrium Aggregation of Colloids

ticle poor phase. The phase separation process is arrested and therefore

irreversible. It leads to a pronounced hindrance of structural reconfigu-

ration and, given the volume fraction of colloids is not too small, perco-

lation becomes possible but is mediated by clusters rather than by single

particles [101,113]. Finally, such gels undergo aging processes and the

micro-structures may collapse over long times [101,114,115].

Figure 1.5: (a) Tunable field directed assembly of chain-like and network-

like fractal aggregates in binary mixtures of magnetic colloids

and the corresponding fractal Dimension Dfas function of an

effective field-strength. [98] (b) Percolated cluster of a chemical

gel at slightly higher volume fraction than (c) where discrete

frozen clusters are not able to percolate [112]. Fractal Particle

aggregates from computer simulations of (d) classical DLA and

(e) slippery DLA [116].

1.3.2 Diffusion limited aggregation

An ideal route towards gelation is described by the concept of diffusion

limited cluster aggregation (DLCA), where each particle or cluster colli-

sion leads to the formation of a rigid and essentially (on the timescale of

the experiment) unbreakable bond with fixed spatial orientation. Clas-

sically, single particles undergoing Brownian motion (DLCA) are consid-

ered but also the aggregation of ballistically moving particles [117,118]

has gathered attention. Systems with DLCA undergo irreversible dynam-

ics and form fractal aggregates with specific fractal dimension Df≈1.71

1Introduction

in continuous two-dimensional space [119,120]. Such colloidal systems

are considered to be ’chemical gels’, while at higher temperatures these

systems become ’physical gels’ where single particles and larger substruc-

tures start to connect and disconnect frequently. This strongly affects (in-

creases) the fractal dimension [121,122] and finally allows the system

to achieve its equilibrium state. Interestingly, it turned out that the ag-

ing properties of a chemical gel, an ongoing compactifaction with time,

can be mapped onto the temperature dependence of the compactness of a

physical gel, at least for low valency patchy particles [121].

A recently introduced new type of DLCA, which accounts for local re-

arrangements via flexible bonds, is slippery diffusion limited cluster ag-

gregation (sDLCA) [123,116]. Slippery bonds allow particles to move or

rotate around each other as long as they stay in contact, meaning that

bonds are still unbreakable but can change their orientation. This addi-

tional degree of freedom generates, at least in three-dimensional simu-

lations [123,116], aggregates of the same fractal dimension as classical

DLCA but with a larger coordination number. The latter means, that the

local structure of aggregates is more compact while the global structure

on large length scales remains in principle the same. In Fig. 1.5(d) and

(e), simulated slippery and classical DLA fractals are shown, together with

images of their local structure in the insets.

DLCA processes have been studied extensively in systems with isotrop-

ically attractive particles [124,119] but also in systems with patchy parti-

cles bearing permanent and/or locally restricted interaction sites on their

surfaces [121,125,126,127,128,129]. In the latter, the spatial orien-

tations of interaction sites can either be free to rotate [127,121,129] or

fixed in space [130,131,98,95]. When the orientations of interaction

sites are fixed in space, the associated ’chemical gels’ undergo anisotropic

diffusion limited aggregation which yields a fractal dimension of Df≈

1.5[132,130,131,98,129], lower than for the isotropic case. This situa-

tion occurs, e.g., in lattice models [132,130,131], where motion is natu-

rally restricted to certain directions. However, there is also the possibility

to direct the DLA process by e.g., external fields [98,33]. In Fig. 1.5(a),

microscopic images of fractal structures stemming from a field directed

diffusion limited aggregation process are shown. In this system it is a

mixture of para and diamagnetic colloids immersed in a ferrofluid and

exposed to an uniform external magnetic field. The red curve gives the

dependency of the fractal dimension on an effective field strength, which

is controlled by the concentration of the ferrofluid. Here, the shape of the

two-dimensional aggregates can be tuned, but only in one direction. A

similar situation is given for the networks formed by the metallo-dielectric

Janus-Particles presented in Fig. 1.3(c), where chaining along the field is

tunable by the field strength, while chain formation and percolation per-

pendicular to the field is determined by the same field strength. Starting

from these systems it is of great interest to design and investigate a system

1.3Non-Equilibrium Aggregation of Colloids

allowing individual control over the chaining properties in several direc-

tions. In the following we will present an experimental setup allowing

field controlled bi-directional chain and network formation.

1.3.3 Aggregated networks of superparamagnetic

colloids

An experimental quasi two-dimensional system of suspended colloidal

particles, each composed of super-paramagnetic iron-oxide aggregates

embedded in a polymer matrix, was designed to undergo field directed

colloidal aggregation and has been investigated experimentally by Dr. B.

Bharti [25,133] at North Carolina State University in the Group of Prof.

Dr. O. D. Velev.

The experimental setup is shown in Fig. 1.6(d) and consists of two gold

electrodes for the application of an AC-electric field (E) and a pair of elec-

tromagnets capable of generating an uniform magnetic field (H). The rel-

ative angle between the external electric and magnetic fields is kept at

constant 90◦such that the electric field is orthogonal to the magnetic field

(E ⊥H) and both lay in the assembly plane. The change in the spatial

distribution of the particles upon the application of fields was monitored

perpendicular to the assembly plane by a bright field microscope (Olym-

pus BX-61). The directional assembly of colloids in external electric and

magnetic field was performed using an aqueous dispersion of electromag-

netoresponsive microparticles. The microspheres (COMPEL TM , Bangs

Labs Ltd.) were composed of superparamagnetic nanoparticles embed-

ded in a matrix of polystyrene with diameter σ=5.7±0.2µm. The particles

were strongly negatively charged (pH <2) because of surface functional-

ization with -COO - groups and hence stable in their aqueous dispersion.

The superparamagnetic microparticle dispersion was transferred to the

assembly chamber on the experimental set-up and different field configu-

rations were applied.

The external field driven assembly of colloids is then governed by the

field-induced dipolar interactions between the particles. Here, two non-

interacting independent electric and magnetic dipoles are induced in each

particle. This configuration of the particles can be termed as double-

dipolar state, where each dipole selectively interacts with the dipoles of

same kind belonging to different particles (electric with electric and mag-

netic with magnetic). The crossed external electric and magnetic fields,

oriented in plane but perpendicular to each other, enforce a directed self-

assembly process of two-dimensional single-particle chain networks along

the field directions. Experimental photographs of the resulting network

structures are shown in Fig. 1.6(b) and (e). Additionally, chain like aggre-

gates when only the electric (Fig. 1.6(a)) or magnetic (Fig. 1.6(c)) field is

present are shown. Interestingly, chain formation due to the presence of

only the magnetic field is different to the case when only an electric field is

1Introduction

Figure 1.6: Field directed aggregation of superparamagnetic polymer par-

ticles under the presence of (a)/(c) electric/magnetic, (b) and

(e) crossed electric (vertical direction) and magnetic fields (hor-

izontal direction). Then, particles form rectangular network-

like structures. (d) Sketch of the experimental setup used

for assembling superparamagnetic microspheres by orthogo-

nal AC-electric and constant magnetic field.

present. In general, magnetically assembled chains are more compact and

partially staggered. This phenomenon will be discussed in more detail in

Sec. 3.4and Sec. 5.1.3.

1.4 Outline of this Thesis

The overall goal of this work is to enhance the understanding of how direc-

tional interactions in general, and induced dipolar interactions in partic-

ular, effect prototype non-equilibrium self-organization processes. In de-

tail, the interplay between complex self-assembly due to anisotropic pair

interactions and lane formation as well as diffusion limited aggregation

is investigated. In both cases, extensive Brownian dynamics simulation

studies are performed to understand the self-assembly processes. Using

models of different complexity we are able to establish quantitative con-

nections to experimental results as well as conceptional explanations of

the observed assembly processes. Besides that, simple tools from density

functional theory are applied to estimate the onset of spinodal decom-

1.4Outline of this Thesis

position. This is of special interest as the non-equilibrium pseudo-phase

behavior turns out to be tightly related to the equilibrium phase behavior

of the systems.

The rest of this work is organized as follows.

First, basic concepts of the theory of simple fluids are briefly intro-

duced in Chapter 2. In Chapter 3we develop several different models

to describe and investigate particular colloidal systems in experiment. We

start with the dipolar microswimmers presented in Sec. 1.2.4and intro-

duce the double dipolar particle model (DDP-model), which is a complex

model with anisotropic (dipole-dipole) interactions. Additionally, we in-

troduce a model of reduced complexity for which the equilibrium phase

behavior is well known. Then we turn towards models for superparam-

agnetic particles in crossed fields as described in in Sec. 1.3.3. Again, we

first present a complex model treating the anisotropic dipole-dipole in-

teractions in detail and then turn towards a complexity reduced generic

model, which allows to tune the anisotropy in pair interactions. Addi-

tionally we introduce a new technique which allows structural analysis

of colloidal network structures on a particle resolved level. In Chapter 4,

lane formation of dipolar microswimmers is investigated. Special atten-

tion is given to the connection between lane formation under the presence

of dipolar interactions and phase separation in the equilibrium system.

Chapter 5investigates the diffusion limited aggregation of superparam-

agnetic colloids in crossed magnetic and electric fields. The main ques-

tion here is how the induced anisotropies change the structural properties

of the networks. Besides, a close connection of the simulated data with

experimental data is presented which allows to map out a state diagram

for different field conditions. Finally, Chapter 6is giving a summary and

an outlook and in the appendix methodological details on the computer

simulations are presented.

Theoretical Concepts and

Methods

In this chapter we first start with a microscopic/particle based de-

scription of colloids dispersed in a liquid. This description results

in a system of coupled differential equations, the so called Langevin

equations. Due to the complexity of the considered many-particle sys-

tems, our investigations are based on computer simulations, which are

a powerful tool to study colloidal systems on a single particle re-

solved level. To this end we introduce the methodology of so called

Brownian Dynamics (BD) Computer simulations in the appendix.

A second approach towards a description of colloidal ensembles is

given by Density Functional Theory (DFT). This method yields a way to

find the equilibrium density distribution of a colloidal system, which

allows analysis of its equilibrium phase behavior. In detail, we intro-

duce some basic concepts and methods from DFT, which are necessary

to understand the occurrence of phase transitions and phase coex-

istence. This chapter ends with an approximated expression for the

isothermal compressibility, which later serves as a tool to estimate

the occurrence of phase instability.

2.1Langevin Equation

2.1 Langevin Equation

A material of micron-sized particles dispersed in a liquid is usually called

a colloidal suspension. Each of the colloidal particles is subject to several

forces which are imposed on it by the solvent and by other colloids. The

forces emerging from the solvent are two-fold and result in dissipation

due to frictional forces and in diffusion due to thermal fluctuations.

First, solvent molecules hitting a colloid transfer momentum on it. This

happens according to their mean kinetic energy d×1/2kBT, where kBis

Boltzmann’s constant, dthe spatial dimension of the system and Tits

temperature. Usually it is not necessary to resolve the momentum trans-

fer from each solvent particle onto each colloid explicitly. A statistical

description turns out to be sufficient. The widely used approach is to de-

scribe the momentum transfer onto the colloid iby an effective random

force FD

iand an effective torque TD

i, which can be seen as vector sums over

all forces and torques resulting from single hitting events. Secondly, the

solvent leads to a frictional force FF

iand a frictional torque TF

iopposing

translation and rotation of the colloid, respectively. Finally, the colloids

also interact with each other according to some forces Fiand torques Ti

stemming from pair interactions, e.g., magnetic interactions. Note that

these forces can also incorporate external influences stemming from, e.g.,

an electric field. Adding all these forces to newton’s equations of motion

yields for a particle iof mass mthe so called Langevin equations

m¨

ri=Fi−γT˙

ri+FD(2.1)

I˙

ωi=Ti−γRωi+TD(2.2)

whereas γTand γRare the translational and rotational friction coeffi-

cients, respectively. The stochastic contributions obey Gaussian distribu-

tions with mean

<FD(t)>=0(2.3)

<TD(t)>=0(2.4)

and with variance

<FD

i(t)·FD

j(t′)>=2kBT γTδij δ(t−t′)

(2.5)

<TD

i(t)·TD

j(t′)>=2kBT γRδij δ(t−t′)

.(2.6)

Usually Eqs. [2.1,2.2] are written in terms of translational and rotational

diffusion coefficients DTand DR. These coefficients are defined by a fluctu-

ation dissipation theorem (FDT) relating temperature and friction via

DT=kBT /γT(2.7)

DR=kBT /γR.(2.8)

2Theoretical Concepts and Methods

and can easily be substituted in Eqs. [2.1,2.2,2.5,2.6]. Due to the cou-

pling between random forces and torques to the friction coefficients in

Eqs. [2.5,2.6], thermal fluctuations effectively compensate for energy

losses due to dissipation in this framework. The natural timescale on

which a single colloid diffuses a distance equal to its own diameter σis

then given by the so called Brownian time scale

τb=σ2/DT.(2.9)

2.1.1 Ovderdamped limit

The Langevin equations (Eqs. [2.1,2.2]) are second order differential equa-

tions accounting for inertial effects. In colloidal systems however, dissipa-

tion of kinetic energy due to friction is usually so strong that once a mov-

ing particle experiences no force its (translational) motion with respect to

the solvent will stop on a very short time scale τm=m/γT[7]. Relating the

time τm, during which momenta of particles relax, to the Brownian time

τballows to estimate the influence of inertia on the particle dynamics. To

this end, the translational Langevin equation1can be rewritten

τm

τb

ri=1

γTτb{Fi−γT˙

ri+FD}(2.10)

As τmis proportional to the mass and hence to the particle volume V∝σ3

it decreases faster with smaller particle diameter than the Brownian time

scale τb, which is proportional to the particle surface S∝σ2. It turns out,

that for typical colloidal systems the coefficient on the left hand side of

Eq. [2.10]

τm

τb≪1(2.11)

becomes very small2. This situation corresponds to the so called over-

damped limit of the Langevin equation, where the inertial parts in

Eqs. [2.1,2.2] become very small and can be neglected. Please note, that in

this limit (O([ τm

τb

]0)) the equations of motion are not able to describe the

particle behavior on time scales smaller than the time τm. The overdamped

equations then read

γT˙

ri=Fi+FD(2.12)

γRωi=Ti+TD,(2.13)

whereas γTand γRare again the translational and rotational friction coef-

ficients and the stochastic contributions obey again Gaussian distributions

according to Eqs. [2.3,2.4,2.5,2.6].

1For simplicity, this consideration is restricted to the translational part. A analogous

argument holds however for the rotational dynamics.

2For example, for a SiO particle with σ=1µm at room temperature suspended in water

is found τm≈10−7sand τb≈2s[7].

2.3Phase Coexistence

The solutions of Eqs. [2.12,2.13] are the trajectories of the colloidal par-

ticles. However, an analytical solution for a many particle system is in

general not possible. An alternative is to integrate Eqs. [2.12,2.13] nu-

merically; a procedure which is usually called Brownian dynamics (BD)

computer simulation [134,135]. Large parts of this work are based on

these computer simulations. In the appendix a detailed description of

these methods is presented.

2.2 Phase Transitions

A phase transition is the transformation of one thermodynamic equi-

librium state of matter towards another one because of a change in the

thermodynamic conditions. Thereby certain macroscopic properties of

the system, such as magnetization, change. From a theoretical point of

view, phase transitions are categorized by the occurrence of a discontinu-

ity in the first derivative of the thermodynamic free energy Fwith respect

to some thermodynamic variable. For instance, when water is heated up

to its boiling point its density decreases instantly. Hence, the vapor-liquid

transition is an example for a (discontinuous) first-order phase transition.

The set of all phase points which yield such a discontinuity separates dif-

ferent phases and is called the phase boundary. Phase transitions without

a discontinuity in any first derivatives of the free energy are called (con-

tinuous) higher-order transitions. Importantly, there exists a critical point

(with the critical temperature Tc) defining the end of the phase bound-

ary. Consequently, a path leading from vapor to liquid at or above the

critical temperature yields a continuous change in density. Furthermore,

decreasing temperature leads towards the triple point representing the in-

tersection of the phase boundaries between solid, liquid and vapor phases.

At the triple point the vapor-liquid phase boundary and the liquid-solid

phase boundary merge and form a lower temperature limit for the liquid

phase.

2.3 Phase Coexistence

In general, a thermodynamic system is in its equilibrium phase if the

Helmholtz free energy

F=U−T S (2.14)

is minimized, whereas Uis the internal energy due to pair interactions

and/or external fields and Sis the entropy. For a mono-disperse system of

Hard-Spheres, meaning that all particles and interactions between them

are equal and solely repulsive, holds U > 0and it is only the entropy Sby

which Fcan be minimized3. Hence, the system seeks to achieve a homoge-

3The temperature Tserves only as a scaling factor here

2Theoretical Concepts and Methods

Figure 2.1: Prototype phase diagram of a simple fluid. Coexistence line C,

and spinodal S indicating the occurrence of meta and instabil-

ity of the homogenous phase. The critical temperature is the

upper boundary of the coexistence region.

neous state, where the mean density is the same everywhere in the system.

In addition to systems forming such homogeneous phases there exist ther-

modynamic systems where two phases, such as liquid and vapor coexist at

the same time. Such a situation is the result of a phase separation process

and characterized by spatial inhomogeneities, meaning that domains of

different densities are present in the system. In a mono-disperse system,

phase coexistence requires in any case attractive particle interactions such

that the internal energy can become U < 0and starts to compete with the

entropy Sand temperature Tin minimizing F.

Interestingly, phase boundaries are the sets of phase points where the ho-

mogeneous phase is either metastable or unstable. Here, density fluctu-

ations can cause the system to phase separate. In Fig. 2.1a schematic

temperature-density phase diagram for a one component fluid is shown.

The homogeneous liquid and vapor phases are separated by the coexis-

tence curve Cenclosing the region of phase coexistence. Within the co-

existence curve lies the spinodal line S, defining the region of instability;

here, infinitesimal density fluctuations are sufficient to induce phase sep-

aration. Between the spinodal and the coexistence curve the system is

metastable and finite fluctuations are needed to induce the phase separa-

tion process.

Finally, the simplest theoretical understanding of first-order phase tran-

sitions and phase separation processes is given by the van der Waals the-

ory [136], which yields an equation of state for a simple fluid expressing

the relation between pressure P, volume V, temperature Tand particle

number N. The solutions PT(V) of the van der Waals equation at fixed

T > Tcare monotonously decaying, while they oscillate for temperatures

2.4Pair Correlation Function

T < Tc. These oscillations are usually called the van der Waals loops and

result in positive slopes of the isotherms PT(V). Consequently, the isother-

mal compressibility, given by

χT=−1

(

∂V

∂P

)T,(2.15)

becomes negative inside these loops. As this means that an increase in

pressure results in an expansion of the fluid, this is an unphysical result

and indicates the instability of the homogeneous phase. In this frame-

work, the unstable region inside the coexistence region is characterized

by χT<0. This provides a way to determine the spinodal line Sby cal-

culating the thermodynamic parameters at which χtchanges its sign. In

the following we will shortly present a way to determine χTvia basic den-

sity functional theory methods. We start with an expression for the direct

correlation function in the random phase approximation. Then we relate

the direct correlation function to the isothermal compressibility via the

Ornstein-Zernike Equation.

2.4 Pair Correlation Function

In a system of Nparticles, the one and two particle densities ρ(1)(r) and

ρ(2)(r,r′) can be defined in terms of delta functions

ρ(1)(r) =<

∑

i=1

δ(r−ri)>(2.16)

ρ(2)(r,r′) =<

∑

i=1

∑

j=1

δ(r−ri)δ(r′−rj)>(2.17)

whereas the brackets indicate an ensemble average. These functions are

normalized via ∫

ρ(1)(r)dr=< N > (2.18)

∫ ∫

ρ(2)(r,r′)drdr′=< N 2>−< N > (2.19)

Eq. [2.18] gives in principle the mean number of particles ¯

Nand (except

of proper normalization) the quantities ρ(n)(r)drncan be interpreted as the

probability to find nparticles in the volume element drn, irrespective of all

other particles. From this follows directly that the two-particle correlation

function is given by

g(2)(r,r′) =

ρ(2)(r,r′)

ΠN

i=1ρ(1)(ri)

(2.20)

2Theoretical Concepts and Methods

which reduces for an homogenous and isotropic fluid to the radial distri-

bution function

g(2)(r) =<1

ρN

∑

i=1

∑

j=1

δ(r+ri−rj)>, (2.21)

here expressed in terms of δ-functions and with the overall number den-

sity ρand a particle distance r=|r−r′|. The radial distribution func-

tion g(2)(r) = g(r) is of special importance for the understanding of fluids.

First, g(r) quantifies how much the structure of a fluid deviates from com-

plete randomness at distance r. This can be seen by writing Eqs. 2.20 as

ρ(2)=ρ2g(r) where g(r) corrects the homogenous density of the ideal gas

ρfor particle correlations. Secondly, it can be measured by scattering ex-

periments, which allows direct insight into the real structure and gives

straight forward possibilities for comparison with results from computer

simulations. In the later, its representation via δ-functions is exploited to

calculate a histogram H(r) counting all pair distances rfalling into cer-

tain bins of width ∆r. The normalized ensemble average of H(r) is then

compared to the corresponding histogram of the ideal gas

Hideal (r) = 4

3πρ[(r+∆r)3−r3] (2.22)

which yields

g(r) =

< H(r)>

N Hideal (r)

.(2.23)

Note that in a two-dimensional setup, the histogram Hideal =πρ[(r+∆r)2−

r2] has to be used for comparison.

Finally, the knowledge of g(r) allows to calculate many thermodynamic

properties, e.g. the isothermal compressibility or the total correlation

function introduced in the next section.

2.5 Direct Correlation Function

In Density Functional Theory (DFT), the Helmholtz free energy Fis

treated as a functional of the particle density ρ(1)(r) and can be separated

into an ideal part and an excess part [136]

F[ρ(1)(r)] = Fid [ρ(1)(r)] + Fex[ρ(1)(r)] (2.24)

accounting for contributions from configuration and interaction, respec-

tively. The ideal part is given by

Fid =kBT

∫

ρ(1)(r)(ln[λ3

Tρ(1)(r)] −1)dr(2.25)

with λT=√h2/2πmkBTbeing the thermal wavelength and hthe Planck

constant.

2.5Direct Correlation Function

The definition of the direct correlation function is related to the second

functional derivative of the excess part of the free energy functional with

respect to the single particle density

c(2)(r,r′) = −β

δ2Fex [ρ(1)(r]

δρ(1)(r)δρ(1)(r′)

.(2.26)

In general, the excess part of the free energy functional is unknown

and has to be approximated, but there exist explicit expression for the

direct correlation function c(2)

HS of the Hard-Sphere system in three dimen-

sions [137]. However, for the two-dimensional Hard-Disc system exist

only approximated expressions [138].

In this work the (quite accurate) expression of Guo and Riebel [139]

c(2)

HS,2D(r, η) = Θ(1−r)[−1−qη2

(1−2η+qη2)2]

{1−a2η+η2

[arccos(r/a)−r/a(1−r2/a2)1

2]}

(2.27)

with

η=ρπσ2/4

q=(4√3π−12)/π2

a=0.3699η4−1.2511η3+2.0199η2−2.2373η+2.1

(2.28)

will be used. More complicated pair interactions can be treated by certain

approximations, like the random phase approximation.

2.5.1 Random phase approximation

The random phase approximation is based on the idea to separate the inter-

action potential u(r,r′) into a reference part ur(e.g., hard sphere interac-

tion) and a perturbation part up(short ranged attraction)

uλ(r,r′) = ur(r,r′) + λup(r,r′)λ∈[0,1] (2.29)

where λcontrols the strength of the perturbation. Then, the pair density

ρ(2)(r,r′) is related to the excess part of the free energy functional [136] via

a functional derivative

ρ(2)(r,r′) = 2

δFex[ρ(1)(r]

δuλ(r,r′)

(2.30)

with respect to the interaction potential u(r,r′).

Integration of Eq. [2.30] allows to express the excess free energy func-

tional as

Fex [ρ(1)(r)] = Fex

r[ρ(1)(r)]

∫ ∫

ρ(1)(r)ρ(1)(r′)up(r,r′)drdr′+Fcorr [ρ(1)(r)]

(2.31)

2Theoretical Concepts and Methods

with

Fcorr [ρ(1)(r)] = 1

∫1

dλ

∫ ∫

ρ(1)(r)ρ(1)(r′)h(2)((r,r′); λ)up(r,r′)drdr′(2.32)

being the contribution to Fex from perturbations. In a mean field ap-

proximation this part can be neglected and forming the second functional

derivative with respect to ρ(1)(r) defines the direct correlation function in

the random phase approximation

c(2)(r,r′) = −β

δ2Fex [ρ(1)(r]

δρ(1)(r)δρ(1)(r′)

≈c(2)

r(r,r′)−βup(r,r′)

(2.33)

where c(2)

ris the reference part for which we use c(2)

HS,2D(r, η) from Eq. [2.27].

It is well accepted that the expression −βup(r,r′) is a good approximation

of the direct correlation function c(2)(r,r′) for large distances. Hence, the

perturbation contains the long-range part of the potential.

2.6 Ornstein-Zernike Equation

The Ornstein-Zernike equation

h(2)(r,r′) = c(2)(r,r′) +

∫

c(2)(r,r′′)ρ(1)(r′′)h(2)(r′′,r′)dr′′ (2.34)

relates the pair distribution function to the direct correlation function c(2)

via the definition of the total correlation function

h(2)(r,r′) = g(2)(r,r′)−1.(2.35)

The idea behind the Ornstein-Zernike equation is that the total correlation

between two particles can be split into a direct correlation between them

and ’higher order contributions’ mediated by a third, fourth and so on

particle. Therefore, the Ornstein-Zernike equation has in principle to be

solved recursively. For a uniform and isotropic fluid Eq. [2.34] simplifies

h(2)(r) = c(2)(r) + ρ

∫

c(2)(|r−r′|)h(2)(r′)dr′(2.36)

and the indirect correlations appear as a convolution integral. Finally,

applying a Fourier transformation on Eq. [2.36] and using the convolution

theorem yields the relation

H(k) = C(k) + ρC(k)H(k) (2.37)

with H(k) and C(k) being the Fourier transforms of h(2)(r) and c(2)(r), re-

spectively. In the long wavelength limit (|k|=0), the Fourier transforma-

tion reduces to

H(0) =

∫

[g(r)−1]dr.(2.38)

2.7Kirkwood-BuffEquation

2.7 Kirkwood-BuffEquation

The isothermal compressibility in a statistical ensemble is defined by [140]

the change of volume Vwith pressure Pat constant temperature Tvia

χT=−1

(

∂V

∂P

)T.(2.39)

In the Grand canonical ensemble, fluctuations <(∆N)2>in the mean

particle number ¯

Ncan be related to the isothermal compressibility

ρkBT χT=

<(∆N)2>

.(2.40)

using the thermodynamic relation −1

V(∂V

∂P ) = 1

N ρ (∂¯

∂µ ) [140].

Again we see, that χT>0has to be fulfilled, because <(∆N)2> / ¯

Nis

a positive quantity. Using Eqs. [2.16,2.17,2.18,2.19] allows to reformu-

late the mean square deviation of the mean particle number <(∆N)2>in

terms of the particle densities

∫ ∫

ρ(2)(r,r′)−ρ(1)(r)ρ(1)(r′)drdr′=< N 2>−< N > −¯

N2(2.41)

and for a homogenous fluid then follows with Eqs. [2.40,2.20] that the

isothermal compressibility can be expressed as an integral over the radial

distribution function g(r) yielding the compressibility equation

1+ρ

∫

[g(r)−1]dr=

<(∆N)2>

< N >

=ρkBT χT.(2.42)

Together with Eqs. [2.37,2.38] a connection to the Fourier transform of the

direct correlation function is given by

ρkBT χT= [1−ρC(0)]−1,(2.43)

which is the Kirkwood-BuffEquation for an one-component system. For

the isothermal compressibility calculated from the Kirkwood-Buffequa-

tion must hold χT>0according to Eq. [2.40]. Note that this result was

deduced from Eq. [2.36] which assumes a homogenous fluid. Hence neg-

ative compressibility indicate a thermodynamical instability of the homo-

geneous phase and using the direct correlation function in the random

phase approximation allows to calculate χT.

The Kirkwood-Buffequation can be generalized towards multi-component

systems with ndifferent particle species. The equation then reads [136]

kBT χT= [

∑

α,β

ραρβ[δαβ /ρα−Cαβ (0)]−1,(2.44)

2Theoretical Concepts and Methods

whereas αand βindicate the considered species, ραis the number density

of particles of species αand Cαβ (0) is the Fourier transform in the long

wave-length limit of the approximated direct correlation function

c(2)

αβ (0)≈c(2)

HS −1/kBT uαβ (r,r′) (2.45)

with the particle type specific interaction potential uαβ and the reference

part c(2)

HS , which is independent of the particle species in a mono-disperse

system.

In case of a two-component system with particle species Aand Band

the corresponding densities ρAand ρBfulfilling ρ/2=ρA=ρB, we find

ρkBT χT= [1−ρ/2(CAA(0) + CAB(0))]−1,(2.46)

because for pair interactions reasonably must hold uαβ =uβα , which im-

plies Cαβ (0) = Cβα (0) in the random phase approximation.

In contrast to an one component system, where the instability of the

homogeneous phase results in phase separation of i.e., liquid and vapor,

one can find demixing processes in thermodynamically unstable multi-

component system. These different types of instabilities can not be dis-

tinguished by considering solely the isothermal compressibility according

to Eq. [2.44]. However, there exist more general theoretical descriptions

allowing such characterization of different instabilities [141].

Theoretical Models of

Complex Dipolar Colloids

In this chapter we introduce common interaction potentials used

for modeling colloidal particles. Based on these potentials we con-

struct more complex models for dipolar particles under the presence

of external inductive fields. In total, it is four different models.

First the DDP model for dipolar microswimmers and a related model

of reduced complexity. Secondly, we model superparamagnetic par-

ticles under the presence of crossed electric and magnetic fields

(CDP model), but also introduce a generic model for colloids with

mutidirectional induced interactions. Additionally, we discuss visu-

alization and structure analysis techniques which we developed to

understand structures formed by dipolar colloids.

In general, colloidal particles are subject to a large variety of forces and

pair interactions e.g., Van der Waals forces, depletion forces or electro-

static forces. Throughout this work we will consider different particle

models, most of them related to dipolar interactions. In all of these

models, it is assumed that the particles interact sterically and bear some

additional (dipolar) interaction properties. However, influences of van der

Waals forces or electrostatic repulsion are not considered, unless stated

otherwise. This is because we want to isolate and analyze the influence

of the dipolar interactions. This approach is justified by the fact that

we can always assume a stable homogeneous phase of solely repulsive

non-dipolar particles as the initial state of our system. Exposing these

particles to external fields and inducing relatively strong dipolar interac-

tions then allows to investigate the influence of the dipolar interactions

3.1Colloidal Interaction Potentials

without being superimposed by other forces e.g., van der Waals. However,

hydrodynamic interactions are not considered here, although they might

influence the systems behavior significantly [7].

3.1 Colloidal Interaction Potentials

In this section we shortly discuss basic models for pair-interactions be-

tween colloidal particles, which are frequently used in computer simula-

tions. The introduced interactions will be used, extended and recombined

in the following parts of this Chapter, where more complex particle mod-

els are constructed. The present section serves as a brief overview of the

basic constituents of these more complex models.

The Lennard-Jones potential

The Lennard-Jones (LJ) potential is an isotropic pair potential with repul-

sive and attractive parts. It is given by

ULJ =4ϵ[(σ /r)12 −(σ /r)6)],(3.1)

whereas ris the particle center-to-center distance and σthe particle di-

ameter. The potential depth, controlling the interaction strength, is given

by ϵ. The r−6dependence is used to model attractive van-der-Waals inter-

actions between neutral atoms, while the r−12 dependence results in very

rapid increase of repulsion for distances r < σ .

The soft-sphere potential

Based on the Lennard Jones potential ULJ (r) a so called Soft-Sphere USS (r)

(or Soft-Disk in two-dimensions) potential can be defined by truncating

ULJ at rc=21/6σand shifting it by ϵ. The resulting interaction

USS (r) = 4ϵ[(σ /r)12 −(σ /r)6)] + ϵfor r < 21/6σ(3.2)

is then solely repulsive and can be used to mimic sterical interactions be-

tween colloidal particles (as done throughout this work). An important

feature of this Soft-Sphere potential is that it decays continuously and

vanishes at the truncation distance rc. This is of special importance for its

utilization in Brownian dynamics computer simulations where the force

FSS

ij (rij ) = −∇riUSS (rij ) with rij =ri−rj(3.3)

imposed on particle iat position riby particle jat position rjis then well

defined and yields no ’jump’ at the truncation distance rc

ij =21/6σ.

3Theoretical Models of Complex Dipolar Colloids

The Yukawa interaction potential

A point-charge Qor an uniformly charged spherical shell with diameter

σand total charge Qgenerates an electric field according to the coulomb

potential

Ucou(r) = 1

4πϵ0

for r > σ , (3.4)

with ϵ0being the electric constant. Such an uniformly charged shell can

be seen as a basic model for a colloidal particle in a fluid. The specific

mechanisms by which the colloidal surface becomes charged are manifold,

e.g. dissociation of surface molecules, and depend on the material and

solvent properties [7].

Importantly, once the particle surface is charged, ionized solvent par-

ticles of opposite electrical charge are attracted towards it. They form

a cloud of counter-ions, the so called the electrical double layer (EDL),

around the colloid and partially screen the Coulomb potential Ucou(r).

The resulting effective potential is then given by the functional form of

the Yukawa interaction [7]

Uyu (rij ) = q2exp (−κrij )

rij /σ

,(3.5)

with the inverse length κ−1describing the strength of the screening and

the interaction strength parameters qi=qj=q∝Q. In experiment, strong

screenings with κ≈10σ−1can be achieved via increasing counter-ion con-

centrations by externally introduced electrolytes [7].

The dipole-dipole interaction

The dipole-dipole interaction between two point dipoles µiand µjat po-

sitions riand rjand distance rij =ri−rjis given by [142]

Udip(rij ,µi,µj) =

µi·µj

ij −3

(µi·rij )(µi·rij )

(3.6)

and is an example for an anisotropic long-ranged pair interaction. Due to

its anisotropy, the dipole-dipole interaction results in torques between the

two dipoles. It can be used to describe magnetic or electric dipoles and is

a standard expression used in models of dipolar colloids.

In these models, the dipole moments µican either be seen as induced

by external fields or as being permanent. Torques are usually neglected

for induced dipole moments while the orientation of permanent dipole

moments is fixed to the geometry of the particles carrying them. Then

torques act on the particle itself. The later means that phenomena like

the Neel relaxation are neglected, which is reasonable for not too small

3.1Colloidal Interaction Potentials

particles [143]. Finally, in computer simulations one usually defines the

so called dipolar coupling strength (here for a two-dimensional system)

λ=µ2/kBT(3.7)

giving the ratio between the strength of dipolar interactions and thermal

energy. Once this ratio becomes λ≈kbT, the dipole-dipole interactions

start to influence the collective particle behavior significantly.

3.1.1 Representations of anisotropic pair interactions in

two-dimensions

In the following, the dipole-dipole interaction, as presented in Eq. [3.6],

will be used to model complex dipolar colloids. Before presenting more

complex models, we want to introduce a simple visualization technique,

which allows to understand intuitively the interaction patterns between

dipolar particles. To this end, we start with a hard sphere of diameter σ

in the center of a two-dimensional coordinate frame with position vector

r1= (0,0). We place a dipole moment of strength µ=√σ3ϵin the center of

this sphere, whereas ϵis an arbitrary unit of energy. The dipole moment

has a fixed orientation and points along the y-direction. Hence, the dipole

moment is given by µ=µey.

Figure 3.1: A dipolar disc of diameter σ(solid circles) in the center of the

two-dimensional x-y plane. The particles dipole moment is in-

dicated by an arrow in its center and points along the y-axis.

The energy Udip is calculated according to Eq. [3.6] for a ficti-

tious second particle of equal kind at various positions rand

given in units of ϵ. This probing particle is indicated by a gray

circle and arrow at an arbitrary position. The color code gives

the interaction energy ’landscape’. The white circle around the

central particle indicates the excluded area.

3Theoretical Models of Complex Dipolar Colloids

The dipolar interaction energy between this particle and another parti-

cle of equal kind at arbitrary positions r2can be calculated from Eq. [3.6].

The set of energies Udip(r12) can be visualized as a color map in the x-y

plane. This results in an energy ’landscape’ as shown in Fig. 3.1. In this

representation, it is easy to see the preference of dipolar particles to form

head-tail arrangements by the low energy regions around the head and

bottom of the central particle.

Such an energy ’landscape’ might seem trivial when considering a dipo-

lar particle with one central dipole moment, but may help a lot in under-

standing interaction patterns between particles carrying several dipole

moments. Please note that such a representation becomes less powerful

in case the orientations of dipole moments are free to rotate or three-

dimensional quantities. However, in this work we are investigating sys-

tems effectively confined to two dimensions with induced, orientationally

fixed, dipolar interactions.

3.2The DDP Model for Dipolar Microswimmers in Experiment

3.2 The DDP Model for Dipolar

Microswimmers in Experiment

In the present section we introduce the double dipole particle (DDP)

model. The experimental particles motivating this work are gold-patched

dielectric spheres confined between two glass plates, forming a quasi-2D

geometry [55]. Application of an in-plane AC electric field induces dipole

moments in both, the gold and the dielectric part of the particles, with

the gold’s dipole being significantly larger due to its larger polarizability.

Additionally, the plane between gold patched and dielectric part become

aligned along the field. At low frequencies these two induced dipoles have

the same orientation (along the field). Placed into an aqueous solution,

one observes [55] a spontaneous motion of each particle in the direction

orthogonal to the field and away from the particle’s gold patch. The un-

derlying mechanism is an asymmetric flow of solvent charges, usually

called induced-charge electrophoresis [55,54]. To examine the collective

behavior of these driven particles, we perform BD simulations in a 2D

quadratic cell of size L2with periodic boundary conditions. The cell con-

tains N=800 spherical particles with equal diameter σdefining a unit

length. Particle positions are denoted by ri=xiex+yiey(i=1, .., N ) with

the unit vectors exand eydefining a 2D coordinate frame. The external,

electric field Eext points along the y-axis. Sterical interactions are modeled

by a soft-sphere (SS) potential

USS (rij ) = 4ϵSS

(

(σ /rij )12 −(σ /rij )6+1/4

)

(3.8)

truncated at rc

ij =21

6σ, where rij =|rj−ri|is the particle distance. The

strength of repulsion is set to ϵ∗

SS =ϵSS /kBT=10 where kBis Boltzmann’s

constant and Tis the temperature. To mimic the impact of the propulsion

force we randomly assign to each particle a fixed vector si=siexwhich

points either to the ’right’ (s=1) or to the ’left’ (s=−1) with probability

0.5, respectively. Thereby a random fifty-fifty mixture of two different

particle ’species’ is created. Each particle is then subject to a constant

but orientation-dependent driving force fd,i =fdsi. Changes of particle

orientation (si) are not considered here.

To incorporate the dipolar interactions we assume that each particle i

bears two point dipole moments µ(1)

iand µ(2)

iwhose orientation is fixed

along the external field , i.e. µ(α)

i=µ(α)

iey,α=1,2. Moreover, the two

dipole-moments are shifted out of the particle center by δ(1)

si=δsiand

δ(2)

si=−δsiwith δ=0.25σ. The values of µ(α)

iare hold constant (that is, we

neglect variations of the local field) and set differently in order to mimic

the strong asymmetry of the metallo-dielectric Janus-particles. Specifi-

cally, we choose µ(2)

i=2µ(1)

iand set the dimensionless dipole strength to

µ∗≡µ(2)

i∗=µ(2)

i/√σ3kBT. Test simulations revealed that the system’s be-

3Theoretical Models of Complex Dipolar Colloids

Figure 3.2: Potential energy ’landscape’ due to the total dipolar interac-

tion (see color code) for two DDPs of diameter σ(solid circles)

with (a) si=sj=−1and (b) si=1, sj=−1in a top-view perspec-

tive for (µ1)∗=1.58. The particle dipole moments are indicated

by arrows with a white [black] head for µ(1)[µ(2)]. The white

circle around the centered particle indicates the excluded area.

3.2The DDP Model for Dipolar Microswimmers in Experiment

havior is only weakly sensitive to the ratio µ2/µ1, whereas changing the

shift parameter δhas a strong effect. Here we stick to a value of δalready

used in Ref. [95] for the case of oppositely oriented dipoles. The distance

between two arbitrary dipole moments µ(α)

iand µ(γ)

jof different particles

is given by

rαγ

sisj=rj+δγ

sj−(ri+δα

).(3.9)

The dipolar interaction between the two of them is calculated via the 3D

point dipole potential

Udip(rij ,µ(α)

i,µ(γ)

j, si, sj) = µ(α)

i·µ(γ)

j(rαγ

sisj)−3

−3(µ(α)

i·rαγ

sisj)(µ(γ)

j·rαγ

sisj)(rαγ

sisj)−5,

(3.10)

and the total dipolar interaction between two particles is given by the sum

UDD (rij , si, sj) =

∑

α,γ=1

Udip(rij ,µ(α)

i,µ(γ)

j, si, sj).(3.11)

We use the standard 2D Ewald summation method to handle the long-

range character of the dipole interactions [144]. Combining UDD and

the formerly introduced SS potential defines the ’double-dipole particle’

(DDP) model

UDDP =UDD +USS .(3.12)

The angle dependency for distances rij > σ is illustrated in Fig. 3.2. It

is seen that two identical DDPs (si=sj) prefer a head-to-tail configura-

tion similar to ”single-dipole” particles with one dipole in their center. In

contrast, for DDPs with si,sj, the most attractive configurations are stag-

gered in the sense that one DDP is shifted towards the gold-patched ’back’

of the other particle. The corresponding minimum energy Umin

DD (si,sj) is

roughly 4kBTlarger than in the head-to-tail configuration Umin

DD (si=sj).

The overdamped BD equations of motion are given by

γ˙

ri=

∑

j=1

∇UX(ij) + fs

d,i +ζi(3.13)

where γis the friction constant, UX(ij) is a specific pair potential (with

X=SS,DDP , or LJ), and ζiis a Gaussian noise vector which acts on par-

ticle iand fulfills the relation ⟨ζi⟩=0and ⟨ζi(t)ζj(t′)⟩=2γkBT δij δ(t−t′).

Hydrodynamic interactions (HI) are neglected, in accordance with earlier

studies [92] revealing that HI do not alter lane formation qualitatively.

The BD equations are solved via the Euler scheme [135] with an integra-

tion step width ∆t=10−5τb, where τbis a Brownian timescale defined by

τb=σ2γ/kBT.

3Theoretical Models of Complex Dipolar Colloids

1.0 1.2 1.4 1.6

r12[

]

−2.0

−1.5

−1.0

−0.5

0.0

U[kBT]

ULJ

≠

DDP

Figure 3.3: Angle-averaged DDP potential for s1=s2and s1,s2at µ∗=

1.58. Included is the LJ potential at ϵ∗=2.3.

3.2.1 The driven Lennard Jones fluid - A generic model

for attractive microswimmers

Besides the anisotropy, another main feature characterizing the DDP in-

teractions is that they are effectively attractive. This is seen from the angle-

averaged potential

uDD (r12) = 1

2π

∫2π

UDD (r12, s1, s2)dΩ12,(3.14)

with Ω12 being the angle of the connecting vector r12 relative to the x-

axis. Numerical data for the cases s1=s2and s1,s2are shown in Fig. 3.3.

Clearly both angle-averaged potentials display a pronounced, attractive

potential well with nearly identical depths. In fact, the angle-averaged

potential somewhat resembles that of an LJ fluid (also plotted in Fig. 3.3),

disregarding differences in the width of the attractive well and the decay

of the potentials with r12. Hence, we also consider a driven LJ fluid char-

acterized by the pair potential ULJ =4ϵ((σ /rij )12 −(σ /rij )6) with cut-offat

2.5σand dimensionless interaction strength ϵ∗=ϵ/kBT. Except of the In-

teraction potential, the general set up is the same as in the DDP model.

Specifically, the equations of motion (Eqs. [3.13]), the distribution of ori-

entations siand the driving fdacting on particle i. In the following the

reduced units used in this chapter are summarized.

Tabular representation of reduced units.

reduced density ρ∗=ρσ 2

reduced driving force f∗

d=fdσ /kBT

reduced dipole moment µ∗≡µ(2)

i∗=µ(2)

i/√σ3kBT

reduced LJ coupling strength ϵ∗=ϵ/kBT

3.2The DDP Model for Dipolar Microswimmers in Experiment

3.2.2 Target quantities

To quantify the degree of laning we introduce for every particle ia slice

in x-direction with lateral width σcentered around yi. We then assign

a variable Φiwhich equals 1if the slice contains no particle of opposite

driving direction and 0otherwise. The ensemble-averaged laning order

parameter then follows as

Φ=⟨

∑

i=1

Φi/N ⟩,(3.15)

where the brackets denote a time average. Non-laned states are char-

acterized by Φ≈0whereas perfectly laned states correspond to Φ=1.

However, the system can visually appear as being laned already at much

smaller values of Φ(e.g., Φ≈0.2), especially at high densities.

To characterize the local structure in x-direction we calculate the radial

distribution function between particles of the same type

gs(x) = 2

ρN ⟨

∑

i=1

∑

j,i

δ(x−|xij |)Θ(σ−|yij |)Θ(sisj)⟩(3.16)

with xij (yij ) being the x(y)-component of rij , and δand Θare the delta-

and the Heavyside step function, respectively. By normalization, gs(x)

decays to 2for x→ ∞ in a single-species fluid and to 1in a completely

mixed binary fluid. By interchanging xand yin Eq. [4.2] we additionally

define the correlation function gs(y) in y-direction.

3Theoretical Models of Complex Dipolar Colloids

3.3 Colloids in Crossed Fields

In this section we are particularly interested in modeling the field-directed

aggregation of colloids with field-induced dipolar interactions. A previ-

ously studied example for such systems is capped (metal-coated) dielec-

tric particles under the presence of an uni-axial electric field inducing

quadrupolar-like interactions [26,95]. Also mixtures of para- and dia-

magnetic colloids under the presence of an uni-axial magnetic field sus-

pended in a ferromagnetic carrier fluid [98] have been considered. In these

studies, it is one external field which allows to tune the aggregated struc-

tures with respect to one direction. Here we consider even more complex

interactions caused by crossed (orthogonal) fields, allowing bi-directional

tuning of particle aggregation.

Before discussing the experimental system presented in see Sec. 1.3.3we

briefly mention another example of a possible experimental realization of

tunable bi-directional aggregation. which is a quasi two-dimensional sys-

tem of suspended dielectric colloidal particles under the influence of two

in-plane orthogonal AC electric fields with a phase shift of π. The fields

will polarize the particles’ ionic layer and/or dielectric material periodi-

cally but at different times due to their phase shift. We assume that by

adjusting the field frequencies and phases to the relevant timescales gov-

erning particle diffusion and the relaxational dynamics of the polarized

ionic layer, two decoupled orthogonal dipole moments in each particle

can be generated by this setup.

The actual experimental setup we are modeling is a quasi two di-

mensional system of suspended colloidal particles, each composed of

super-paramagnetic iron-oxide aggregates embedded in a polymer ma-

trix, which has been investigated experimentally by Dr. Bharti [25,133].

In this case, crossed external electric and magnetic fields, oriented in

plane but perpendicular to each other, can be used to induce independent

electric and magnetic dipole moments in the colloids, which leads to a

directed self-assembly process of two-dimensional single-particle chain

networks. An experimental photograph of the resulting structures is

shown in Fig. 1.6(e) and experimental details are described in Sec. 1.3.3.

Please note that the oscillation of the AC electric field does not induce a

magnetic field in the assembly plane. Thus, magnetic and electric inter-

actions are decoupled in this setup. This follows directly from the fourth

Maxwell equation (Ampere’s circuital law)

∫

∂Σ

H·dl=

∫

(j+∂tE)·ds(3.17)

where Eis the oscillating electric field, jis the electric current density and

dsdenotes the differential vector element of the assembly plane (surface)

Σ. As dsis normal to the surface and thus also to the electric field E

and the current density j, the right hand side of Eq. [3.17] vanishes inside

3.3Colloids in Crossed Fields

the plane. In fact, the experimental setup corresponds to a current sheet,

and it can be shown that the strength of the induced magnetic field grows

linear with increasing lateral distance from the plane [145]. Thus, for a

quasi two-dimensional experimental assembly chamber of lateral size in

the range of a few micrometers the magnetic field which is induced by the

oscillation of the external electric field can be neglected.

In the following, two different models for field-directed aggregation are

presented. But before, we shortly comment on general properties of the

induced dipole moments.

Interaction potential of crossed point dipoles

The crossed orthogonal external fields induce orthogonal dipole moments

µm=µemand µe=µee, which we term for simplicity as ’magnetic (m)’

and ’electric (e)’ dipoles (although one might also think of two electric mo-

ments). The coordinate frame is adjusted to coincide with the directions of

these moments so that em=exand ee=ey. In general, these moments can

have different absolute values, but for simplicity they are assumed to be

equal in this preliminary consideration. The two types of dipole moments

are also assumed to be independent from each other, meaning that they

interact only with dipole moments of the same type on other particles.

Intuitively, one would model the interaction energy between dipoles of

particles 1and 2by the point-dipole potential

Uα

dip(r12) =

µα

1·µα

12 −3

(µα

1·r12)(µα

2·r12)

,(3.18)

where αindicates the dipole type as being either eor m. Due to the con-

straint µα

1∥µα

2it follows that

Uα

dip(r12) =

µα

1µα

(1−3(r12 ·eα)2

).(3.19)

The resulting total dipolar interaction between two particles is the sum of

the dipolar potentials stemming from the magnetic and electric dipoles,

respectively. Using µ=|µα

i|and the relation (r12 ·ee)2+ (r12 ·em)2=r2

(which holds since µeand µmare orthogonal) we obtain

dip(r12) + Um

dip(r12) = −µ2

.(3.20)

The resulting interaction on the right side of Eq. [3.20] is an isotropic,

purely attractive interaction that lacks any kind of directional character.

Therefore, the potential defined in Eq. (3.20) can not generate any directed

and/or rectangular structures as observed in experiments (see Fig. 1.6(a)-

(d) and [25,133]). Underlying reasons for the more complex character

of the true interactions might be many-body effects like mutual depolar-

ization [146,147,148], and/or nonuniform intra particle properties e.g.,

3Theoretical Models of Complex Dipolar Colloids

the distribution of magnetic material inside particles [149]. Hence, the

following parts of this chapter are concerned with more complex models.

3.4The CDP Model for Superparamagnetic Particles in Experiment

3.4 The CDP Model for Superparamagnetic

Particles in Experiment

In the present section we introduce the crossed dipole particle (CDP)

model, to describe the superparamagnetic particles of the experimental

setup introduced in Sec. 1.3.3. There, superparamagnetic polymer parti-

cles in a quasi two-dimensional setup are exposed to crossed electric and

magnetic fields. In Fig. 1.6(a) and (c) it can be seen that the influences

of the magnetic and electric fields on the self-assembly processes differ

strongly. While the presence of an electric field ( Fig. 1.6(a) ) results in

straight single particle chains oriented along the electric field, the exter-

nally induced magnetic interactions result in more compact (staggered)

chain-like aggregates pointing along the magnetic field ( Fig. 1.6(c) ) when

no electric field is applied.

On the origin of induced magnetic interactions

The bidirectional assembly of particles in external fields is governed by

induced dipolar interactions and results in the formation of a variety of

particle structures. In the case of AC-electric field driven assembly of mi-

croparticles in aqueous dispersions, the dipoles are induced by partial po-

larization i.e redistribution of ions in the ionic double layer around the

particles. Therefore the particles can be assumed to have a single dipole

which overlaps with the center of mass of the particles. These dipole-

dipole interactions lead to assembly of particles into linear chains. How-

ever, in an uniform magnetic field, dipoles are induced by polarization

of local magnetic domains. In our case of composite microspheres, the

magnetic domains are the superparamagnetic nanoaggregates embedded

in the particles. Hence, the net magnetic dipole moment in a particle

is the vector sum of all the individual dipoles induced in the embedded

nanoparticles. Depending upon the spatial distribution of the individual

magnetic domains, the geometric center of the sum over single magnetic

moments may not overlap with the center of mass of the particles. This

has been shown experimentally for similar superparamgentic beads [149].

The dislocation of the net magnetic interaction site from the center of the

particle may result into partially staggered chain configurations instead of

linear ones. We believe that this is the case with our superparamagnetic

microparticles, where linear chains are formed in external AC-electric

field and partially staggered chains are formed in an uniform magnetic

field (see Fig. 1.6).

The theoretical model

To this end, the electric interactions are modeled by placing a point dipole

moment µel

ioriented along the electric field in the center of each particle

3Theoretical Models of Complex Dipolar Colloids

i, whereas a shifted and truncated (12,6) Lennard-Jones Potential is used

to model sterical interactions between particles iand j

USS (rij ) = 4ϵ

(

(σ /rij )12 −(σ /rij )6+1/4

)

.(3.21)

Here, rij =ri−rjis the center to center distance, ϵand σset the units

of energy and length, respectively and a spherical cut offrc,SS

ij =21/6σ

is applied. This setup corresponds to the well established dipolar soft

sphere model [22]. To account for magnetic interactions, each particle

bears additionally a magnetic dipole moment µmg

iwhich is pointing along

the magnetic field. In contrast to the electric case, the magnetic dipole

moments are shifted out of the particle centers: For each particle this shift

is given by a vector δiwith an orientation taken from an uniform ran-

dom distribution and a constant absolute value δ=0.2σ. Furthermore,

the absolute value of each magnetic moment is chosen from a Gaussian

distribution with mean ¯

µmg and variance σmg =¯

µmg /2under the condition

that µmg

i∈[0,2¯

µmg ]. The latter interval is chosen to prevent unphysical

negative moments and to avoid the occurrence of very strong moments,

which is important for computational efficiency.

This model is motivated by the fact that experimental particles consist of

polymer droplets incorporating numerous superparamagnetic iron-oxide

aggregates. The underlying assumption is that different particles carry

different amounts of magnetic material and that this magnetic material

is inhomogeneously distributed throughout the particle1and forms local-

ized magnetic interaction centers. The underlying reasons might be vari-

ations in the production process and polydispersity of the particles. Note

that the increase of magnetic material goes cubic with the particle diam-

eter σ, which makes larger differences in the particles magnetic response

rather reasonable, even if the the distribution of particle sizes is relatively

sharp. Also (localized) loss of magnetic material through the particle sur-

face during the experiment or in storage could be considered; a process

which was observed during Dr. Bharti’s experiments. These assumptions

are reflected in the distribution of the absolute magnetic moments µmg

and in the random spatial shift δof the magnetic interaction site, respec-

tively. However, the actual values for the particle parameters were not

optimized or derived from experimental or theoretical insights. Instead,

their choice was done by intuition. Please note that we are not aiming on

the most accurate description of single particles but rather on an accurate

description of the many particle system. Hence, we do not claim that the

magnetic material is in reality distributed throughout the particles in ex-

actly the way we model them. Furthermore, an accurate description of the

magnetic properties of various interaction sites would in principle require

1Recently, an experimental study concluded that the magnetic properties of similar

particles result from an inhomogeneous internal distribution of the magnetic mate-

rial [149].

3.4The CDP Model for Superparamagnetic Particles in Experiment

Figure 3.4: Direction-dependent pair interaction U(rij ) [see Eq. 3.24] be-

tween a particle with horizontal (a) and vertical (b) shift of

magnetic interaction in the center of the coordinate frame

and a second (reference) particle at various positions rij with

no shift of the magnetic moment. Positions of the mag-

netic/electric dipole moments are indicated by red/blue dots

and corresponding arrows. In both cases holds δ=0.2σand

µel =µmg =√ϵσ 3. Sterically excluded areas are indicated by

white circles while the color code gives the interaction energy

U(rij ) in units of ϵ. These are two realizations for the random

position of the magnetic moment yielding different anisotropic

interaction patterns.

to calculate the magnetization self-consistently, which is again not feasi-

ble in the many particle systems we are interested in. Instead, we provide

a physically motivated and reasonable model, which is reproducing the

structures observed in experiment and allows to overcome the difficulties

associated with the control of global and local particle density in the ex-

periments.

Finally, both moments are induced and have therefore a fixed orien-

3Theoretical Models of Complex Dipolar Colloids

tation. For simplicity, we neglect torques due to off-centered magnetic

interaction sites. With this particle setup, the distance between dipole

moments µα

iand µα

jof species α= (el, mg) is given by

rij =rj−ri+δj−δi(3.22)

whereas α=el corresponds to the electric case with δi,j =0.The dipolar

interaction between particles iand jis then calculated via the usual three-

dimensional point dipole potential

Uα

dip(rij ,δi,δj) = µiµjr−3

ij −3(µirij )(µjrij )r−5

ij (3.23)

and the total interaction potential is given by

U(rij ) = USS (rij ) + Uel

dip(rij ) + Umg

dip(rij ,δi,δj).(3.24)

Standard two-dimensional Ewald summation method is used to handle

the long-range character of the dipole interactions [150,144]. In Fig. 3.4,

the interaction energy map for to random realizations of such particles

is shown. Clearly, the model result in anisotropic interactions which are

different for different particle pairings. In consequence, we consider a

multi-component system of particles with anisotropic attractive interac-

tions.

We perform overdamped Brownian dynamics simulations in a two-

dimensional quadratic simulation box at reduced number density ρ∗=

ρ/σ 2with Nparticles of equal diameter σ. The equations of motion are

given by

γ˙

ri=

∑

j=1

∇rij U(rij ) + ζi(3.25)

where γis the friction constant and ζiis a Gaussian noise vector acting

on particle ifulfilling ⟨ζi⟩=0and ⟨ζi(t)ζj(t′)⟩=2γT ∗δij δ(t−t′) with

T∗=kBT /ϵ being the temperature times Boltzmann’s constant in units

of ϵ. All simulations are performed at the same constant temperature

T∗=0.01. The Eqs. [3.25] are solved via the Euler scheme (see Sec. 6) with

an integration step width ∆t=10−5τb, where τbis a Brownian timescale

defined by τb=σ2γ/ϵT ∗. In the following the reduced units used in this

chapter are summarized.

Tabular representation of reduced units.

reduced density ρ∗=ρσ 2=0.35

reduced temperature T∗=kBT /ϵ =0.01

reduced dipole moment µ∗=µ/√σ3ϵ

3.5Characterization of Network Structures - Energy Histograms

3.5 Characterization of Network Structures -

Energy Histograms

Under the presence of both, magnetic and electric fields, the superpara-

magnetic particles described in the previous section form network-like

structures as shown in Fig. 1.6. These structures can be thought of as ag-

gregates of basic building blocks, each consisting of a central particle with

a specific arrangement of nearest neighbors at distance ≈σ. In Fig. 3.5all

rectangular and linear configurations with field aligned connection vec-

tors between nearest neighbors are shown. Note that this means that the

number of neighbors is restricted, Nb≤4. These configurations corre-

spond to basic building blocks for chains, end of chains/dimers and junc-

tions (yellow, magenta light blue and black central particle). In the fol-

lowing, a method which allows to identify these building blocks in the ex-

perimentally observed colloidal networks is presented. To this end, each

particle is associated with two fictitious dipole moments mαof absolute

value mα=√ϵσ 3placed in its center. Here ϵis again an arbitrary unit

of energy. One moment points along the electric field (α=el), and one

along the magnetic field (α=mg). Note that these moments are not phys-

ical; Being fictitious quantities these moments serve in the following as

Figure 3.5: Different local arrangements of particles with nearest neigh-

bor distances σin rectangular network structures and in lin-

ear chains. The legend gives the fictitious energies uel (umg )

in units of an arbitrary energy ϵfor particles with dipole

strengths mα=√ϵσ 3. Note that horizontal chains have posi-

tive electric energies uel (red frame), while vertical chains have

positive magnetic energies umg (blue frame).

3Theoretical Models of Complex Dipolar Colloids

’tools’ for structure analysis only. Introducing these moments allows to

calculate corresponding fictitious electric and magnetic energies between

the central particle of a building block, denoted by i, and all its nearest

neighbors jvia the summed dipole-dipole potential

uα

∑

j=1

mα

imα

je−3

ij −3(mα

ieij )(mα

jeij )e−5

ij (3.26)

with eij =rij /rij . Hence, each particle ican be associated with two ficti-

tious energy values, the electric one uel

iand the magnetic one umg

i. Cer-

tain energy pairs correspond to certain basic building blocks as it can be

seen in in Fig. 3.5. In fact, knowledge of both values, uel

iand umg

i, allows

unique identification of the building block belonging to particle i: knowl-

edge of one energy contribution is insufficient to distinguish between end

of chains (dimers) and junctions (configurations with four neighbors).

Applying this identification scheme to network structures from sim-

ulation or experiment requires to define a cut offdistance for nearest

neighbors, which is chosen here to rc=1.15σ. Because σ≲rcthe ener-

gies uαcan reasonably be calculated by using the unit distance vector eij

(see Eq. [3.26]) instead of the true distance. Here we assume that nearest

neighbors are (almost) in contact. A certain rectangular network struc-

ture is properly characterized by the distributions of the energies uel

iand

umg

i, which means by the distribution of the basic building blocks. In

Fig. 3.6(d), the distributions of uel

iand umg

iare shown for the experimen-

tal network given in Fig. 3.6(a) as well as for a network resulting from

simulation, which is shown in Fig. 3.6(e). The histograms are normalized

by the number of particles in aggregates A. Certain (integer valued) peaks

of these distributions correspond to the populations of certain building

blocks, which are placed as cartoons beneath these peaks. Closer inspec-

tion of Fig. 3.5shows that both energies are negative or equal zero( uα≤0

and uγ,α≤0) for particles in junctions or more compact structures. In

contrast to that, chain-like configurations generally yield positive values

of one energy type and negative energies of the other type (uα>0and

uγ,α≤0). In fact, particles in horizontal chains have positive electric in-

teraction energies uel >0, while particles in vertical chains have positive

magnetic energies umg >0. This can also be seen in Fig. 3.6(b), where each

particle iis color-coded according to its electric energy uel

Hence it is straight forward to determine the ’role a particle plays’. In

the following, particles in horizontal chains (uel >0) are colored in red,

particles in vertical chains are colored in blue (umg >0) and node particles

are colored in black (uel <0and uel <0). Particles without nearest neigh-

bors are indicated by black circles. Applying this identification scheme to

the network structure from experiment (Fig. 3.6(a)) results in a re-color

coded representation given in Fig. 3.6(c). This scheme works surprisingly

well and allows proper identification of horizontal and vertical chains as

3.5Characterization of Network Structures - Energy Histograms

Figure 3.6: Colloidal network in experiment at E=37Vand H=3.5V

and simulation at µel =0.4,¯

µmg =0.4and density ρ∗=0.35.

Experimental microscopic image (a) in natural colors, (b) with

particles colored according to the distribution uel and (c) col-

ored according to the ’role’ a particle plays in the network. (d)

Distributions of uel (blue) and umg (red) for the experimental

structures in (a) and for a simulated network colored according

the ’role a particle plays’ in (e).

3Theoretical Models of Complex Dipolar Colloids

well as nodes or clustered aggregates.

In addition, once the role a particle pays in the aggregate is found, statis-

tics of populations of node or chain particles can be collected. In the

following, the number of particles in vertical chains is denoted by v, in

horizontal chains by hand in nodes by n. By Awe denote the number of

particles with at least one neighbor and it holds

v+h+n=A. (3.27)

3.6The EDP Model - A Generic Approach towards Mutidirectional Field

Induced Dipolar Interactions

3.6 The EDP Model - A Generic Approach

towards Mutidirectional Field Induced

Dipolar Interactions

In the present section we introduce the extended dipole particle (EDP)

model. Our aim is to construct a model which captures essential features

of the interactions between colloids in crossed fields. However, we do not

claim to model one specific (electric and/or magnetic) system in its details,

but rather provide a generic and computationally convenient model. To

this end, we consider a two-dimensional system of Nsoft spheres of equal

diameter σ. The soft sphere interactions are repulsive and are modeled by

a shifted and truncated (12,6) Lennard-Jones Potential

USS (rij ) = 4ϵ

(

(σ /rij )12 −(σ /rij )6+1/4

)

(3.28)

which is cut offat rc,SS

ij =21/6σ. Here, rij =|rj−ri|is the particle center-to-

center distance and ϵsets the unit of energy.

The crossed orthogonal external fields induce orthogonal dipole mo-

ments µm=µemand µe=µee, which we term for simplicity as ’magnetic

(m)’ and ’electric (e)’ dipoles (although the model is also appropriate for

two electric moments). The coordinate frame is adjusted to coincide with

the directions of these moments so that em=exand ee=ey. In general,

these moments can have different absolute values, but for simplicity they

are assumed to be equal in this preliminarily consideration. The two types

of dipole moments are also assumed to be independent from each other,

meaning that they interact only with dipole moments of the same type on

other particles. The occurrence of rectangular particle arrangements in

experiments [25] suggests an effective ’four-fold valency’ of pair interac-

tions, irrespective of other details.

Here we want to take into account the four-fold valency but also the

overall attractiveness (no repulsion) of the crossed point-dipole setup [see

Eq. (3.20)]. We thus introduce, as detailed below, artificial ’dipole mo-

ments’ composed of charges with short-ranged interactions. Note that the

fixed orientation of dipole moments and the overall attractiveness con-

trasts the ’classical’ theoretical concepts of patchy particles [4,103,102,

105,151], which are able to rotate and are characterized by localized at-

tractive and repulsive interactions.

To be specific, each dipole moment µα(with α=e, m) is replaced by two

opposite charges −qα1=qα2which are shifted out of the particle center by

a vector δαk= (−1)kδeα, with k=1,2. The vector δαkpoints either par-

allel (k=2) or anti-parallel (k=1) along the corresponding point dipole

moment µα. Independent of their type, all charges have the same abso-

lute value q=|qαk|=2.5(ϵ/σ )−1/2and shift |δ|for the sake of simplicity. In

principle though, this ’extended’ dipole model allows also to vary the val-

ues of qand δfor different interaction types. Also, the choice of the value

3Theoretical Models of Complex Dipolar Colloids

Figure 3.7: Distribution of externally induced fictitious ”charges” qinside

a particle. Positions of charges are determined by the vec-

tors δαk∈[−δex, δex,−δey, δey]) pointing either parallel or anti-

parallel to the corresponding fields.

q=2.5is essentially arbitrary, as we will later normalize the interaction

energy to eliminate the dependence of its magnitude on the charge sepa-

ration δ[see Eq. (3.32) below]. A schematic representation of the model

with its internal arrangement of ’charges’ is shown in Fig. 3.7. Mimicking

magnetic dipoles via spatially separated ’charges’ is clearly artificial from

a physical point of view, but in the spirit of the generic character of our

model. Charges kand lon different particles i and j interact via a Yukawa

potential

Uαkαl

ij (rij ) = −q2

exp(−κrαkαl

ij )

rαkαl

(3.29)

with rαkαl

ij =|rj−ri+δαl−δαk|. The inverse screening length is chosen

to κ=4.0σ−1and a radial cutoffrc=4.0σis applied, which ensures in-

teraction energies smaller than 10−6at cut-offdistance. Using a screened

potential between the charges has mostly computational reasons; correct

treatment of the true, long-ranged coulomb potential requires specific

simulation methods [144]. In the present model the effort is enhanced

by the fact that each particle has four charges. Still, the directional depen-

dence of the interactions does not change due to the screening. We also

note that previously similar models with comparable interaction ranges

have been used to describe dipolar colloids in the framework of discon-

tinuous molecular dynamics simulations [108,152]. The arrangement of

charges inside particles then results in a pair-interaction UDIP (rij ) given

3.6The EDP Model - A Generic Approach towards Mutidirectional Field

Induced Dipolar Interactions

Figure 3.8: Normalized direction-dependent pair interaction U(rij ) [see

Eq. (3.34)] between a particle in the center of the coordinate

frame and a second particle (indicated as black circle in (a))

at various positions rij for three different charge separations

δ=0.1,0.21,0.3σcorresponding to (a), (b), (c). Sterically ex-

cluded areas are indicated by white circles. (d) Interaction

energy at distance rij =σas function of φ, the angle mea-

sured in multiples of πagainst the x-axis, for δ=0.1σ(yellow),

δ=0.21σ(purple) and δ=0.3σ(black).

UDIP (rij ) =

∑

k,l=1

[Uekel

ij (rij ) + Umkml

ij (rij )].(3.30)

In principle, UDIP (rij ) is a function of qand δ. To facilitate the compar-

ison between the interactions at different δ(qis chosen to be constant), we

normalize UDIP (rij ) according to

UDIP (rij ) = UDI P (rij )×u/UDIP (σeα) (3.31)

where the constant u=−2.804ϵis calculated from the non-normalized

energy UDIP (σeα) with model parameters δ=0.3σand q=2.5(ϵ/σ )−1/2.

This procedure ensures that the normalized energy between two particles

at contact (rij =σ) and direction rij =σeα(pointing along one of the fields)

has the constant value ufor all δ, that is

UDIP (σeα) = u. (3.32)

3Theoretical Models of Complex Dipolar Colloids

Therefore we can reformulate the dipolar coupling strength in Eq. [3.7]

for this model towards

λ=u/kBT . (3.33)

The full pair interaction of our model is then given by

U(rij ) = USS (rij ) + ˜

UDIP (rij ).(3.34)

The resulting potential is illustrated in Fig. 3.8(a)-(c) for a particle in

the center of the coordinate frame and a second particle at various dis-

tances rij and angles φ= arccos(r12 ·ex/r12) with ’charge’ separations δ=

0.1,0.21,0.3σ. The value δ=0.21σis motivated by our simulation re-

sults presented in Sec. 5.2.2. Sterically-excluded areas are shown in white

and energy values are color coded in units of ϵ. The weak anisotropy

of the resulting particle interactions at small δ(where one essentially

adds two dipolar potentials, (see Eq. [3.20]) transforms to a patchy-like

pattern [103,102] by increasing δ. Energy minima become more and

more locally restricted and interactions reveal an increasing four-fold (i.e.,

’patchy’) character, although remaining their attractiveness in general.

This is also seen in Fig. 3.8(d) which gives the energy between two par-

ticles in contact as function of φfor different δ. From Fig. 3.8(d) we also

see that, independent of the ’charge’ separation δ, the minima of the full

interaction potential (see Eq. [3.34]) occur for connection vectors rij =σee

and rij =σem(i.e., pointing along the fields). Note that this already holds

for the non-normalized energy given in Eq. [3.30].

Tabular representation of reduced units.

reduced density ρ∗=ρσ 2

reduced temperature T∗=kBT /ϵ

reduced interaction strength u∗=u/ϵ =−2.804

Brownian time scale τb=σ2γ/kBT

Simulations are performed with N=1800 to 3200 particles at a range

of reduced number densities ρ∗=ρσ 2and temperatures T∗=kBT /ϵ, in

a square-shaped simulation cell with periodic boundary conditions. The

equations of motion

γ˙

ri=−

∑

j=1

∇U(rij ) + ζi(t) (3.35)

are solved via the Euler scheme with an integration step width ∆t=

10−4τb, where τb=σ2γ/kBTis the Brownian timescale, γis the friction

constant and ζi(t) is a Gaussian noise vector which acts on particle iand

fulfills the relations ⟨ζi⟩=0and ⟨ζi(t)ζj(t′)⟩=2γkBT δij δ(t−t′) [135]. We

perform simulations for up to 103τb.

Lane Formation of Dipolar

Microswimmers

Using Brownian Dynamics (BD) simulations we investigate the non-

equilibrium structure formation of a two-dimensional (2D) binary

system of dipolar colloids propelling in opposite directions. Despite

of a pronounced tendency for chain formation, the system displays a

transition towards a laned state reminiscent of lane formation in sys-

tems with isotropic repulsive interactions. However, the anisotropic

dipolar interactions induce novel features: First, the lanes have

themselves a complex internal structure characterized by chains or

clusters. Second, laning occurs only in a window of interaction

strengths. We interpret our findings by a phase separation process

and simple force balance arguments.

4.1 Background

In the present chapter we present results previously published in [153].

Based on particle-based Brownian Dynamics (BD) simulations, we investi-

gate a prototype of field-propelled complex colloids, that is, spheres with

dipolar interactions. Our model is inspired by real systems of metallodi-

electric ”Janus” spheres with two dielectric parts acquiring different (in-

duced) dipole moments in an external electric field [15]. Previous exper-

imental studies in a quasi-two dimensional (2D) setup with an in-plane

field have shown that these particles perform straight motion perpendicu-

lar to the field [55], with two possible directions depending on the orien-

tation of the hemispheres. Ensembles of such particles therefore should

4.1Background

exhibit lane formation, with the additional feature of strongly anisotropic

interactions favoring the formation of staggered chains. Based on a suit-

able model system, we find indeed several new phenomena: First, de-

spite a pronounced tendency to aggregate into chains perpendicular to

the driving force, the dipolar interactions can induce laning at densities

where purely repulsive systems are mixed. At larger interaction strengths,

however, we observe a breakdown of lane formation. Moreover, by com-

paring the lane formation with that in a much simpler system governed

by isotropic, attractive Lennard-Jones (LJ) interactions, we show that the

main laning mechanism is, in fact, the effective (angle-averaged) attraction

between the dipoles. Indeed, for both systems the onset of laning roughly

occurs at coupling strengths related to equilibrium phase separation. Fi-

nally, lanes disappear when the maximum pair attraction corrected by the

thermal energy exceeds the work done by the driving force.

Tabular representation of reduced units.

reduced density ρ∗=ρσ 2

reduced driving force f∗

d=fdσ /kBT

reduced dipole moment µ∗≡µ(2)

i∗=µ(2)

i/√σ3kBT

reduced LJ coupling strength ϵ∗=ϵ/kBT

Target quantities To quantify the degree of laning along the driving

force fs

d,i =sifdex, we introduce for every particle ia slice in x-direction

with lateral width σcentered around yi. We then assign a variable Φi

which equals 1if the slice contains no particle of opposite driving direc-

tion and 0otherwise. The ensemble-averaged laning order parameter then

follows as

Φ=⟨

∑

i=1

Φi/N ⟩,(4.1)

where the brackets denote a time average. Non-laned states are char-

acterized by Φ≈0whereas perfectly laned states correspond to Φ=1.

However, the system can visually appear as being laned already at much

smaller values of Φ(e.g., Φ≈0.2), especially at high densities. To charac-

terize the local structure in x-direction we calculate the radial distribution

function between particles of the same type

gs(x) = 2

ρN ⟨

∑

i=1

∑

j,i

δ(x−|xij |)Θ(σ−|yij |)Θ(sisj)⟩(4.2)

with xij (yij ) being the x(y)-component of rij , and δand Θare the delta-

and the Heavyside step function, respectively. By normalization, gs(x)

decays to 2for x→ ∞ in a single-species fluid and to 1in a completely

mixed binary fluid. By interchanging xand yin Eq. (4.2) we additionally

define the correlation function gs(y) in y-direction.

4Lane Formation of Dipolar Microswimmers

0.1

0.3

0.5

0.7

0.9

(a)

0 10 20 30 40 50 60 70 80 90

∗

0.1

0.3

0.5

0.7

0.9

(b)

∗

Figure 4.1: Laning order parameter as function of driving force for (a) the

DDP system (µ∗=1.58) and (b) the SS system (ϵ∗

SS =10kBT) at

different densities.

4.2 Lane Formation of Dipolar Microswimmers

In Fig. 4.1(a) we plot the laning order parameter Φof DDP systems for dif-

ferent densities ρ∗=ρσ 2as functions of the dimensionless driving force

f∗

d=fdσ /kBT. At low driving forces (f∗

d≤10) the order parameter is essen-

tially zero for all densities considered, reflecting the absence of lanes. The

corresponding local structure at f∗

d=0and ρ∗=0.2is illustrated by the

simulation snapshot shown in Fig. 4.2(a). One observes the formation of

’staggered’ chains, consistent with the most attractive pair configurations

illustrated in Fig. 3.2(b), and in qualitative agreement with experiments of

metallo-dielectric particles under field conditions where self-propulsion

is absent [15]. At very small driving forces (f∗

d<10) staggered chains are

driven against each other and form one large aggregate consisting of both

species. This corresponds to a jammed state.

Increasing f∗

dto larger values, all of the systems considered in Fig. 4.1(a)

display laned ’states’ characterized by non-negligible values of Φ. For

small densities (ρ∗<0.4) this lane formation occurs gradually and signif-

icant values of Φare reached only at high driving forces f∗

d>50. In con-

trast, dense systems display a rather steep laning ”transition” at a driving

force f∗

d≈15. A visualization of exemplary laned states at f∗

d=80 and two

densities is given in fig. 4.2(b) and (c). Here, the driving force fs

d,i =sifdex

points horizontally either to the left, for ’yellow’ particles (si=−1) or to

the right for ’purple’ particles (si=1). To elucidate the impact of dipolar

forces on the functions Φ(f∗

d) we plot in Fig. 4.1(b) corresponding data for

the driven SS system. At low densities ρ∗≤0.4the systems behave similar

4.2Lane Formation of Dipolar Microswimmers

Figure 4.2: Snapshots of the DDP system in the x-y plane at (a) f∗

d=0,

ρ∗=0.2, (b) f∗

d=80,ρ∗=0.2(Φ≈0.2) and (c) f∗

d=80,ρ∗=0.5

(Φ≈0.75). The driving force acts horizontally to the right (left)

on dark (bright) particles: s=1(−1), µ∗=1.58.

4Lane Formation of Dipolar Microswimmers

0 2 46 8 10 12

]

gs(x)

12 3 4 5 6

]

gs(y)

∗

Figure 4.3: Pair correlation functions between double dipolar particles of

the same species along the x-direction (main plot) and the y-

direction (inset). The data pertain to f∗

d=80 and different

densities. In the inset, curves for different densities have been

shifted by one σto enhance visibility.

to their DDP counterparts. However, at high densities (ρ∗>0.4) there is

no lane formation in the (purely repulsive) SS system, in striking contrast

to the behavior of the DDP systems. We will come back to this point be-

low. Here we first consider the internal structure of the lanes in the DDP

systems.

4.2.1 Complex lanes - Internal structure

Visual inspection of Figs. 4.2(b) and (c) suggests that lanes consist of clus-

ters of particles of the same type.

At low densities (see Fig. 4.2(b) ) we find mostly short chains oriented in

y-direction, which reminds of the equilibrium structure of single dipole

particles in external fields [22]. Clearly, such chains are absent in sim-

pler (isotropic) lane-forming systems. At higher densities (see Fig. 4.2(c))

the particles tend to form clusters along the driving force but the local

arrangement is not crystalline in the sense of i.e., a hexagonal Bravais lat-

tice. However, the corresponding (hexagonal) translational order param-

eter [65] (see Eq [5.5]) is larger than that observed at low f∗

For a more quantitative characterization we consider the correlation

functions gs(y) and gs(x) plotted in Fig. 4.3at a high driving force f∗

d=80

and at different densities ρ∗∈[0.2,0.6]. For the lowest density considered

(ρ∗=0.2)gs(y) reveals a pronounced peak at y=1σand a second, smaller

4.2Lane Formation of Dipolar Microswimmers

one at y=2σ, indicating chains of up to three particles along the field di-

rection. Furthermore, we observe a ’hump’ appearing directly before the

first peak. This ’hump’ results from bended and perturbed chains as well

as from a few clusters elongated in x-direction (see Fig. 4.2(b)). Increasing

the density the hump transforms into a pronounced peak, which finally

overtakes the peak at y=1σat ρ∗=0.4. At this density, linear chains

in y-direction have lost their dominance as structural elements. Instead,

clusters spread out in x-direction.

This interpretation is supported by the behavior of the function gs(x)

which displays a similar qualitative change at ρ∗=0.4. At densities ρ∗≤

0.3,gs(x) reveals only one relatively small peak at a distance of x=1σ

showing that a minor fraction of chains is bended, staggered or has ad-

ditional particles attached in x-direction. Increasing ρ∗results in an in-

crease of this first peak and the emergence of a second peak at x≈2σ.

Again this peak reaches relevant values (>2) for ρ∗=0.4and indicates

thereby cluster growth in x-direction and finally a change from linear

chains in y-direction to horizontally elongated clusters. At even higher

density ρ∗=0.5,0.6elongation of clusters in x-direction becomes compa-

rable to the system size.

We conclude that there are two qualitatively different regimes for lane

formation in the DDP-system. The first one (chain regime) is characterized

by short chain-like aggregates inside lanes and occurs for relatively high

driving forces f∗

d>50 and low densities ρ∗≤0.3. These parameters are

comparable to the parameters under which lane formation is observable

in the HS-model and SS-model (see Fig. 4.1(b) and [91,80]). The second

one (cluster regime) is characterized by in x-direction elongated clusters

arranged in lanes and is restricted to larger densities ρ∗>0.4. A similar

intra lane structure has been reported by Menzel in [89] for a confined

two-dimensional setup of active particles with an alignment mechanism.

However, once the density is larger than 0.4and f∗

d>20, the degree of lane

formation is more or less independent of ρ∗and f∗

d. The fact that at very

small driving forces the system undergoes a transition towards a laned

state for all densities ρ∗>0.3at the same driving force is qualitatively

different to the reentrant behavior of the HS-model [91].

We note that these structural details are quite sensitive to the shift pa-

rameter δcharacterizing the position of the dipole moments inside the

particles: Indeed, as revealed by test simulations, smaller values of δtend

to suppress the lane formation in the ”chaining regime” and enhance the

occurrence of chains in the ”cluster regime”.

4.2.2 Complex lanes - Characteristic lane widths

The fact that lanes in the DDP-system reveal certain internal structures,

raises the question whether these internal structures translate into charac-

teristic lane widths.

4Lane Formation of Dipolar Microswimmers

Figure 4.4: Pair-correlation functions , g(y) (black) and gs(y) (red), of the

double dipolar system in y-direction for laned states at den-

sities ρ∗=0.2(a) and ρ∗=0.5(b) and common driving force

f∗

d=80.

Figure 4.5: Pair-correlation functions gs(y) in y-direction for laned states

of the DDP-Model at density ρ∗=0.5and various driving

forces f∗

d∈[10,70]. First ’long-ranged’ minima of gs(y) are in-

dicated by dots. The different functions gs(y) are subsequently

shifted by 4σin vertical direction for better illustration.

4.2Lane Formation of Dipolar Microswimmers

To answer this question we show in Fig. 4.4(a) and (b) the particle type

specific pair distribution functions in y-direction gs(y) as well as the usual

pair distribution function in y-direction g(y) at f∗

d=80 for the two den-

sities ρ∗=0.2,0.5, corresponding to the snapshots in Fig.4.2(b) and (c).

The presence of particle type specific lanes is reflected in the long-range

oscillation of gs(y≥5σ) whereas the periodicity corresponds roughly to

the widths of lanes. In contrast to that, the pair distribution function in

y-direction g(y) yields no long-ranged oscillations because it can not dis-

tinguish between particle species and particle type specific lanes. In con-

sequence, the position of the first minimum of gs(y≥5σ) can be used to

estimate an averaged lane width.

In Fig. 4.5we show gs(y) for the intermediate density ρ∗=0.5at differ-

ent driving forces whereas all of these cases represent laned states. The

first long-ranged minima of gs(y) is indicated by a dot. The different func-

tions gs(y) are subsequently shifted by 4σin vertical direction for better

illustration. Clearly, a tendency to form thinner lanes at higher driving

forces can be observed ranging from 10σat the lowest driving force to

5σat the highest driving force (see also Fig. 4.4(b) for the first minima of

g∗s(y) at f∗

d=80.). Only, the minima for f∗

d=40 does not fit properly to

this trend which might be the case because of too short simulation times.

Interestingly, also the local structure varies with the strength of the

driving force. While at low driving forces there exist up to five peaks in

g∗s(y) this reduces to two or three peaks at the higher driving forces. This

means, that aggregates forming the ’backbone’ of the lane become smaller

(in y-direction) in the same way as lanes themselves become thinner. Also

the absolute peak heights decrease with f∗

d. In analogy to the how melt-

ing process of a solid by increasing temperature it appears here that the

internal structure of lanes ’melt’ by increasing the drive.

However, results are subject to large statistical variations and the func-

tional form of the dependency of the lane width on the driving force re-

mains unclear. Importantly, test simulations with a fully phase separated

initial state consisting of two homogenous lanes revealed that these states

are stable, at least for f∗

d>20. Therefore, the lane width depends not only

on the driving force but also on the initial conditions. As we are consid-

ering a non-equilibrium system here it is not surprising that the system

can be trapped in several meta-stable states depending on the initial con-

ditions. Also the exceptional behavior of g∗s(y) at f∗

d=40 in Fig. 4.5might

result from such meta-stable initial conditions.

Therefore it remains an open question whether there exist characteris-

tic lane widths (in the DDP-system). Previous work on lane-formation in

solely repulsive systems did also not report on characteristic lane widths.

To our best knowledge there is only Liu et. al.[93] who briefly states that at

very high densities ρ∗≈1of repulsive spheres there occurs a full separa-

tion of particle species into two lanes, which corresponds to a lane width

of half the system size. Therefore we shortly want to comment on these

4Lane Formation of Dipolar Microswimmers

Figure 4.6: Pair-correlation functions in y-direction, g(y) (black) and gs(y)

(red), for a laned state of the SS-Model at density ρ∗=0.2and

driving force f∗

d=80.

solely repulsive HS- and SS-models.

The particle type specific pair correlation function in y-direction gs(y)

for a laned state of the SS-model at ρ∗=0.2and f∗

d=80 is shown in

Fig. 4.6. In the SS-system particles are organized in lanes, but not in clus-

ters. This can be seen by the fact that there is no local structure inside

lanes but only one peak (y≈1σ) of gs(y) for distances smaller than the ac-

tual lane width (≈6σin Fig. 4.6). Furthermore, gs(y) shows a less regular

long-range order than in the DDP system (see Fig. 4.4(a)). Importantly,

this lack of clear periodicity of gs(y) contrasts the idea of a characteristic

lane width in SS-systems. At low densities (ρ∗∈[0.2,0.4]), we find that

states with a clear periodicity of gs(y) do not occur systematically. This

means that, under fixed conditions (f∗

d,T∗,ρ∗), lanes have significantly

different widths inside and between different simulation runs.

In the DDP-system, aggregates are assumed to form (due to their own

rigidity) supportive intra lane structures, similar to a backbone. In con-

trast to that, the SS-system reveals no aggregation, which might reduce its

capability to pertain certain lane widths. At least, the presence of such

structures is a major difference between lanes in the SS-model and in the

DDP-model.

4.3 Mechanisms of Lane Formation

In the following we concentrate on the question why the dipolar interac-

tions strongly enhance lane formation compared to the purely repulsive

4.3Mechanisms of Lane Formation

SS model, as revealed by the order parameter plots in Fig. 4.1. We con-

centrate on the density ρ∗=0.5, where the differences are particularly

pronounced.

4.3.1 Lane formation in the isotropic model

To understand the role of the effective attraction in our DDP system on the

lane formation we therefore consider, as a first step, the corresponding be-

havior of a LJ fluid. For simplicity, we choose the same attraction strength

ϵ∗for both, different and same species. Results for the order parameter Φ

of a driven LJ system as function of ϵ∗at fixed f∗

dare shown in Fig. 4.7(a).

For very small values of ϵ∗the order parameter is negligible, consistent

with the behavior of the pure SS system in Fig. 4.1(b). Upon increase of ϵ∗,

lanes first appear at a certain value and then disappear again at a substan-

tially larger coupling. A similar reentrance of the non-laned (mixed) state

occurs at other driving forces, as seen from the laning ”state diagram” (for

ρ∗=0.5) presented in Fig. 4.7(b). The state diagram moreover reveals that

the onset of lane formation occurs at a coupling strength of ϵ∗≈2.5(see

dotted green lines in figs. 4.7(a)-(b)) quite independent of f∗

d(taking the

value Φ=0.02 as a lower limit for laning), while the coupling related to

the disappearance of lanes appears to be a linear function of f∗

d. Given

the lower boundary ϵ∗≈2.5, it is interesting to make a connection to the

equilibrium phase diagram of the 2D LJ fluid: This system has a critical

point (gas-liquid condensation) at ρ∗

c=0.35 and T∗

c≈0.46, correspond-

ing to a critical coupling strength ϵ∗

c≈2.17 [154,155]. The triple point

(gas-liquid-solid) coupling strength is given as ϵ∗

t≈2.5(with correspond-

ing liquid density ρliquid

t≳0.6[154]). From this we can conclude that the

non-driven LJ system at ρ∗=0.5and ϵ∗=2.5is in a strongly correlated

state which is, in fact, thermodynamically unstable, i.e., it lies within the

coexistence curve of the gas-liquid transition. Moreover, revisiting again

Fig. 4.7(a)-(b) we see that ϵcprovides a lower limit of laning. These obser-

vations suggest that the equilibrium phase separation is a prerequisite for

lane formation in the driven LJ system.

We now turn to the breakdown of laning at large coupling strengths.

Here, the external force driving two unlike particles (si,sj) away from

one another competes with the attractive LJ forces.

To quantify this competition we construct an effective pair interaction

ULJ

ef f (xij ) between unlike particles, which are initially in contact and then

driven apart in x-direction. A schematic sketch of this setup is shown in

Fig. 4.8. In this configuration particles are initially in contact (|r12|=σ)

but displaced in y-direction. To account for different vertical positions yij ,

we first average the x-component of the LJ force

FLJ,x =−∂xiULJ (4.3)

4Lane Formation of Dipolar Microswimmers

246 8 10

∗

0.0

0.2

0.4

0.6

0.8

1.0

(a)

1.8 2.3 2.8

∗

0.02

0.06

0.10

12 3 4 5 678

∗

(b) 0

0.1

0.2

0.3

0.4

>0.5

Figure 4.7: (a) Laning parameter versus coupling strength for the LJ sys-

tem at f∗

d=7,ρ∗=0.5. Inset: Enlarged view for small ϵ∗. (b)

Laning ”state diagram” of the LJ system. In (a) and (b), red,

dotted green and black lines correspond to ϵ∗

c,ϵ∗

t, and ϵ∗

max,

respectively.

4.3Mechanisms of Lane Formation

Figure 4.8: Schematic sketch for a yellow particle moving to the left and

a purple particle moving to the right. In this configuration

particles are initially in contact (|r12|=σ) but displaced in y-

direction.

, over the angle φbetween rij and ey(at fixed distance) yielding

FLJ,x =π−1

∫π

sin φFLJ dφ =2FLJ /π (4.4)

with FLJ =−∂rULJ . Integration (setting now yij =0) yields the potential

ULJ (xij ) = 2/πULJ (xij ) (4.5)

. Adding the linear potential associated to the external force yields

ULJ

ef f (xij ) = ¯

ULJ (xij )−fdxij (4.6)

. For small driving forces, ULJ

ef f displays a local minimum Umin and max-

imum Umax, and thus a potential barrier ∆U=|Umax −Umin|as shown

in Fig. 4.9. Unlike particles tend to stick together (against the drive) if

the mean kinetic energy per particle 2×kBT /2is smaller than ∆U(fd, ϵ).

Solving numerically the equation ∆U=kBTfor various f∗

dwe obtain val-

ues ϵ∗

max(f∗

d) indicated by the dotted black line in Fig. 4.7(b). Clearly, our

simplified model describes the breakdown of lane formation very well.

Moreover, having introduced ULJ

ef f , we can also understand the fact that

the onset of laning occurs at coupling strength slightly larger than the crit-

ical coupling in equilibrium, ϵc. The driving force effectively reduces the

attraction between unlike particles, yielding a shift of phase separation.

A similar effect of a shifted onset of phase separation induced by activity

has recently been described by Schwarz-Linek et. al. [69] for swimming

bacteria in polymer suspensions.

4Lane Formation of Dipolar Microswimmers

Figure 4.9: Effective interaction at different driving forces f∗

dfor the LJ-

system. The LJ-interaction strength is chosen to ϵ=4kBTand

minima and maxima of the potential barrier opposing lane for-

mation are indicated by triangles and dots.

4.3.2 The laning state diagram for dipolar swimmers

Given this background, we now consider in Fig. 4.7(c) the state diagram of

the full, anisotropic DDP fluid (ρ∗=0.5) in the parameter plane spanned

by driving force and coupling strength. The latter is now given by µ∗(the

soft-sphere repulsion is fixed). Clearly, the structure of the state diagram

resembles that of the LJ system. In particular, the onset of laning depends

only weakly on f∗

d, and there is an upper boundary for µ∗beyond which

laning disappears.

This prompts the question whether the onset of laning in the driven

DDP system can be related to an equilibrium fluid-fluid phase separation,

as in the LJ system. We note that already the equilibrium DDP system

is a true binary mixture, since the full pair interactions depend on si,

see Fig. 3.2. Therefore the possible fluid-fluid transitions (if existent at

all) are, in general, combinations of condensation and demixing. In the

present study we did not carry out simulations to explore these questions

properly. Still, one can perform some estimates based on mean-field den-

sity functional theory (DFT). We focus on the occurrence of condensation

(rather than demixing), because the angle-averaged (i.e. mean-field) in-

teraction between unlike DDPs is nearly the same as that between like

particles (see Fig. 3.3). The corresponding stability condition of the ho-

mogeneous, mixed phase is that the isothermal compressibility, χT, has

4.3Mechanisms of Lane Formation

to be positive. According to Kirkwood-Bufftheory [156] one has (for a

symmetric binary mixture composed of species Aand B)

χ−1

T∝1−(ρ/2)(˜

cAA(0) + ˜

cAB(0))(4.7)

, where ˜

cAA(AB)(0) are the Fourier transforms of the direct correlation func-

tions (DCFs) cAA(AB)(r12) in the limit of long-wavelengths (k→0). In our

case, AA (AB) corresponds to s1=s2(s1=−s2). Furthermore, we approxi-

mate the DCFs according to a random phase approximation, that is,

cs1s2(r12) = cHS (r12)θ(σ−r12)−(kBT)−1UDD (r12, s1, s2)θ(r12 −σ) (4.8)

, where cHS is the Percus-Yevick DCF of a pure hard-sphere fluid [139].

The Fourier transforms of the second, mean-field like contribution to the

DCFs yield essentially the spatial integral over the effective potentials de-

fined in Eq. (3.14). Numerical investigation of the resulting expression

for χTat ρ∗=0.5reveals that, upon increasing µ∗from zero (where χT

reduces to the hard-sphere compressibility), χTbecomes indeed negative

at µ∗

s=1.31, indicating an instability (”spinodal point”) related to conden-

sation. Thus, simple mean-field DFT predicts the existence of a conden-

sation transition in the equilibrium DDP fluid. Further, for ρ∗=0.5, the

transition should occur at a coexistence value µ∗

coex ≲µ∗

s; its actual value

is, however, more elaborate to determine. BD test simulations of the non-

driven system at ρ∗=0.5and dipole moments larger than µ∗

sreveal indeed

a phase separated structure consisting of thick columns (involving both

particle species) and large voids in between. The question now is, does

µ∗

splay a decisive role for the driven DDP system? Considering Fig. 4.10

we find that µ∗

s(indicated by a white line) yields indeed a good estimate

for the onset of laning at low values of the driving force (f∗

d≲30). This

suggests that, similar to the LJ system, equilibrium phase separation is a

prerequisite of laning at the density considered. Only for larger driving

forces (f∗

d≳30) the estimate worsens (this effect was not observed in the

LJ system where, however, calculations where restricted to small f∗

d).

The breakdown of laning at large µ∗can be estimated, similar to the

LJ system, by constructing an one-dimensional (x-dependent) potential,

in which the DDP interaction between unlike particles competes with the

driving force. We consider exemplary configurations yij =y0(a simple

force average over different configurations yij is not appropriate due to

the interaction anisotropy) and obtain the potential

UDDP

ef f (xij ) = UDDP (x;y0)−fdxij (4.9)

which displays a barrier ∆U(y0, fd, µ). Figure 4.10 includes the numeri-

cal solution µ∗

max(f∗

d) (dotted red line) of the equation ∆U=kBTat y0=

0.645σ, corresponding to the most attractive, ”staggered” configuration at

contact (see Fig. 3.2(b)). Clearly, this line describes the breakdown qualita-

tively, but overestimates the influence of attraction. Quantitatively better

4Lane Formation of Dipolar Microswimmers

0.2 0.8 1.4 2.0

∗

0.2

0.4

0.6

0.8

Figure 4.10: State diagram for lane formation in the DDP system. The dot-

ted white line indicates the spinodal ’dipole strength’ µ∗

s. Red

and black lines indicate the breakdown condition for lane for-

mation with µ∗

max(f∗

d) at y0=0.645(0.9)σ, framing the true

breakdown of lane formation.

results are obtained with y0=0.9σ(dotted black line in Fig. 4.10, cor-

responding to a less attractive initial configuration. The ”true” potential

should be seen as a (weighted) average over different y0. Interestingly, the

reduction of attraction expressed in the effective potential also explains

the shift of the onset of laning with larger f∗

Finally, we briefly want to comment on how the parameter combina-

tions (f∗

d, µ∗) and the associated complex phase behavior (onset and break-

down of lane formation) discussed here relate to experimentally accessi-

ble combinations of swimming speed vand external field strength Eext.

This is important because the swimming speed vof the particles is pro-

portional to E2

ext and therefore it is also proportional to the strength of the

dipolar interactions (see Eq. [1.1] and the discussion below in Sec. 1.2.4).

Hence, the quantities f∗

dand µ∗are not independent for a given particle

type. In fact, it holds fd∝µ2. Thus, the accessible state points and the

corresponding laning behavior are in principle given by a parabola in the

state diagram shown in Fig. 4.10. Hence, it remains an open question

whether the onset and/or breakdown of lane formation can be observed

in experiment, as this parabola may remain inside or outside the region of

lane formation. The answer to this latter question will of course strongly

depend on the different parameters, e.g., permittivity and viscosity of the

fluid, or particle diameter, relating the strength of the external field with

the strength of the dipole moment and the swimming velocity.

4.4Summary

4.4 Summary

In the present section we investigated a system of colloidal microswim-

mers under the presence of an external electric field. The experimental

setup motivating this work has been introduced in Sec. 1.2.4. The complex

model used for this study is a double dipolar Janus particle with two in-

duced dipole moments of different strengths and an orientationally fixed

propulsion direction (see Sec. 3.2). In a two-dimensional setup this means

that the system is effectively binary; there exist two particle species with

opposite orientations and propulsion directions. Such configurations are

generally assumed to undergo lane formation if the propulsion force is

strong enough. We have shown that this is indeed the case for our dipo-

lar model and that the induced dipolar interactions stabilize the forma-

tion of lanes over a wide range of interaction strengths. This is surpris-

ing, as the anisotropic interactions show a strong tendency of the parti-

cles to self-assemble into chains which, in the present model, are oriented

perpendicular to the force. These self-assembly processes result in com-

plex staggered chains, when no driving force is present and the observed

structures are in qualitative agreement with experimental observations.

The underlying mechanisms for lane formation were then identified by

investigating a simple binary Lennard-Jones fluid with oppositely driven

particle species. Here, the onset of lane formation is closely related to the

phase separation of the corresponding , well known, equilibrium system.

By simple theoretical considerations we could also estimate the spinodal

point in the double dipole model and show that the same connection is

present there. Importantly, the (effective) attractiveness of the particle in-

teractions in both models enables the system to undergo lane formation

at propulsion speeds approximately one order of magnitude smaller than

in previously known systems. In addition, the presence of attractive inter-

actions can also suppress lane formation. Using force balance arguments,

the breakdown of lane formation due to ’freezing’ was estimated in very

good agreement with the simulation results.

Mutidirectional Colloidal

Assembly

In this chapter, we are particularly interested in the field directed

aggregation of complex dipolar colloids into chain and network

structures. To this end we consider particles with two different

kinds of dipolar interactions caused by crossed (orthogonal) fields.

Based on a complex double dipolar model we establish a quantita-

tive comparison between simulation and experiment and a theoretical

state diagram for different field conditions.

Furthermore, a detailed analysis of a generic model with a tunable

anisotropy of the pair interactions reveals a transition in the systems

aggregation behavior from an anisotropic DLA regime towards a slip-

pery DLA regime by decreasing the anisotropy. The latter transition is

accompanied by a strange compactifaction phenomena, which we also

find for the more complex model. Hence, it is the non-equilibrium

assembly of transient aggregates which is investigated here.

Outline

A previously studied example for network forming systems is the metallo-

dielectric Janus particles under the presence of an uni-axial electric field

inducing dipolar interactions [26,95]. These structures are shown and

discussed in Fig. 1.3(c) and Sec. 1.2.4. In Chapter 4we investigated the

collective behavior of these particles in the swimmer regime.

In this Chapter we are also interested in field directed network forma-

tion but investigate a different setup of superparamagnetic particles in

crossed magnetic and electric fields (see Sec. 1.3.3). The results presented

in the first Section (Sec. 5.1) of this Chapter are part of a dominantly ex-

perimental study and are based on the model presented in Sec. 3.4. Hence,

this section has a strong emphasis on the comparison between simulated

structures and experimental ones. To this end, a simple method which

allows to determine the ’role’ a particle plays in the observed networks

(see Sec. 3.5) is applied. This method can be used for experimental as well

as simulation data and is based on the fundamental symmetries of the

point-dipole potential. The overall aim is to provide a theoretical non-

equilibrium state diagram for different field configurations at constant

density, because this is not achievable from the experimental side.

The second section (Sec. 5.2) of this Chapter is a purely theoretical study

of the generic model introduced in Sec. 3.6. In principle, this section al-

lows a connection to network formation of the metallo dielectric Janus

particles, but here we focus on conceptional aspects of the aggregation

process for the crossed dipole setup. We identify different diffusion lim-

ited aggregation regimes and connect them to the anisotropy of the inter-

actions. Finally we discuss connections in the aggregation and compacti-

faction behavior between the generic and the complex model (see Sec. 5.3).

5Mutidirectional Colloidal Assembly

5.1 Field Controlled Assembly of Colloidal

Networks - Experiment and Simulation

Using Brownian dynamics simulation we investigate the aggregation

behavior of superparamagnetic polymer particle under the presence

of crossed magnetic and electric fields. The concurrent application

of electric and magnetic fields is an unusual method to organize col-

loids into bidirectional chains and clusters. We find that the mor-

phology of the assembled structures in bi-axial fields is highly depend

on the number density of particles and their initial spatial distribu-

tion. However, in experiments, the number density of the assembling

particles is subject to stochastic variations between different samples.

In the present study, computer simulations allow us to overcome this

difficulty and investigate the state of assembly at a given time with a

precise control over the particle density. Hence, we are able to con-

struct a theoretical non-equilibrium state diagram of the assembled

morphologies under various field configurations.

5.1.1 Background

The experimental data and photographs presented in this section have

been produced by Dr. Bhuvnesh Bharti in the group of Prof. Dr. Or-

lin D. Velev at the North Carolina state university. Theoretical calcula-

tions and evaluation of simulated and experimental data were done by

my self. This Section is based on the double dipolar model for superpara-

magnetic particles in crossed external fields (CDP model) introduced in

Sec. 3.4. Externally induced dipolar interactions are model by two decou-

pled point dipoles, whereas the magnetic dipole moment is shifted out of

the particle center in a random direction. The overall aim is to establish

a close connection between the experimental observations and simulation

results and to provide a theoretical non-equilibrium state diagram for dif-

ferent field configurations. The latter is done by means of extensive Brow-

nian dynamics simulations at constant temperature T∗=0.01 and density

ρ∗=0.35 but at various dipole strengths corresponding to dipolar cou-

pling strengths λ∈[0,50]. The rest of this section is organized as follows:

First, experimental observations are described. Then, a detailed compari-

son of quantitatively comparable structures in simulation and experiment

is performed. Finally, the full field controlled state diagram is presented,

and analyzed. The section ends with concluding remarks.

Tabular representation of reduced units.

reduced dipole moment µ∗≡µ(2)

i∗=µ(2)

i/√σ3ϵ

reduced density ρ∗=ρσ 2

reduced temperature T∗=kBT /ϵ =0.01

5.1Field Controlled Assembly of Colloidal Networks - Experiment and

Simulation

5.1.2 Structure formation in experiment

In experiments we find that at very low electric and/or magnetic field

strengths, the thermal energy in the system overcomes the dipolar inter-

action energy and restricts the formation of colloidal assemblies. Upon in-

creasing the strength of one of the fields, the formation of predominantly

linear chains aligned in the direction of the respective field takes place.

These states can be regarded as linear string fluids, where the assembly is

solely dominated by the single type dipole-dipole interaction (either elec-

tric or magnetic). At intermediate field strength of both fields, a bidirec-

tional chain formation is observed, which results into the formation of fine

networks or cross-linked particle chains. Upon increasing the absolute

magnitude of both field strengths further, these fine networks transform

into coarser networks with partially crystallized colloidal domains. This

formation of particle networks in bi-axial electric and magnetic fields is

a non-equilibrium process and the initially formed metastable structures

dynamically reconfigure with prolonged exposure to the fields. Thereby

they undergo an aging process characterized by ongoing compactifaction.

The analysis of experiments is accompanied by complications. First, ex-

traction of particle positions from microscopic images was automatized.

To this end we used the functionalities of Origin (TM), which requires im-

ages of high contrast. Large-scale images as shown in Fig. 1.6(e) turned

out to be of insufficient quality for this procedure. Our analysis of the

structure formation processes is based on single particle positions and

therefore only small scale images will be considered as they yield higher

contrasts. Secondly, controlling the area fraction or number density in ex-

periment was difficult. Even for a given sample of constant global area

fraction (usually φ≈0.23), we found that the particle density was signif-

icantly higher or lower in its different regions. Besides the fact that the

system is subject to density variations between different samples, this also

means that it is characterized by quite large internal density variations.

The latter is a consequence of the size of the considered particles. They

are relatively large (≈6µm in diameter) and at the given room temper-

ature, they are not homogeneously distributed throughout the samples.

Finally, this means that the observed experimental structure formation

depends on the initial local density of the investigated region. As we are

interested in the formation and structure of particle aggregates, we focus

on regions where particles assemble. Hence, the local density presented

in the microscopic photographs is usually higher than the mean density

in the sample.

5.1.3 Comparing simulation and experiment

In simulations we therefore restrict our considerations to an intermediate

density ρ∗=0.35, which is slightly higher than the mean number den-

sity in experiment (ρ∗

exp ≡4/πφ ≈0.3). Occasionally we consider systems

5Mutidirectional Colloidal Assembly

of number density ρ∗=0.3when comparing with microscopic images of

obviously low particle density.

Importantly, an one-to-one comparison between simulation and exper-

iment could not be established. Although a ’master curve’ mapping the

experimental parameters (e.g., current in the coils) onto dipole strengths

in simulations would be of great help in understanding the systems dy-

namics, the uncertainties associated with the internal structure (distri-

bution of dipole moments) of the particles and the imperfect control of

density in the experimental setup made this unfeasible. Instead, we show

in Fig. 5.1the ’extreme cases’ of states corresponding to vertical chaining

(a), horizontal chaining (b) and coarse network formation (c) in order to

investigate the validity of our simulations. These results are compared be-

tween simulation and experiment by microscopic images, snapshots from

simulation and the previously (see Sec. 3.5) introduced distributions of

fictitious energies uel and umg . The results from simulation are tempo-

ral averages in the range of 8τBto 12τB, while experiments are evaluated

between 8sand 15safter applying the fields.

In the presence of solely the electric field the dominant structural el-

ements are fine linear particle chains oriented along the electric field as

can be seen in Fig. 5.1(a). The distributions of uel and umg show two dis-

tinct peaks, corresponding to particles within vertical chains (uel =−4and

umg =2) and corresponding to particles forming ends of vertical chains

(uel =−2and umg =1). Here, the agreement between simulation and ex-

periment is very good, although the theoretical peak at uel =−4is higher

and hence indicates a tendency to overestimate the amount of particles

within chains at the given simulation parameters. In principle, a system-

atic search for values of the dipole strength µel and density ρ∗would allow

to optimize the mapping between experiment and simulation. However,

as stated previously, this procedure would have to be performed sepa-

rately for each experimental image (due to the lack of a ’master curve’)

and would yield no further insights.

In contrast to the electric case we show in Fig. 5.1(b) particle chains

formed under the presence of solely the magnetic field. These aggregates

are in general more compact and partially staggered, which we assume

to be the consequence of dislocations of dipole moments from the parti-

cles centers of mass. Here, the distributions of uel and umg yield only one

distinct peak at uel =1and umg =−2, respectively. These peaks are as-

sociated with particles forming the end of chains and appear pronounced

and sharp. In contrast to the electric case, simulations underestimate the

frequency of ’chain ends’. However, the distributions are in general signif-

icantly broader than in the electric case and yield significant contributions

for perturbed configurations, which is the frequency of states around the

major peaks. Interestingly, there is an surprisingly accurate agreement be-

tween simulation and experiment with respect to these perturbed particle

configurations. This indicates that the deviations from linear chaining are

5Mutidirectional Colloidal Assembly

captured properly by our model. Furthermore we want to note that the

model parameter (δ=0.2σ) defining the dislocation of the magnetic inter-

action center and the simulation parameters ρ∗and ¯

µmg have not been op-

timized systematically. This is important, because it means that the novel

feature of our model, namely the unusual treatment of magnetic interac-

tions, describes the general properties of the real particles, irrespective of

the actual values. Besides being a justification for the idea of analyzing the

real systems behavior via the theoretical model, these results also support

our assumptions about the nature of the real particles (see Sec. 3.4).

The third case of strong electric and magnetic field presented in Fig. 5.1

(c), shows the formation of coarse particle networks. Here the distribu-

tions of the fictitious energies uel and umg become very broad and flat and

yield only minor peaks at uel =−2and umg =1indicating chaining along

the electric field. In addition, an even less pronounced peak at uel =1

and an associated region of relatively high frequency at umg =−2indicate

perturbed chaining along the magnetic field. Note that the peak heights

are two orders of magnitude smaller here than in the previous examples.

Again, a quite accurate agreement between simulation and experiment is

given by these distributions, although the theoretical curve overestimates

magnetic energies umg in [−1.5,0.5]. This means that the real system has

less staggered or perturbed magnetic (horizontal) aggregates than struc-

tures from simulation.

Finally, Fig. 3.6gives the same information for network formation at

smaller strengths of both fields. Here, the microscopic image of the col-

loidal network formed in experiment (see Fig. 3.6) is re-color coded on the

basis of particle roles and is compared to the particle assembly formed in

the simulations. The distributions of the fictitious energies uel and umg ap-

pear similar to the ones of the coarser network in Fig. 5.1(c), although the

minor peak at umg =−2, which is associated with chaining along the mag-

netic field, is slightly more pronounced and higher in its absolute value.

Hence, this network structure is less compact than the one at higher field

strengths, although the difference may appear small. In the next section,

the difference between these two network types will be discussed in more

detail.

In general we can state that the novel way of modeling magnetic in-

teractions presented here results in simulated colloidal structures which

are visually indistinguishable and quantitatively comparable to the exper-

imental observations. Furthermore, the characteristics of all qualitatively

different states in experiment are also found in simulations. In the next

section we will present a state diagram generated from simulations which

yields the dependency of the assembly process on the field configuration

at constant density and temperature.

5.1Field Controlled Assembly of Colloidal Networks - Experiment and

Simulation

(I) h>A−n−vhorizontal chain regime

(II) v > A −n−hvertical chain regime

(III) h+v > A −nfine network regime if neither (1) nor (2)

(IV) n>A−h−vcompact network regime

Table 5.1: Definitions of the criteria for states presented in Fig. 5.2. The

number of aggregated particles (having at least neighbor) is de-

noted by A.

5.1.4 State diagram

The limitations in control over local and global density make it impossi-

ble to extract the influence of the field strengths on the self-assembly of

the colloidal aggregates from experimental side. The surprisingly realis-

tic behavior of our model allows to overcome these difficulties by means

of BD-simulations. According to the identification scheme introduced in

Sec. 3.5each particle can be associated with a ’role’ and sorted into the

populations of node particles n, vertical chain particles vor horizontal

chain particles h, which are properly defined in the text above Eq. 3.27.

These populations sum up to the total amount of aggregated particles A

A=h+v+n. (5.1)

Based on this equation, four categories for different types of networks and

chain-like structures can be defined. These states correspond to (I) vertical

(electric) string fluid , (II) horizontal (magnetic) string fluid, (III) networks

of fine single particle chains and (IV) networks of more compact clusters

interconnected by chains. The idea is that if either hor vdominate the sys-

tem we consider it to be string fluid. This is the case when the population

hor the population vcorrespond to more than 50% of all aggregated par-

ticles. The actual threshold of 50% is arbitrary, but chosen for simplicity

here. Accordingly, Eq. [5.1] can be reformulated and yields the definitions

h>A−v−nand v > A −h−nfor the magnetic and electric string fluid

respectively. A compact network or isolated cluster phase corresponds to

a dominant population of node particles (n > A −v−h). The finer network

structures require the string fluid structures to dominate (h+v > A −n).

But they are also combinations of different string fluids and hence the

cases when either horizontal h>A−v−nor vertical chaining v > A −h−n

dominates have to be excluded. The precise definitions of these states are

summarized in Table. 5.1

According to these criteria, Fig. 5.2shows four sketches of particle ag-

gregates whereas particles belonging to different populations (h, v, n) are

occasionally indicated. In addition, the non-equilibrium state diagram

5Mutidirectional Colloidal Assembly

for chain and network aggregation at ρ∗=0.35 is shown for various elec-

tric and magnetic field strengths. Only states with more than 70% of the

particles being aggregated (A > 0.7N) are presented. Different colors cor-

respond to the four different states. The presence of these four types of

Figure 5.2: Non-equilibrium state diagram for the aggregation of super-

paramagnetic particles in crossed fields at density ρ∗=0.35.

Only states in which more than 70% of particles are members

of dimers or larger aggregates are considered.

particle states is governed by the relative and absolute electric and mag-

netic field strengths. At small dipole moments µα≤0.25 (or very low field

5.1Field Controlled Assembly of Colloidal Networks - Experiment and

Simulation

strengths), no significant assembly was observed. In the borderline case,

where one of the fields is much stronger than the other one, the assembled

configuration is a string fluid phase (States I or II) oriented in direction of

the dominating field. However, as discussed previously the local particle

assembly in these states is not the same in external electric and magnetic

fields. In the former, aggregates are rather linear and in the latter partially

staggered.

A more versatile behavior is observed when the two moments are simi-

lar in magnitude (µel ≈µmag ). When the absolute values of both moments

are comparable but obey µα<0.4, a fine bidirectional low-density net-

work or gel-like structure is obtained (state III). Interestingly, the region

of fine network states in the state diagram is not symmetric. If the electric

moment remains sufficiently small, fine networks can be realized even at

relatively strong magnetic interactions. We assume this is the case because

the magnetic interaction prefer staggered arrangements and prevent par-

ticles from strong compactifaction.

However, if both moments are increased in their absolute magnitudes

µα>0.4, a more coarse structure with compact particle packing is ob-

served (state IV). In this state, small domains of colloidal clusters are in-

terconnected via (short) particle chains.

Here it should be mentioned that the distinction between the fine and

compact network structures is given by a gradual change in the popula-

tions of particles with different roles. The actual position of this transition

depends on the arbitrary definitions of populations given in Table 5.1.

Nevertheless, our characterization is sufficient to show the existence of

different states. Moreover, the state boundaries shown in Fig. 5.2are not

very distinct and interchange between the states is gradual. Hence, detec-

tion of this smooth crossover in experiment is not possible, especially as it

is very sensitive to (local) density variations.

The network states might be regarded as gel-like structures. However,

the percolation transition, which is in principle a prerequisite for gela-

tion [101], was not systematically evaluated. Nevertheless, many of the

simulated networks turned out to be percolated. Also, the relaxational

dynamics, which are needed to identify gelation [101], have not been in-

vestigated; still, the presented network structures are transient in charac-

ter and undergo subsequent compactifaction in time. In Fig. 5.3the tem-

poral evolution/collapse of the coarse network from Fig. 5.1(c) is shown.

The number of particles in nodes n(black curve) continuously increases

with time. At these strong interaction strengths, nearly all particles are

aggregated A≈1(yellow curve). Interestingly we see that the number of

particles in vertical chains dominates the one in horizontal chains. In fact,

this is the case in most of the network structures observed, also in the fine

networks from experiment and simulation (see Fig. 3.6(c) and (e)). The

population of particles in horizontal chains becomes dominant, only for

significantly stronger magnetic fields. Importantly, the categorization of

5Mutidirectional Colloidal Assembly

Figure 5.3: Collapse of a network structures in simulation at ρ∗=0.35 and

µel =0.45, µmg =0.45. Shown is the time evolution of the num-

ber of particles in nodes n(black), the number of particles in

horizontal/vertical chains h(red/blue) and the number of par-

ticles in aggregates A(yellow). This state point corresponds to

a coarse network.

certain state points is therefore time dependent.

Such aging processes are known in systems undergoing diffusion lim-

ited aggregation, which makes an interpretation as a (chemical) gel rea-

sonable. A final conclusion on whether this system undergoes gelation

can not be given here. It is mostly computational reasons why the ques-

tion on gelation can not be addressed. Calculating percolation transitions

as well as temporal correlation functions requires large particle numbers

and very long simulation times, which are not accessible with this model.

Importantly, the compactifaction of network structures with increasing

field strength is counter intuitive from a conceptional point view. In gen-

eral, increasing the interaction strength is equal to decreasing the temper-

ature. This relation is usually expressed in the definition of the dipolar

coupling strength

λ=µ2/kBT . (5.2)

In this study, the compactifaction is observed at λ≈20-50. At such strong

couplings, the system should turn towards some sort of diffusion limited

aggregation and form less compact fractal structures. And indeed, visual

inspection of the large scale assembly in experiment ( Fig. 1.6(e)) makes it

reasonable to interpret the structures as fractals. However, locally we find

compactifaction by increasing the coupling strength.

In Sec 5.2.4-5.2.5and in Sec. 5.3the relaxational behavior and com-

pactifaction processes will be discussed with respect to results of a com-

5.1Field Controlled Assembly of Colloidal Networks - Experiment and

Simulation

putationally more convenient generic model allowing deeper insights into

the underlying mechanisms of these assembly processes.

5Mutidirectional Colloidal Assembly

5.2 Diffusion Limited Aggregation in

Mutidirectional Fields

Based on Brownian Dynamics computer simulations in two dimen-

sions we investigate aggregation scenarios of colloidal particles

with directional interactions induced by multiple external fields.

To this end we propose a model which allows continuous change in

the particle interactions from point-dipole-like to patchy-like (with

four patches). We show that, as a result of this change, the non-

equilibrium aggregation occurring at low densities and temperatures

transforms from conventional diffusion-limited cluster aggregation

(DLCA) to slippery DLCA involving rotating bonds; this is accompa-

nied by a pronounced change of the underlying lattice structure of

the aggregates from square-like to hexagonal ordering. Increasing

the temperature we find a transformation to a fluid phase, consistent

with results of a simple mean-field density functional theory.

5.2.1 Background

In this Section we present results previously published in [111]. Here,

we turn towards general properties of the structure formation in colloidal

systems with externally induced mutidirectional interaction anisotropies.

This is done in a conceptional fashion by means of two-dimensional Brow-

nian dynamics (BD) simulations of a generic particle model, the extended

dipole particle model (EDP model), which is inspired by the general setup

of the previously presented system and presented in Sec. 3.6. However,

the model does not aim on a specific system e.g., magnetic and electric

interactions or two temporarily present electric interactions as described

in Sec. 3.3, but rather on unveiling the underlying mechanisms of directed

aggregation in colloidal systems. Furthermore, the model is designed to

be rather simple. The aim is to be able to perform large-scale computer

simulations with reasonable effort while still capturing certain character-

istic features of the real system. The EDP model does not aim on being a

replacement for the crossed dipole particle (CDP) model discussed in the

previous section. Rather it is an additional approach

• to investigate large-scale structures (fractal dimension of aggre-

gates),

• to investigate long-time behavior (stability of bonds over long times)

• to analyze the influence of the anisotropy of pair interactions.

• and to identify conceptional aggregation regimes (slippery and

anisotropic DLCA)

5.2Diffusion Limited Aggregation in Mutidirectional Fields

Indeed, of special interest to us is the interplay between diffusion

limited aggregation and interaction anisotropies.

To this end, externally-induced dipole moments are mimicked via

pairs of screened Coulomb potentials. The two charges associated

with each dipole are shifted outward from the particle center, one

parallel to its corresponding field and the other one anti-parallel.

A sketch of the internal arrangement of interaction sites in such a

particle is shown in Fig. 3.7for the case of two orthogonal (decou-

pled) fields. In general, this model allows to investigate also the

influence of a third or fourth field. For simplicity, we restrict our

considerations here to the case of two decoupled fields. By chang-

ing the charge separation, we systematically investigate the (tran-

sient) structural ordering and aggregation behavior predicted by this

model. Our large-scale Brownian dynamics simulations show that

the system is very sensitive to changes in temperature T∗, number

density ρ∗, and charge separation δ. In this large parameter space

we find a variety of different states ranging from small fractal ag-

gregates and single-chain structures at low temperatures to coarser,

isolated or interconnected clusters at higher temperatures.

Highlights of our results are the following: At very high interac-

tion energies and large charge separations we find that the parti-

cles undergo anisotropic diffusion limited cluster aggregation with

rectangular local particle arrangements. Lowering the charge sepa-

ration shifts the model behavior to a slippery diffusion limited ag-

gregation (sDLCA) regime accompanied by a sharp transition of the

lattice structure from rectangular to hexagonal. In the proximity

of this transition we observe long-lived or arrested frustrated struc-

tures consisting of strongly interconnected hexagonal and rectangu-

lar lattice domains connected with each other. We also show that,

upon increase of the temperature, the systems enter a fluid state.

The corresponding ’fluidization’ temperature turns out to be very

close to the spinodal temperatures obtained from a mean-field den-

sity functional theory.

The rest of this section is organized as follows. First we present our

model and explain different target quantities which will be calcu-

lated from the simulations. Secondly, the numerical results are de-

scribed. In the sections 5.2.2-5.2.4we first discuss the local and

global structures and the temporal stability of aggregates at a low

temperature and an intermediate density, focusing on the impact of

the model parameter δ. In section 5.2.5and 5.2.6we then turn to the

impact of temperature and density. Finally, the conclusions summa-

rize the results.

5Mutidirectional Colloidal Assembly

Target quantities To characterize the structure of the systems

we consider several quantities. The first one is the mean coordina-

tion number

z=1

∑

i=1

zi,(5.3)

where ziis the number of neighbors of particle iand the sum is over

all particles. In the following, two particles are considered to be

nearest neighbors if their center-to-center distance is smaller than

rb=1.15σ.

To identify local particle arrangements, the orientational bond order

parameter is of special importance. For particle kit is given by

φn

k=1

∑

l=1|exp(inθkl

λ)|(5.4)

with zkbeing the number of neighbors and θkl

λ= arccos(rkl ·rkλ/(rkl rkλ))

being the angle between the bond of particle kand its neighbor l

measured against a randomly chosen bond of particle kto one of its

neighboring particles λ. Hence, φn

k=0for zk<2. The integer value

ndetermines the type of order which is detected by this parameter.

We concentrate on φ4and φ6to identify square (rectangular) and

hexagonal lattice types. Its ensemble average is calculated via

Φn=1

∑

i=1

φn

i.(5.5)

The reversibility of ’bond’ formation and slipperiness of existing

bonds can be characterized by the bond and the bond-angle auto-

correlation functions cb(t) and ca(t). To evaluate cb(t) we assign a

variable bij (t) to each pair of particles at each time step which is 1

if the particles iand jare nearest neighbors or zero otherwise. The

bond auto-correlation function is then defined as

cb(t, t0) = ⟨bij (t0)bij (t)⟩,(5.6)

where the brackets indicate an average over all pairs that are bonded

at time t0. The bond-angle auto-correlation function ca(t, t0) is de-

fined similarly by defining the unit vector

aij (t) = rij (t)/rij (t),(5.7)

such that

ca(t, t0) = ⟨1−arccos (aij (t)·aij (t0))/π⟩(5.8)

where we average again over all pairs. While cbgives the information

on how stable bonds are over time, catells how stable their direction

5.2Diffusion Limited Aggregation in Mutidirectional Fields

is over time. Note that in contrast to the typical definition of corre-

lation functions for stationary systems [134,136], here the functions

cb(t) and ca(t) are not independent of the time origin t0, although

this dependency will not be denoted in the following.

Finally, we consider the fractal dimension Dfof particle clusters,

which is particularly important in the context of DLCA. Clusters are

defined as a set of particles with common next neighbors. The size

of a cluster is then quantified by its radius of gyration

g=1

Ncl

∑

i=1

(ri−¯

r)2,(5.9)

where Ncl is the number of particles in the cluster, and ¯

ris the po-

tion of its center-of-mass. By plotting ln Rgagainst ln Ncl for differ-

ent clusters, we extract the fractal dimension Dfvia the relationship

Rg∼N1/Df

cl (see Ref. [124,98]).

Tabular representation of reduced units.

reduced density ρ∗=ρσ 2

reduced temperature T∗=kBT /ϵ

reduced interaction strength u∗=u/ϵ =−2.804

Brownian time scale τb=σ2γ/kBT

5.2.2 The interaction anisotropy - Effect on local

order

At first we study the system at low temperature T∗=0.05 and

intermediate density ρ∗=0.3for different charge separations δ.

In Fig. 5.4simulation snapshots for δ=0.1σ , 0.21σ , 0.3σat t=

300τb[see. Eq. (3.35) below] are shown, where τbis the Brownian

timescale. The color-code reflects the orientational bond order pa-

rameter φ4

iof each particle i. All three cases are characterized by

clusters with irregular shapes. However, local particle arrangements

differ strongly. While for δ=0.1σthe particles aggregate in a hexag-

onal fashion, at δ=0.3σthey aggregate into rectangular structures.

At the intermediate charge separation δ=0.21σ, hexagonal order

dominates the system; however, some clusters also reveal subsets

of particles in rectangular arrangements. A more quantitative de-

scription is given by the orientational bond order parameters Φ4(6)

shown in Fig. 5.5(a) as functions of δ. By increasing δ, one observes

a sharp transition at δ≈0.21σfrom hexagonal towards rectangular

(square) order.

5Mutidirectional Colloidal Assembly

Figure 5.4: Simulation snapshots at ρ∗=0.3and T∗=0.05 for (a) δ=0.1σ,

(b) δ=0.21σand (c) δ=0.3σ. Particles are colored according

to their value of φ4

5.2Diffusion Limited Aggregation in Mutidirectional Fields

Figure 5.5: Results for simulations with N=1800 at temperature T∗=

0.05 and density ρ∗=0.3. (a) Orientational bond order param-

eters Φ4for square (black) and Φ6(yellow) for hexagonal parti-

cle arrangements. (b) Mean coordination number ¯

zas function

of charge separation δat times t=100,200,300τb.

5Mutidirectional Colloidal Assembly

Figure 5.6: Minimum energy of a particle with six neighbors in hexagonal

arrangement as function of δ(black) and energy for a particle

in rectangular arrangement with 4neighbors (red).

This transition turned out to be independent of the considered par-

ticle numbers as test simulations revealed. Physically, it can be inter-

preted as a reduction in valency of a ’patchy’ particle from six-fold

(isotropic interaction) to four-fold.

The very presence of such a sharp transition can be explained via

energy arguments based on the δ-dependent pair potential plotted

in Figs. 3.8(a)-(d). To this end, we calculate the energy Uhex

i(δ) =

∑6

j=1U(rij ) of a particle iwith six neighbors j, which are located in

a hexagonal arrangement at ’contact’ distance σaround i. Note that

not all hexagonal configurations do have the same contact energy.

This is due to the anisotropy of interactions, see Fig. 3.8(d). There-

fore we consider a hexagonal configuration in which the contact en-

ergy is as low as possible (this configuration was found numerically).

The dependence of this lowest contact energy Uhex

i(δ) on the charge

separation parameter is plotted in Fig. 5.6. Also shown is the corre-

sponding energy Usq

i(δ) =

∑4

j=1U(rij ) = 4×uof a particle with four

neighbors jlocated at distance σin a rectangular arrangement, i.e.,

in the energy minima around i(the quantity uwas defined below

Eq. (3.31)). Note that the energy Usq

i(δ) does not depend on δac-

cording to Eq. (3.32). As shown in Fig. 5.6, the two curves intersect

at a ”critical” value of δ=0.24σ. Thus, the simple energy arguments

already suggest a transition between states with local hexagonal and

square order, even though the predicted critical value is somewhat

larger than the value of δ=0.21σseen in the actual simulations at

finite temperature and density [see Fig. 5.5(a)].

Further information is gained from the behavior of the mean coor-

5.2Diffusion Limited Aggregation in Mutidirectional Fields

dination number as a function of δplotted in Fig. 5.5(b) for three

different times t=100τb,200τband 300τb. At all times considered,

zundergoes a steep decrease at δ≈0.21σfrom a nearly constant

value, ¯

zhex ≈4.5, to a value ¯

zsq ≈3.5. This behavior reflects, on

the one hand, again the presence of a sharp transition; on the other

hand, the actual values of ¯

zhex (¯

zsq ) reveal the ”non-ideal” character

of the aggregates in terms of coordination numbers. For example,

for δ > 0.21σwe find that ¯

zand Φ4decrease with δ, while Φ6in-

creases. However, this does not indicate a decline of the rectangular

order; it rather results from an increasing amount of particles resid-

ing in chains oriented either in x- or y-direction. The coordination

number ziof a particle iin such a chain is ≤2, leading to a mean

coordination number ¯

z < 4. Furthermore, the parameters φ4

iand φ6

[see Eq. (5.4)] become unity for a particle forming exactly two bonds

under an angle of π(straight chain). This does not affect Φ4, which

is already large at δ > 0.21σ, but significantly increases Φ6. Finally,

the counter-intuitive decrease of Φ4with δresults from the increas-

ing amount of particles with only one neighbor (e.g., ends of chains

appearing white in Fig. 5.4(c)). These particles yield no contribution

to Φ4[see Eq. (5.4)].

The ”non-ideal” values of ¯

zhex and ¯

zsq also explain why our energy

argument for the location of the hexagonal-to-square transition,

which was based on ideal arrangements with six and four neigh-

bors, respectively, does yield the transition value δ=0.24σrather

than δ=0.21σobtained from simulation. We can now reformulate

the argument by using the actual mean coordination numbers ex-

tracted from our simulations, ¯

zsq =3.5(instead of 4) and ¯

zhex =4.5

(instead of 6). Following the calculations for the ideal arrange-

ments described before, the energy of the square-like arrangement

is Usq =3.5×u. For the hexagonal arrangement, we use the average

minimum energy with either zi=4or zi=5neighbors, yielding

Uh(¯

zhex , δ)=(Uhex (4, δ) + Uhex(5, δ))/2. The resulting critical value of

the charge separation is δ≈0.21σ, which coincides nicely with the

transition value observed in our simulations.

5.2.3 Transient character of aggregates

Although the local structures characterized by ¯

zand Φ4(6)persist, in

general, over the simulation times considered, we are still facing a

transient (out-of-equilibrium) structure formation as seen, e.g., from

the slight increase of ¯

zwith time in Fig. 5.5(b). This raises a question

about the typical ”lifetime” of the aggregates.

To this end we now consider dynamical properties, namely the bond

and bond-angle auto-correlation functions, cb(t) and ca(t). It is not

5Mutidirectional Colloidal Assembly

Figure 5.7: Time correlation functions obtained from simulations with

N=1800 at temperature T∗=0.05 and density ρ∗=0.3.

(a) [(b)] Time evolution of the bond [angle] auto-correlation

function cb(t) [ca(t)] for three different charge separations

δ=0.1σ , 0.21σand 0.3σcolored in yellow, purple and black

respectively.

reasonable to extract decay rates from these functions (as it is usually

done) because in transient states, decay rates are, strictly speaking,

functions of time themselves. Still, it is interesting to see whether

the temporal correlation of bonds (bond angles) for different δal-

lows us to distinguish between qualitatively different aggregation

regimes.

Numerical results for cb(t) and ca(t) are plotted in Figs. 5.7(a) and (b),

respectively, where we consider a large time range up to t≈103τb.

The time axis starts at the finite time when all the systems have

formed stable aggregates. The data in Figs. 5.7(a) and (b) pertain

to three representative values of the charge separation parameter re-

lated to the hexagonal structures (δ=0.1σ), rectangular structures

(δ=0.3σ), and to the transition region (δ=0.21σ). In the square

regime (δ=0.3σ) the decay of both cb(t) and ca(t) is almost identical

5.2Diffusion Limited Aggregation in Mutidirectional Fields

and very slow. From this we conclude that the square regime is char-

acterized by almost unbreakable bonds with fixed orientations. This

is different in the hexagonal regime (δ=0.1σ) where cb(t) remains

nearly constant even after long times (meaning that bond-breaking

is very unlikely), while ca(t) decays much faster. Thus, the direc-

tions of bonds are less restricted. We interpret this behavior as evi-

dence that two particles, though being bonded, are still able to rotate

around each other to some extent. This is a characteristic feature of

slippery bonds. Finally, in the transition regime (δ=0.21σ) both

functions cb(t) and ca(t) decay significantly faster than in the other

cases, with the decay of the bond-angle correlation function being

even more pronounced. In that sense we may consider the bonds in

the transition region also as slippery (although less long-lived than

in the other cases).

We conclude that the different structural regimes identified in the

preceding section are indeed characterized by different relaxational

dynamics. Moreover, all of the observed aggregates have lifetimes of

at least several hundred τb. Such long-lived bonds are indicative of

diffusion limited cluster-cluster aggregation. In the next section we

therefore consider the fractal dimension.

5.2.4 The interaction anisotropy - Effect on

large-scale fractal structure

In Fig. 5.8the fractal dimension Dfis shown as a function of δat

time t=250τb, density ρ∗=0.3and temperature T∗=0.05. We find

that Dfincreases slightly with δbut remains in a range between 1.4

and 1.5, except at δ=0.21σ. There, the fractal dimension exhibits a

bimodal distribution, taking values between Df≈1.48 and Df≈1.6

(dashed line in Fig. 5.8).

Despite these variations and taking into account the error range, the

values of Dffound here are significantly smaller than the fractal

dimension Df=1.71 observed in earlier studies of DLCA in two-

dimensional continuous (off-lattice) systems [124,119]. Except for

the case δ=0.21σ, the values in Fig. 5.8are comparable with previ-

ous findings for DLCA in two-dimensional lattice systems and sys-

tems with spatial or interaction anisotropies [130,131,132]. The

present system is indeed anisotropic in the sense that the external

fields impose preferences on the directions of particle bonds and

therefore also on the orientations of aggregates. This effect is most

pronounced in the rectangular regime (δ=0.3σ). Therefore, it is

plausible that our system undergoes a special case of anisotropic

DLCA, in (quantitative) accordance with experimental results [98]

and theoretical predictions [120,130,131,127]. We should note

5Mutidirectional Colloidal Assembly

Figure 5.8: Fractal dimension Dfas a function of charge separation at ρ∗=

0.3and T∗=0.05. At δ=0.21σwe find a bimodal distribution

of fractal dimension with peaks at Df=1.48 (solid line) and

1.6(dashed line).

that, due to our simulation method, the cluster sizes (typically in-

volving 101−103particles) are relatively small compared to the par-

ticle numbers considered in the literature (106particles) [119,157,

120] and therefore most probably subject to finite size effects. A

more accurate study of (the impact of anisotropic interactions on)

the fractal dimension is beyond the scope of this study. Still, our

results do indicate a non-typical diffusion limited aggregation be-

havior.

We also relate our findings to the newer concept of slippery DLCA [123,

116], where the bonds are essentially unbreakable but able to rotate.

Indeed, as discussed in section 5.2.3, bonds are slippery in nature

for small δin the hexagonal regime. For three-dimensional systems

it has been reported [123,116] that the fractal dimension Dfre-

mains the same for slippery and classical DLCA, while the mean

coordination number ¯

zdiffers. Specifically, ¯

zis significantly higher

for sDLCA [123,116]. The same observation emerges when we con-

sider our values of ¯

zplotted in Fig. 5.5(b), from which one sees a

pronounced decrease of ¯

zupon entering the square (DLCA) regime.

However, in contrast to earlier studies we find Dfto slightly in-

crease with δ, especially in the hexagonal regime. We interpret this

behavior as a consequence of the fact that binding energies in the

hexagonal regime decrease with increasing values of δ, while they

remain constant in the square regime (see Fig. 5.6). The correspond-

ing stability of bonds should be correlated to the binding energies

which explains the slightly increasing values of Dfin the hexagonal

5.2Diffusion Limited Aggregation in Mutidirectional Fields

regime. Note that the increase of Dfwith δturns out to be larger

(but still comparable) than the error range in Fig. 5.8. Hence, the

interpretation given above remains somewhat speculative.

Finally, in the transition region (δ=0.21σ) we found a bimodal dis-

tribution of the fractal dimensions Dfwith maxima at Df≈1.48 and

Df≈1.6. This second maximum corresponds to only ≈25% of the

considered cases (twelve independent simulation runs). The first

maximum at Df≈1.48 therefore clearly dominates and fits nicely

to the functional dependence of Dfon δ(see Fig. 5.8). We assume

that the less frequent peak results from a ’switching’ of the local

structures between hexagonal and rectangular arrangements, which

is accompanied by a significantly larger bond-breaking probability

(see Fig. 5.7(c)). Again this allows compactifaction of aggregates and

increases the fractal dimension in the transition regime.

5.2.5 Beyond DLCA - Higher temperatures

Diffusion limited aggregation is restricted to systems with attrac-

tive particle interactions much stronger than kBT. By increasing

the temperature sufficiently, thermal fluctuations become able to

break bonds which results in a faster decay of the the bond auto-

correlation functions and a compactifaction of aggregates. Indeed,

for square lattice models it was found that Dfis a monotonically in-

creasing function of temperature [132,129]. In Fig. 5.9(a) the fractal

dimension Dfof the present model is plotted as a function of tem-

perature T∗for charge separations δ=0.1σ , 0.21σand 0.3σ.

We first concentrate on the case δ=0.3σ, corresponding to the

square regime at low T∗. In the range of very low temperatures

T∗<0.25, the fractal dimension is small and stays essentially con-

stant. Increasing T∗towards slightly larger values then leads to

an increase of Df, reflecting the (expected) compactifaction. This

increase of Dfis accompanied by an increase of the mean coor-

dination number ¯

z[see Fig. 5.9(b)] within the temperature range

considered, indicating the growing number of bonds due to local

and global structural reconfigurations. The corresponding changes

in the stability of the bonds are illustrated in Fig. 5.10, where we

have plotted the time evolution of cb(t) for several temperatures (at

δ=0.3σ). Clearly, the decay of cb(t) becomes faster for higher tem-

peratures. This is the reason why structural reconfigurations and, in

consequence, compactifaction of aggregates becomes possible.

These trends persist until T∗

f ,sq ≈0.375, beyond which the system

at δ=0.3σstarts to behave in a qualitatively different way. The

mean coordination number ¯

zdisplays a maximum and subsequently

a rapid decay. We also find that the fractal dimension has not yet

5Mutidirectional Colloidal Assembly

Figure 5.9: Temperature dependence of the system properties at density

ρ∗=0.3for charge separations δ=0.1,0.21,0.3σcolored in yel-

low, purple and black, respectively. (a) Fractal dimension Df

evaluated at t≈250τb, (b) Mean coordination number, (c) Ori-

entational order parameters Φ4and Φ6.

5.2Diffusion Limited Aggregation in Mutidirectional Fields

Figure 5.10: Bond auto correlation function cb(t) for different tempera-

tures T∗at charge separation δ=0.3σand density ρ∗=0.3.

reached its maximum value at T∗

f ,sq ; this maximum occurs at the

slightly larger temperature T∗≈0.42 (see Fig. 5.9(a)). This ’delay’ of

Dfcan be understood from the fact that, upon the entrance of bond-

breaking, filigree parts of the aggregates are more likely affected

than more compact ones. Hence, the fraction of ’compact’ small

aggregates still grows. Even more important, the function Φ4(T∗)

in Fig. 5.9(c) displays a pronounced decay of rectangular order for

T∗> Tf ,sq. From the sum of these indications we conclude that, at

temperatures higher than T∗

f ,sq ≈0.375, the system transforms into

a (stable or metastable) fluid phase. In this fluid phase, the over-

all structure starts to become homogeneous and isotropic, while the

local structures involve only a small number of bonds with short

bond-life times.

For the system at δ=0.1σ(hexagonal structure at low T∗), an esti-

mate of the ”fluidization” temperature T∗

f ,hex based on the behavior

of order parameters, coordination number and fractal dimension is

more speculative. Nevertheless, the data suggest that T∗

f ,hex > T ∗

f ,sq .

This is indicated, first, by the fact that Φ6(T∗) decays only very slowly

with temperature until T∗≈0.6(see Fig. 5.9(c)). Second, the mean

coordination number shows only a weak maximum (and no fast de-

cay after-wards) compared to the case δ=0.3σ. Third, the fractal

dimension keeps increasing with T∗for all considered temperatures

T∗<0.6. Therefore we conclude that T∗

f ,hex >0.6. We understand this

higher fluidization temperature at δ=0.1σfrom the fact that bind-

ing energies in hexagonal structures are larger; therefore, higher

coupling energies must be overcome.

To further justify these interpretations, particularly the emergence

5Mutidirectional Colloidal Assembly

Figure 5.11: Simulation snapshots at ρ∗=0.3with δ=0.21σat (a) T∗=

0.15, (b) T∗=0.3, and (c) T∗=0.45. Particles are colored

according to their value φ4

5.2Diffusion Limited Aggregation in Mutidirectional Fields

of fluid phases, we performed a stability analysis of the homogenous

isotropic high temperature state based on mean-field density func-

tional theory (DFT). Specifically, we consider the isothermal com-

pressibility χT. Positive values of χTimply that the homogeneous

(fluid) phase is stable, whereas negative values indicate that this

phase is unstable. Specifically, the instability arises against long-

wavelength density fluctuations, i.e. condensation. According to

Kirkwood-Bufftheory [156] one has

χ−1

T∝1−ρ˜

c(k=0),(5.10)

where ˜

c(0) is the Fourier transform of the direct correlation function

(DCF) c(r12) in the limit of long-wavelengths (k→0). We approx-

imate the DCF for distances rij > σ according to a mean field (MF)

approximation, that is

cMF (r12) = −(kBT)−1U(r12), r12 > σ , (5.11)

and use the Percus-Yevick DCF cHS (r12) of a pure hard-sphere

fluid [139] for |r12|≤σ. The full DCF is then given by

c(r12) = cHS (r12) + cMF (r12).(5.12)

In Fig. 5.12 we present numerical results for the expression 1−ρ˜

c(0)

at ρ∗=0.3as function of temperature. At low T∗, all systems are

characterized by negative values of 1−ρ˜

c(0). This indicates that the

homogeneous isotropic phase is unstable, consistent with the results

Figure 5.12: Numerical solutions to Eq.5.10 as function of T∗for density

ρ∗=0.3and charge separations δ=0.1,0.21,0.3σcolored in

yellow, purple and black, respectively.

5Mutidirectional Colloidal Assembly

of our simulations. Upon increasing T∗the mean-field compress-

ibility χTthen becomes indeed positive for all charge separations

considered. Specifically, for δ=0.3σthe change of sign (related to

a ”spinodal point”) occurs at T∗

f ,sq =0.325 and for δ=0.1σat the

much higher temperature T∗

f ,hex =0.6. These values are in surpris-

ingly good agreement with our estimates for the ”fluidization” tem-

peratures based on the order parameter plots.

The case δ=0.21σis again different. Here we find [see Fig. 5.9(b)]

that, starting from low temperatures inside the DLCA regime, the

mean coordination number monotonically decreases. However, this

does not indicate ”fluidization” but rather a gradual transition from

a state with dominant hexagonal order towards a mixed state com-

prised of coexisting clusters with local hexagonal and square-like

order. Indeed, [see Fig. 5.9(c)], the orientational order parameters

Φ4and Φ6reveal that the fraction of particles bound in square clus-

ters increases with T∗and finally overtakes the fraction of particles

involved in hexagonal clusters at T∗≈0.35. Corresponding snap-

shots of simulation results are shown in Fig. 5.11. At all tempera-

tures considered one observes separated clusters. With increasing

temperature their shape becomes more regular, while the local rect-

angular order becomes more pronounced. Finally, at T∗=0.45 the

fractal dimension Dfand the square order parameter Φ4reach their

maximum values, suggesting a ”fluidization” similar to the behavior

observed at other values of δ. Interestingly, our stability analysis [see

Eq. (5.10)] indicates an instability at the same temperature T∗

f=0.45.

With this surprisingly accurate agreement between theory and sim-

ulation, we conclude that in the transition regime (δ=0.21σ), in-

creasing thermal fluctuations first push the system from a domi-

nantly hexagonal state into a rectangular one, which then enters a

metastable fluid phase after passing the ”spinodal point”.

5.2.6 Spotlight on higher densities

In this section we revisit the systems behavior at the low tempera-

ture T∗=0.05, but consider different densities in the range ρ∗≤0.7.

Whereas low-density systems at T∗=0.05 display DLCA as dis-

cussed in section 5.2.4, this aggregation mechanism is expected to

disappear at higher densities: here, the particles are just unable to

diffuse sufficiently freely. Rather, the particles will very frequently

collide and then immediately form rigid bonds. A typical structure

at the highest density considered, ρ∗=0.7, and separation param-

eter δ=0.21σis shown in Fig. 5.13. Clearly, the system is perco-

lated, that is, the particles form a single, system-spanning cluster.

Interestingly, this cluster is composed of extended regions charac-

5.2Diffusion Limited Aggregation in Mutidirectional Fields

Figure 5.13: System at T∗=0.05, density ρ∗=0.7and δ=0.21σ. The

color-code gives the orientational bond-order parameter φ4

of each particle i.

terized by either square-like order or hexagonal order. We note that,

at δ=0.21σ, simultaneous appearance of clusters with both types of

order also occurs at low densities and higher temperatures (see sec-

tion 5.2.5). However, at the high density considered here the regions

of each type are larger and the particle arrangements are much more

regular (i.e., there are less defects).

To better understand the impact of the density on the cluster struc-

tures we plot in Fig. 5.14(a) the orientational bond order parameters

Φ4and Φ6as functions of ρ∗for δ=0.21σ(at δ=0.1σand δ=0.3σ

the order parameters are essentially independent of the density).

From Fig. 5.14(a) it is seen that the amount of rectangular (hexago-

nal) order sharply increases (decreases) at a density of ρ∗≈0.45. This

is a surprising result as one would expect that, upon compressing

the system, close-packed, hexagonal structures rather become more

likely. However, at the low temperature considered here, structural

reorganization is strongly hindered.

We also note that all of the systems investigated at densities ρ∗>

0.45 turned out to be percolated (suggesting that the value ρ∗=0.45

is indeed related to the percolation transition). It thus seems that

the percolation tends to stabilize the initially formed square-lattice

symmetry, as the subsequent reorganization is hindered by the lack

of mobility. In effect, we are faced with quenched states that could

100

5Mutidirectional Colloidal Assembly

Figure 5.14: (a) Orientational bond order parameter Φ4and Φ6as func-

tion of density ρ∗for δ=0.21σat T∗=0.05. (b) Mean coor-

dination number ¯

zas function of ρ∗at T∗=0.05 for different

δ=0.1σ , 0.21σand 0.3σcolored in yellow, purple and black,

respectively.

101

5.3Small and Large-Scale Structures

not density within the time domain studied. This interpretation is

also consistent with the decrease of the mean coordination number

once the system is percolated (ρ∗>0.45) as shown in Fig. 5.14(b).

5.3 Small and Large-Scale Structures

In this final section we discuss the connection between our find-

ings on field directed network formation in simulation studies of

the extended dipole particle model (EDP) and the crossed dipole

particle model (CDP). To this end, we first want to mention the vi-

sual similarity between the aggregates found in simulations of the

EDP model in its rectangular regime (see Fig. 5.15(a)) and the large

scale structures found in experiment (see Fig. 5.15(b)). However, it

is clear that the local structures differ strongly between these two

cases. Considering the CDP model, we find that it properly repro-

duces the local structure of the true experimental network structures

as shown in Fig. 5.15 (c) and (d) and discussed (quantitatively) in

Sec. 5.1.3. Hence, both models seem to be reasonable approaches to-

wards a theoretical description of the real system, although on dif-

ferent length scales.

Interestingly, we found that the network structures formed by the

CDP model undergo compactifaction by increasing the coupling

strength. As discussed, from the ideal concept of diffusion limited

aggregation the opposite behavior is expected, where an increase

in the coupling strength results in less compact large-scale fractal

aggregates [122]. However, it is not clear whether the observed com-

pactifaction affects the fractal dimension on large length-scales. Our

analysis was based on local properties, and the observed compacti-

faction is therefore a local one. Please note that it is well known that

gels with long-ranged interactions and formed by a quench into the

two-phase region undergo aging dynamics [114]. However, in this

work we were investigating network structures at constant given

times and it is not the temporal compactifaction which is discussed

here. Revisiting the EDP model, we see that the global structure

of the aggregates does not undergo compactifaction with increas-

ing coupling strength, which is reflected in the constant values of

the fractal dimension Dfby varying T∗(see Fig. 5.9(a)). Consider-

ing the local structure, which can be done via the mean coordination

number ¯

z, we find that structures become less compact with increas-

ing coupling strength as ¯

zcontinuously decreases (see Fig. 5.9(b)).

Hence, the EDP model behaves as it is expected from diffusion lim-

ited aggregation processes (but different than the CDP model).

The remaining question is why these models yield different behavior

regarding compactifaction?

102

5Mutidirectional Colloidal Assembly

Figure 5.15: Small- and large-scale images of network structures in exper-

iment and Simulation. Large-scale image from (a) Simulation

of the EDP-model at ρ∗=0.3,T∗=0.05 and (b) Experiment

at field strengths E=20V /mm, H =90A/m. Local network

structures in (c) Simulation of the CDP-Model at ρ∗=0.35,

µel =µmg =0.4and (d) microscopic image at the same condi-

tions as in (b). Particles in (c) and (d) are color-coded accord-

ing to the method introduced in Sec. 3.5

103

5.3Small and Large-Scale Structures

Unfortunately, a general answer to this question can not be given

here. Nevertheless, the following arguments and interpretations

might provide an explanation, although a somewhat speculative

one.

5.3.1 Strange compactifaction - A hybrid of

anisotropic and slippery DLA?

The EDP model undergoes either anisotropic or slippery diffusion

limited aggregation depending on the interaction anisotropy param-

eter δ(at strong coupling strengths λ > 10 or low temperatures T∗≤

0.2). The transition between these aggregation regimes is very steep

and solely controlled by δ. This is because the EDP model forms

an one-component system of identical particles. In case the charge

separation δwould be different for different particles, one would ex-

pect a different behavior. For example, if δis distributed around

the hexagonal-square transition parameter δsq

hex ≈0.21σ, some parti-

cles would clearly prefer rectangular arrangements, while other ones

would prefer hexagonal arrangements. Building on our previous

analysis, this means that some particles will form slippery bonds

with their nearest neighbors, while other particles form orientation-

ally fixed bonds. Hence, the system should reveal an hybrid aggrega-

tion process composed of anisotropic and slippery DLA. This means,

that locally one could observe slippery DLA as well as anisotropic

DLA, depending on the actual particles one is looking at. However,

on large length-scales, it is uni-directional chains along one of the

fields, which are the dominant structural elements. Inside the chains

bond rotations are strongly suppressed, because such bond rotations

involve massive structural reconfigurations and even bond-breaking

events. For example, two particles connected by a slippery bond and

forming a chain which is connecting two compact aggregates can

only rotate around each other if either the chain breaks, or the ag-

gregates move against each other. Hence, on larger length-scales the

system should look like anisotropic DLCA.

However, one might expect that for stronger interaction energies

these two particles might be able to induce or guide the motion of

larger aggregates, bend the chain and maybe form more compact

structures.

Coming back to the crossed dipole particle (CDP) model we find in-

deed a similar situation. The ’energy landscapes’ for two random re-

alizations of particles with randomly shifted magnetic moments are

shown in Fig. 3.4and reveal different interaction patterns. In a many

particle system (simulation), various pairings of particles with dif-

ferent interaction patterns will be present. Therefore the CDP model

104

5Mutidirectional Colloidal Assembly

corresponds in principle to a multicomponent system, similar to the

EDP-model variation discussed above. The irregular assembly of

magnetic chains (see Fig. 5.1(c)) results from that multi-component

character. This becomes obvious when comparing the magnetically

assembled chains with the linear electrically assembled chains.

In consequence, it appears to us that the observed local compacti-

faction behavior with increasing interaction strength results from

the ’multicomponent’ character of the system. The underlying ag-

gregation mechanism can be interpreted as a hybrid of slippery and

anisotropic diffusion limited aggregation. While the EDP-model

serves as a conceptional model which allows to identify the influ-

ence of anisotropic pair interactions on diffusion limited aggrega-

tion, it is not able to capture (in its present status) all features of the

true system. Still, it allows to interpret them.

5.4 Summary

In this Chapter we presented the computer simulated aggregation

of complex superparamagnetic colloids in crossed external fields.

A detailed comparison between experiments and simulations of

crossed dipole particle (CDP) model introduced in Sec. 3.4was per-

formed and yielded surprisingly similar structures. This analysis

was done by a simple but powerful method previously introduced

in Sec. 3.5. The central result of this section was a state diagram

for the aggregation behavior under various field conditions but con-

stant density. Our simulations revealed, that the system undergoes

counter-intuitive local compactifaction by lowering the temperature

and demonstrated the usefulness of using computer simulations as

supporting tools for understanding experiments, where certain pa-

rameters (here the number density) can not be controlled properly.

In addition to the CDP model, our theoretical study of the extended

dipole particle (EDP) model focused on understanding the forma-

tion of transient aggregates at low-temperatures. Performing large-

scale BD simulations we found that, depending on the patchiness of

particles (which is governed by the distribution of the field-induced

attractive ”sites” in the particles) different aggregation mechanisms

arise. These have been analyzed via appropriate structural order pa-

rameters, bond time-correlation functions as well as by the fractal

dimension. Our BD results demonstrated that by varying the patch-

iness of pair interactions, that is, the distribution of attractive sites,

the systems transform from DLCA (essentially rigid bonds) towards

sDLCA (slippery bonds). Moreover, we showed that the change of

aggregation behavior is accompanied by significant changes of the

local cluster structure.

105

5.4Summary

Indeed, the cluster structure can be easily manipulated by exploiting

the interplay between temperature, density and model parameter

δ. This allows formation of unexpected structures e.g., pronounced

rectangular packing instead of closed packed hexagonal structures

by increasing density. This unusual behavior appears to be dictated

by the inability of the originally formed lattices with hexagonal sym-

metry to rearrange into less dense square lattices.

Finally, the two models were compared with respect to the small and

large-scale structures they form. The local compactifaction of net-

work structures with decreasing temperature was then interpreted

as a hybrid of slippery and anisotropic diffusion limited aggrega-

tion although these final comments remained speculative and fur-

ther studies are needed for definite conclusions.

106

Conclusion and Outlook

In this work, we have used computer simulations to study the

non-equilibrium self-assembly of complex colloidal particles

with field induced dipolar interactions. The actual systems in-

vestigated are motivated by recent experimental observations in

quasi two-dimensional setups of colloids confined to a surface.

Theoretical models for the experimental particles and systems

have been developed and compared to results from experiments.

But also more generic models have been designed to allow in-

sights into the underlying mechanisms of the colloidal assembly.

After giving a general introduction to non-equilibrium colloidal sys-

tems and presenting in detail two experimental examples in Chapter

1, we briefly reviewed some basic theoretical descriptions of simple

fluids in Chapter 2. Special attention was paid to the concept of

phase separation.

In Chapter 3we then developed and discussed four different mod-

els for the two experimental systems presented in Chapter 1. To

this end, we used ’energy landscapes’ to represent two-particle in-

teraction patterns, which allows an intuitive understanding of the

complex dipolar character of our models. Furthermore, we took ad-

vantage of the basic symmetry of the dipole-dipole interaction and

introduced an method to analyze colloidal networks on a single par-

ticle level. This method was exemplary applied to an experimental

and a simulated network structure and turned out to be a simple but

powerful tool.

In this method, two decoupled types of fictitious dipole moments

are associated with each particle and are used to calculate fictitious

energy spectra of the considered structures, which allows a proper

characterization of the systems state. Furthermore, the energy spec-

tra allow to identify for each particle the ’role’ it is playing inside the

network, which is either an edge or a node. Given this information

it is in principle possible to deduce the adjacency matrix of the col-

loidal networks. From a mathematical point of view, this means that

the topological properties of the network become accessible [158]

and it is an interesting question how true physical properties (e.g.,

thermal and electrical conductivity) are related to topological prop-

erties (e.g., scale freeness or bipartiteness and of course percolation).

Although designed to analyze rectangular real space network struc-

tures in two-dimensions this method can be extended towards other

types of structures, which do not necessarily correspond to Bravais

lattices i.e., non-Bravais clusters as formed by the dipolar Janus-

particles and described in Sec. 4.2.1.

In the following Chapters we investigated the self-organization and

self-assembly processes of the different particle models by means of

Brownian dynamics computer simulations, basic concepts of density

functional theory and heuristic theoretical considerations.

Lane formation of dipolar microswimmers

In Chapter 4we investigated a system of colloidal microswimmers

under the presence of an external electric field. The experimental

setup motivating this work has been introduced in Sec. 1.2.4. The

complex model used for this study is a double dipolar Janus par-

ticle with two induced dipole moments of different strengths and

an orientation ally fixed propulsion direction (see Sec. 3.2). In a

two-dimensional setup this means that the system is effectively bi-

nary; there exist two particle species with opposite orientations and

propulsion directions. Such configurations are generally assumed to

undergo lane formation if the propulsion force is strong enough. We

have shown that this is indeed the case for our dipolar model and

that the induced dipolar interactions stabilize the formation of lanes

over a wide range of interaction strengths. This is surprising, as the

anisotropic interactions show a strong tendency of the particles to

self-assemble into chains which, in the present model, are oriented

perpendicular to the driving force. These self-assembly processes re-

sult in complex staggered chains, when no driving force is present

and the observed structures are in qualitative agreement with exper-

imental observations. The underlying mechanisms for lane forma-

108

6Conclusion and Outlook

tion were then identified by investigating a simple binary Lennard-

Jones fluid with oppositely driven particle species. Here, the onset

of lane formation is closely related to the phase separation of the

corresponding , well known, equilibrium system. By simple theoret-

ical considerations we could also estimate the spinodal point in the

double dipole model and show that the same connection is present

there. Importantly, the (effective) attractiveness of the particle inter-

actions in both models enables the system to undergo lane formation

at propulsion speeds approximately one order of magnitude smaller

than in previously known systems [91]. In addition, the presence of

attractive interactions can also suppress lane formation. Using force

balance arguments, the breakdown of lane formation due to ’freez-

ing’ was estimated in very good agreement with the simulation re-

sults. Finally, our observation of lane formation should be directly

measurable in real systems of field-propelled metallo-dielectric par-

ticles, for which the model has been designed.

Starting from the present study, there is several further intriguing

questions to be explored.

Outlook on lane formation

The model we were investigating contains several parameters, the

dislocation of the dipole moments δand the ratio between the dipole

strengths µ1/µ2which have not been varied systematically. As the

values chosen in this work are ’educated guesses’ but have not

been derived quantitatively from microscopic particle properties,

it would be of interest to investigate their influence on the lane for-

mation process. Importantly, the dipole moments are symmetrically

shifted out of the particle center. Decreasing the shift parameter δ

strongly influences the anisotropy of interactions and increases the

systems tendency to form chains perpendicular to the drive, as test

simulations revealed. At smaller dipole shifts, the competition be-

tween lane formation and (perpendicular) chain formation should

be more pronounced and the internal structure of lanes might be

tuned.

Importantly, as the intra lane structures serve as ’backbones’ of

lanes, tuning these structures would most probably affect the width

of lanes. Unfortunately, it is and remains unclear whether there

exists a characteristic lane width at all. It would be of great interest

to answer this question. Due to the more pronounced competition

between laning and chaining it might become possible to quantify

lane widths.

Also, is the behavior observed here related to a Raleigh-Taylor (RT)

instability occurring in driven, macroscopically phase-separated

109

mixtures? In the context of RT, the critical wavelength [159] can

be estimated by a simple criterion involving the surface tension be-

tween the phases; this might provide an additional way to quantify

the lane width of our system and in general.

Our results on lane formation in attractively interacting systems

may open up the door towards a deeper understanding of the lan-

ing transition as they show that laning in systems with attractive

interactions is intimately related to phase separation, irrespec-

tive of the detailed form of the interaction potential. In recent

years, the study of pseudo-phase separation processes in ’classi-

cally self-propelling’ systems, where the propulsion direction is

subject to rotational diffusion, has become a very lively field of

research [69,16,58]. Especially hard sphere systems, which do

not undergo phase separation in equilibrium, undergo a separation

process into high and low density pseudo-phases if the propulsion

forces are strong enough. A deeper understanding of this process

is based on mimicking the influence of self-propulsion by an ef-

fectively attractive potential [69]. Interestingly, adding attractive

interactions to the active hard-sphere system can enhance as well as

suppress pseudo-phase separation [70,71] similar to the effect on

lane formation presented here. And the same behavior has also been

reported for driven magnetic mixtures [42]. Obviously, it would be

of major interest to have a general theory for pseudo-phase transi-

tions in active systems, which, as our study indicates, should then

connect pseudo-phase transitions to true equilibrium phase transi-

tions in a conceptional way. A promising approach towards such an

understanding might be based on the driven Lennard Jones fluid.

This simple system allows large scale computer simulations which

can be accompanied by standard methods from (dynamical) density

functional theory. First studies using such techniques have been

performed [91,42] on soft-sphere systems. However, the (binary)

Lennard Jones system presented here has not been investigated so

far, but may serve as a generic model. In fact, in collaboration with

Christopher W¨

achtler and Sabine H. L. Klapp at TU Berlin, an ex-

tensive study on lane formation in the binary driven Lennard-Jones

fluid has been performed and will be published soon [160].

Accordingly, we expect our findings to be transferable to other

driven colloidal systems with direction-dependent interactions,

examples being ”patchy” particles which, similar to our system,

display both equilibrium aggregation and condensation [103], or

dipolar colloids under shear flow [43]. In this context it is an in-

teresting question whether lane formation can be interpreted as a

directed spinodal decomposition. Here, an effective Cahn Hillard

theory might be a reasonable approach [75]. The latter then would

110

6Conclusion and Outlook

Figure 6.1: Simulation snapshots at ρ∗=0.3,T∗=1and µ∗=3.5for driv-

ing forces (a) f∗

d=1, (b) f∗

d=3and (c) f∗

d=5. Brighter dots in

particles indicate the particle orientation.

raise again the question about dynamical characteristic length scales

(lane widths) usually found in decomposing systems.

Finally, we shortly want to propose an interesting model system

for an active dipolar fluid or one might call it an active ferrofluid. In

experiment, self-propelling Janus Particles (e.g., metal coated) are

often activated by light or localized chemical reactions [53,52,16].

Then the propulsion direction is fixed to the particle orientation,

which itself is subject to rotational diffusion. Additionally, the

metal-coated surface part might be made of a magnetic material

e.g., iron oxide. Hence, a system of active magnetic/dipolar par-

ticles is realized and applying an external magnetic field should

in principle allow to orientate these particles and their propulsion

directions as the propulsion direction is fixed with respect to the

particle body. Starting from the soft-sphere dipole fluid [22] (soft-

sphere particles with centered dipole moments µi), one possibility

of modeling activity is to associate a driving force fi=fiµi/µiwith

each particle i. Thus, the particle moves along its dipole moment,

although other geometries are also of interest. In fact, we performed

two dimensional test simulations allowing the dipoles to rotate in

three dimensions according to the equations of motion for a dipolar

particle with constant propulsion force fdand interaction potential

Ui=

∑N

j=1USS (rij ) + Udip(rij ) (see Eq. [3.6])

ri=−DT

kBT∇iUi+fd

µi

+ζT

i(6.1)

µi=−DR

kBT

(µi×∇µiUi) + ζR

i(6.2)

with rotational and translational diffusion coefficients DRand DT

and white noise fulfilling the Eqs. [2.5,2.6]. In Fig. 6.1, we show

111

Figure 6.2: Mean square displacement in the active dipole fluid at ρ∗=

0.6,T∗=1and f∗=2for different dipole strengths.

simulation snapshots at constant density and dipole strength for dif-

ferent driving forces.

The overall scenario these test simulations indicate is the following.

Given the dipolar coupling is strong enough, the self-propulsion

enhances the systems ability to form a quasi-ferromagnetic state,

where dipole moments mostly point in the same direction. Further-

more, this transition enhances the diffusion properties and allows

the system to be super-diffusive in the long-time limit. Correspond-

ing results showing the mean square displacement [134] for differ-

ent dipole strengths are shown in Fig. 6.2. Despite the details of

this transition, we want to note that in equilibrium there is no fer-

romagnetic transition of the two-dimensional dipole fluid [22]. The

general interest of this system is given by its connection to various

other model systems. On the one hand, it yields an alignment mech-

anism similar to the Vicsek-model [59,57] (which also shows the

motility induced ferromagnetic transition) but it is based on true

physical interactions (steric and dipole-dipole interaction). On the

other hand, it is closely connected to non-equilibrium structure for-

mation of dipole fluids as it can be tuned by external fields. In gen-

eral, external fields allow a tremendous variety of structure forma-

tion in passive dipole fluids, e.g., chaining [2] or layering [41]. The

interplay between these phenomena and active motion most proba-

bly opens the door towards a new field of research concerned with

tunable active dipole fluids.

112

6Conclusion and Outlook

Field directed aggregation

The effect of anisotropic pair interactions on colloidal self-assembly

processes is itself an interesting question and in Chapter 5a sys-

tem of double dipolar (patchy) particles has been investigated. The

corresponding experimental setup and the observed self-assembled

structures have been introduced in Sec. 1.3.3and a paper publica-

tion is preparation [161].

Here, the assembly of chain and network structures under the pres-

ence of crossed magnetic and electric fields was tunable by the re-

spective field strengths. Importantly, it is not Janus particles which

were considered here, but polymer particles with an unknown inter-

nal distribution of superparamagnetic (nanometer scale) iron-oxide

aggregates. Again, a complex model (CDP-model) with explicit

dipole-dipole interactions has been developed in Sec. 3.4. The

model is based on the assumption that different particles carry

different amounts of magnetic material and that the distribution

of the magnetic material inside the particles is not homogenous

but rather characterized by an off-centered effective magnetic in-

teraction site. Based on a new and rather simple method to an-

alyze colloidal networks, a detailed comparison of simulated and

experimental colloidal aggregates was performed. The simulated

structures turned out to be in surprisingly good agreement with the

experimental ones. A precise control of the local and global density

was not possible in experiment and no clear understanding of the

non-equilibrium state diagram at different external field strengths

could be established. The advantage of the theoretical approach was

that it allowed to close this gap in the experimental understanding.

Simulations revealed that at fixed intermediate density there exist in

principle four different states, corresponding to magnetic and elec-

tric field aligned string fluids and fine and coarse colloidal networks.

The main result here is the non-equilibrium state diagram from sim-

ulation and the newly developed method to characterize colloidal

networks. We found that there exists a smooth crossover from fine to

coarse networks by increasing both field strengths, which could not

be explained at this point. It also remained unclear whether these

network states do undergo gelation. This is mostly due to computa-

tional reasons because the analysis of the prerequisites of gelation,

namely very slow relaxational dynamics and percolation, requires

very long simulation times and large particle numbers (finite size

scaling). Such procedures were not feasible with this model.

In addition, a generic model (EDP model) with tunable direction-

ality of effective dipole-like interactions was developed in Sec. 5.2.

In an extensive study of its aggregation behavior we investigated the

113

influences of temperature and of the ’degree’ of patchiness of pair in-

teractions. The focus of the study of the generic model was to under-

stand the formation of transient, aggregated structures appearing at

low-temperatures. Performing large-scale BD simulations we have

found that, depending on the patchiness of particles, which is gov-

erned by the distribution of the field-induced attractive ”sites” in the

particles, different aggregation mechanisms arise. These have been

analyzed via appropriate structural order parameters, bond time-

correlation functions as well as by the fractal dimension. Our BD

results demonstrate that by varying the charge separation param-

eter, that is, the distribution of attractive sites, the systems trans-

form from DLCA (essentially rigid bonds) towards sDLCA (slippery

bonds). Moreover, we show that the change of aggregation behavior

is accompanied by significant changes of the local cluster structure.

Based on our understanding of these ideal aggregation regimes of a

one component system, we interpret the strange compactifaction ob-

served on the local scale in CDP model as a hybrid of slippery and

anisotropic DLA in a multi component system.

Outlook on field directed aggregation

Although this interpretation remains somewhat speculative, as dis-

cussed in Sec. 5.3, we assume that it can be tested by introduc-

ing a random distribution of the shift parameter δin the generic

model such that different particles have different charge separations.

This will most probably result in a smoother transition between the

hexagonal and rectangular regime and lead to a situation, where

rectangular chaining and branching persist while the local order

is already hexagonal and the generic (EDP) model could be related

much more closely to the complex (CDP) model. Then, its compu-

tational and conceptional advantages could be exploited more ef-

ficiently. For example, it remained an open question whether the

system undergoes gelation. Computationally it should be possible

to determine the percolation transition and characterize the relax-

ational dynamics in the generic model. These results should then be

transferable to the complex model and to the real system. Besides,

it remains an open question whether it is the range of interactions

which governs the compactifaction process. Finally, having access

to large-scale network structures allows to extract topological infor-

mation (adjacency matrix of the underlying graph) via the structure

analysis method discussed above. A large zoo of potentially interest-

ing correlations emerges here. For instance, the interplay between

fractal dimension and graph connectivity [158].

From a technological point of view the presented multi-directional

114

6Conclusion and Outlook

field approach yields strong potential for the design of adjustable

materials. In fact, the cluster structures can be easily manipulated

(except of particle density) by exploiting the interplay between dif-

ferent fields of different strengths and orientations. This allows for-

mation of unexpected structures e.g., pronounced rectangular pack-

ing instead of closed packed hexagonal structures by increasing den-

sity, as observed in the generic model.

It has potentially important consequences for colloidal assembly,

as is points out the ability to use mutidirectional field-driven as-

sembly for the making of lower-density, yet highly interconnected,

phases. The formation of particle networks with multiple percola-

tion directions can find application in a range of new materials with

anisotropic electrical and thermal conduction, magnetic or electric

polarizability or unusual rheological properties. The aggregated

clusters can be dispersed in liquid, while the percolated networks

can be embedded in a polymer or gel medium [162]. The key to

the fabrication of such novel classes of materials containing particle

clusters and networks is the control of the process parameters to

obtain the desired inter-connectivity, density and structure. In addi-

tion, the general setup allows also to change the angle between the

electric and magnetic field which should result in new interaction

patterns and exotic structures. Future research should also focus on

a more detailed investigation of the interplay between the aggrega-

tion mechanisms observed here (anisotropic and slippery DLCA),

and the equilibrium phase behavior, particularly the location of a

condensation transition. This includes investigation of the influence

of entropy which we did not discuss but may strongly influence the

aggregation behavior [163]. Furthermore, connections to transient

and directional cluster formation mediated by DNA-links [164],

long-ranged repulsion [165], anisotropic particle shapes [10,1,12]

or other non-equilibrium mechanisms such as activity [68,166]

and/or hydrodynamics [167] are of interest.

In this work, two types of non-equilibrium processes have been

considered, namely ’Hit and Stick’ aggregation and lane formation,

which are characterized by strong anisotropic interactions and self-

propulsion, respectively. The combination of these particle proper-

ties is a next step towards colloidal systems of increased complexity.

Such systems might result in a ’super-diffusion limited aggrega-

tion’ process, meaning that particles are able to self-propel and to

undergo field directed aggregation. By now, ’Hit and Stick’ aggrega-

tion of ballistically moving particles has been considered [117,118],

but the interplay of self-propulsion and ’Hit and Stick’ aggregation

remains an open question. However, our findings on lanes formed

by dipolar microswimmers point in this direction.

115

Appendix

Brownian Dynamics Computer Simulations

The solutions of Eqs. [2.12,2.13] are the trajectories of the col-

loidal particles. However, an analytical solution for a many par-

ticle system is in general not possible. An alternative is to integrate

Eqs. [2.12,2.13] numerically; a procedure which is usually called

Brownian dynamics (BD) computer simulation [134,135].

Numerical Integration

The overdamped Eqs. [2.12,2.13] are of first order which allows to

use the Euler integration-scheme [134,135]. Throughout this work,

directional pair interactions with fixed orientations have been con-

sidered. Therefore, torques and rotational motion of particles has

not been calculated. For completeness we present here the full so-

lution of the translational and rotational equations of motion. The

numerical integration of Eqs. [2.12,2.13] then reads

ri(t+∆t) = ri(t+∆t) +

kBTFi∆t+˜

i(6.3)

ei(t+∆t) = ei(t) +

kBTTi∆t×ei(t) + ˜

i×ei(t),(6.4)

with an integration step width ∆tand orientational unit vectors ei(t).

In this numerical integration scheme, the stochastic contributions

iand ˜

iobey again Gaussian distributions with mean

<˜

i(t)>=0(6.5)

<˜

i(t)>=0(6.6)

and with variance

<˜

i(t)·˜

j(t+∆t)>=2DTδij ∆t

(6.7)

<˜

i(t)·˜

j(t+∆t)>=2DRδij ∆t

.(6.8)

The forces and torques Fiand Tiare stemming from interactions

with other colloidal particles or might result from external poten-

tials e.g., an electric field exerting the force Fext and torque Text) on

particle i. Therefore the total force and torque on particle idue to

all other particles (j,i) is given by the superposition of the forces

Fij and torques Tij imposed on particle iby the particles j

Fi=ΣN

j=1Fij +Fext.(6.9)

Ti=ΣN

j=1Tij +Text (6.10)

whereas Nis the number of particles in the system.

Accordingly, their calculation (in a computer simulation) requires

knowledge about the pair interactions between two specific colloids.

In fact, these interactions define the colloidal models and important

examples have been considered in Sec. 3.1.

Importantly, in case the pair interactions are long-ranged special

methods have to be utilized to overcome difficulties related to the

periodic boundary conditions and computational efficiency.

Long-range interactions

In single particle resolved computer simulations, pair interactions

are calculated for particle separations smaller than a certain, in-

teraction dependent, cut-offdistance rc. Under periodic boundary

conditions, this cut-offhas to be smaller than half the size of the

central simulation box. However, there are certain types of interac-

tions which decay very slow with the particle separation: Examples

for such long-ranged interactions are the Coulomb potential or the

dipole-dipole interaction. Hence, these potentials cannot be simply

truncated because otherwise major errors would be introduced. In

the following we present a common approach to handle this diffi-

culty via the Ewald-summation method.

Ewald summation of the Coulomb potential

In this section we introduce the Ewald summation technique for

ionic systems in three dimensions, which then will allow a straight

forward generalization for dipolar systems in the next section.

Thereby we follow closely the derivation of Klapp and Sch¨

on in [150].

A detailed review of special aspects of the two-dimensional Ewald

summation is found in [144].

The pair-interaction Ucou(r) between two charged particles is given

118

6Conclusion and Outlook

by Eq. 3.4. By scaling prefactors into the charges q, the total energy

due to the Coulomb interaction between Nparticles is given by

Uc=1

∑

i=1

qiΦ(ri) with Φ(ri) =

∑

j=1

|ri|.(6.11)

In the following we assume that the system is neutral, meaning that

∑N

i=1qi=0, and that it is a three dimensional bulk system con-

structed of a (cubic V=L3) central simulation cell of size Lwith

periodic boundary conditions. The later condition allows to write

the electrostatic potential as

Φ(ri) =

∑

{n}

′N

∑

j=1

|rij +n|,(6.12)

with rij =rj−riand {n}being the set of lattice vectors naming the pe-

riodic images of the central cell, meaning that n= (nx, ny, nz)Lwith

nα∈Z. In the central cell, denoted by n=0, the contribution j=i

is excluded from the summation, which is indicated by the prime

in the summation over nin Eq. 6.12. In this general setup we now

derive the Ewald summation technique.

To this end, we consider the charge density ρi(r) which is connected

to the electrostatic potential Φ(ri) via Poisson’s equation [142]

Φ(ri) =

∫

dr′ρi(r′)|ri−r′|−1.(6.13)

The distribution of charges seen by particle ican be written a sum

of delta-functions

ρi(r′) =

∑

{n}

′N

∑

j=1

qjδ(r′−rj+n).(6.14)

Importantly, the decay of the potential Φresulting from this charge

density is rather slow (proportional to the inverse distance). Hence,

also the lattice sum converges (for a given ordering) slowly. Fur-

thermore, the sum is only conditionally convergent, meaning that the

order of summation determines the result. The idea of Ewald sum-

mation is now to split the charge distribution ρi(r′) into three parts

ρa

i(r′), ρb

i(r′), ρc

i(r′) which generate a electrostatic potential converg-

ing in a more favorable way. This is done by superimposing each

δ-function in Eq. 6.14 with a continuous distribution of opposite

point-charges qj. Usually, one uses a Gaussian distribution

ρj,n(r′) = −qj(α/√π)3exp [−α2(r′−rj+n)2].(6.15)

Here, αacts as a parameter controlling the width of the distribution,

similar but inverse to the variance. The overall number of charges

119

in the system thereby increases because the artificially introduced

distributions are normalized by

−qj=

∫

dr′ρj,n(r′).(6.16)

Adding Eq. 6.15 to Eq. 6.14 results in the artificial charge density

ρ(a)

i(r′) =

∑

{n}

′N

∑

j=1

qjδ(r′−rj+n+ρj,n(r′)) (6.17)

which can be interpreted as a density of ’screened’ charges. The as-

sociated potential [150]

Φ(a)(ri) =

∑

{n}

′N

∑

j=1

qjerfc(α|rij +n|)/|rij +n|(6.18)

shows a much more rapid decay than Φ(ri), because of the mono-

tonically decreasing complementary error function erfc(). Here, α

controls the decay or screening. And indeed, the expression for

Φ(a)(ri) reminds of the yukawa interaction, whereas the screening

constant κcorresponds to αand the exponential decay is replaced

by the erfc().

By now, we have artificially introduced further charges into system.

Therefore we now reformulate the charge density ρj,n(r′)) of these

fictitious charges to increase its convergence properties and add it

later as an effectively negative charge density to Φ(a). Hence, we will

finally pertain the original setup. From Eq. 6.17 we see that

ρi(r′)−ρ(a)

i(r′) = −

∑

{n}

′N

∑

j=1

ρj,n(r′) (6.19)

where again the prime in the lattice sum indicates that the terms i=

jare excluded for n=0, which is also the reason why ρi(r′) depends

on i. Now we introduce the charge density

ρ(b)(r′) = −

∑

{n}

∑

j=1

ρj,n(r′)

=−

∑

{n}

∑

j=1

qjα3√π−3exp[−α2(r′−ri+n)2].

(6.20)

Note that the lattice sum is carried out here without any restric-

tions (no prime). Therefore ρ(b)(r′) is independent of iand peri-

odic in space and contains the contribution from the central cell

120

6Conclusion and Outlook

(i=jwhen n=0). The later is given by

ρ(c)

i(r′) = ρi,0(r′) = −qiα3√π−3exp[−α2(r′−ri)2].(6.21)

This is an important difference to the right hand side of Eq. 6.19,

and allows the charge density ρ(b)(r′) to be calculated in reciprocal

space. The later is spanned by the lattice vectors

k=2π/L(mx, my, mz) with mα∈Z.(6.22)

It can be shown [150] that the (discrete) Fourier transform of the

charge density ρ(b)(r′) results in the potential

Φ(b)(ri) = 4π

∑

k,0

∑

j=1

k2exp[−k2/4α2] exp[−ik·rij ] + Φ(b)

LR (ri).(6.23)

The term Φ(b)

LR (ri) is the contribution for zero wave vectors k=0and

given by

Φ(b)

LR (ri) = 4π/(V(2ϵ′+1))

∑

j=1

qjri·rj(6.24)

if the system is seen as a large sphere surrounded by a dielectric

continuum with dielectric constant ϵ′. The advantage of this repre-

sentation is, that damping is controlled by the wave-number k=|k|

and sufficient convergence is already achieved for relatively small

wave-numbers k≈20 −30 if αis chosen in such a way, that only

the central simulation cell n=0has to be evaluated in Φ(a)(ri) (see

Eq. 6.18).

The potential according to ρ(c)

i(r′) compensates for the unphysical

self-interaction hidden in Φ(b)(ri) and is given by [150]

Φ(c)=−qi2α/√π. (6.25)

At this point, we have split up the original electrostatic potential

Φ(ri) (see Eq. 6.12) into various contributions with controllable con-

vergence properties. Making use of Eq. 6.11 allows to write down

the corresponding parts of the Ewald potential energy UEW

cou as:

·the contribution from real space

Ureal

cou =1

∑

i=1

qiΦ(a)(ri)

∑

i=1

∑

{n}

′N

∑

j=1

qiqj

erfc(α|rij +n|)

|rij +n|

(6.26)

121

·contribution from the Fourier sum

Uf ourier

cou =1

∑

i=1

qi[Φ(b)(ri)−Φ(b)

LR (ri)]

=2π

∑

i=1

∑

k,0

∑

j=1

qiqj

k2exp[−k2/4α2] exp[−ik·rij ]

(6.27)

·contribution from long-range interaction

Ulong

cou =1

∑

i=1

qiΦ(b)

LR (ri) = 2π

V(2ϵ′+1)|

∑

j=1

qjri·rj|2(6.28)

·the contribution from self-interaction

Uself

cou =1

∑

i=1

qiΦ(c)=−α/√π

∑

i=1

i.(6.29)

The total potential energy is then given by

UEW

cou =Ureal

cou +Uf ourier

cou +Ulong

cou +Uself

cou .(6.30)

Ewald summation of the dipole-dipole potential

With this background, we are now able to formulate the Ewald sum

for dipolar systems. Interestingly, the dipole-dipole potential Udip

(see Eq. 3.6) is related to the functional form of the Coulomb poten-

tial of a unit point charge

Ψ(rij ) = r−1

ij (6.31)

via

Udip(rij ,µi,µj)=(µi·∇i)(µi·∇j)Ψ(rij ).(6.32)

Formally, this means nothing more than replacing the charges qγin

Ucou by the operators µγ·∇γresults in the dipole-dipole potential

Udip. Applying this transformation to the explicit expressions inin

Eqs. 6.26,6.27,6.28 and 6.29 allows to write down the correspond-

ing parts of the Ewald dipole-dipole energy UEW

dip as:

·the contribution from real space

Ureal

dip =1

∑

i=1

∑

{n}

′N

∑

j=1

(µi·µj)B(|rij +n|, α)

−[µi·(rij +n)][µj·(rij +n)]C(|rij +n|, α)

(6.33)

122

6Conclusion and Outlook

·the contribution from Fourier space

Uf ourier

dip =2π

∑

k,0

k2exp[−k2/4α2]F(k)F∗(k)(6.34)

·the contribution from long-range interaction

Ulong

dip =2π

V(2ϵ′+1)

(

∑

j=1

µi)2(6.35)

The functions B(|rij +n|, α), C(|rij +n|, α) and F(k) are defined as:

B(|rij +n|, α) = 1

r3[2αr

√π

exp −α2r2+ erfc(αr)] (6.36)

C(|rij +n|, α) = 1

r5[2αr

√π

(3+2α2r2) exp −α2r2+3erfc(αr)] (6.37)

F(k) =

∑

i=1

(µi·k) exp(−ik·ri).(6.38)

The self-part of the dipolar Ewald sum does not follow directly by

applying the transformation qγ→µγ·∇γ. For details see Ref. [150].

Uself

dip =−2α

3√π

∑

i=1

µ2

i.(6.39)

The total potential energy is then given by

UEW

dip =Ureal

dip +Uf ourier

dip +Ulong

dip +Uself

dip .(6.40)

Ewald summation for quasi two-dimensional dipoles

Most of this work is concerned with the behavior of complex dipo-

lar colloids at interfaces. Such quasi two-dimensional systems are

treated by restricting translational motion to a plane of area A=L2,

while allowing the dipole moments to rotate in three dimensions.

The Ewald summation for dipolar interactions has to be adapted

to these geometrical restrictions. To distinguish between the three-

dimensional and quasi two-dimensional cases, the lattice vectors

of the Fourier contributions are denoted by g=2π(mx, my)Lhere,

with mx,my∈Z. Specifically, the total energy can be split up in

contributions [22,144]

123

parallel

U∥

dip(rij ,µ∥

i,µ∥

j) = −1

∑

i,j=1

[Bq2d(rij )µ∥

i·µ∥

j+Cq2d(rij )(µ∥

i·rij )(µ∥

j·rij )]

∑

g,0

erfc(g/2α)

F∥(g)F∗

∥(g)−2α3

3√π

∑

i=1|µ∥

i|2

(6.41)

and perpendicular

U⊥

dip(rij ,µ⊥

i,µ⊥

j) = −1

∑

i,j=1

[Bq2d(rij )µ⊥

i·µ⊥

∑

g,0

[2α

√g

exp −g2

4α2−gerfc(g/2α)]Fq2d

⊥(g)Fq2d

⊥∗(g)

−2α√π

∑

i,j=1

µ⊥

i·µ⊥

j−2α3

3√π

∑

i=1|µ⊥

i|2

(6.42)

to the plane, whereas µ⊥

γand µ∥

γdenote the perpendicular and par-

allel components of dipole moment µγ, respectively.

(6.43)

The functions Bq2d(|rij +n|, α), Cq2d(|rij +n|, α) and Fq2d

⊥,∥(g) are defined

as:

Bq2d(|rij +n|, α) = −erfc(αr)

r3−2αr

√π

exp −α2r2

r2](6.44)

Cq2d(|rij +n|, α) = 3erfc(αr)

r5+2αr

√π

(3/r2+2α2)

exp −α2r2

r2(6.45)

Fq2d

⊥(g) =

∑

i=1

µ⊥

iexp(ig·ri)(6.46)

Fq2d

∥(g) =

∑

i=1

(µ∥

i·g) exp(ig·ri).(6.47)

In Brownian Dynamics computer simulations, it is not the interac-

124

6Conclusion and Outlook

tion energy but rather the forces

fq2d

i=−∇ri(U∥

dip +U⊥

dip)

∑

j=1

∑

{n}

′{Cq2d(rij )[(µi·µj)rij + (µi(µj·rij ) + (µj(µi·rij )]

−Dq2d(rij )(µi·rij )(µj·rij )rij }

+2π

∑

g,=0

erfc(g/2α)

(g·µ∥

i)g

×[sin(g·ri)ReN(g)−cos(g·ri)ImN(g)]

+2π

∑

g,0

[2α

√π

exp(−g2/4α2)−erfc(g/2α)g]g

×[sin(g·ri)ReO(g)−cos(g·ri)ImO(g)]

(6.48)

which are or interest. Here we have defined the functions

Dq2d(rij ) = 1

[2αrij

√

(15 +10α2r2

+4α4r4

ij ) exp(−α2rij2) + 15erfc(αrij )]

(6.49)

N(g) =

∑

j=1

(g·µ∥

j) exp(ig·rj)(6.50)

O(g) =

∑

j=1

(µ⊥

j) exp(ig·rj)(6.51)

In this work, explicit expressions for torques are not needed, as only

induced dipole moments are simulated.

Computational aspects of Ewald Summation

As previously stated, Ewald summation is a technique used to over-

come difficulties associated with the interaction range, system size

and convergence of interaction sums in many-particle computer

simulations. These difficulties are also closely related to the compu-

tational efficiency. Here we shortly comment on these aspects.

The convergence of the Ewald sum is controlled by the parameter α,

which is usually chosen in such a way that only interactions between

particles with distances smaller than half the central simulation cell

rEW

c=L/2have to be calculated in real space. The actual value

α=6.5/L we are using also determines the maximum number of lat-

tice vectors mmax =

√

x+m2

ywe had to take into account. Including

125

lattice vectors up to mmax ≈30 (depending on the studied particle

numbers N∈[512,1024]) ensures, that the dipole-dipole interaction

decays to values smaller than 10−6kBT.

Transferring parts of the summation into reciprocal space is, in

addition to the absolute convergence, also much faster than calcu-

lating everything in real space. In the two-dimensional setups we

are considering, one can additionally take advantage out of certain

symmetries. For example, the periodic image with g= (1,0)2π/L

yields the same contribution to the Ewald sum as the one with

g= (0,1)2π/L.

Furthermore, the quantities N(g) and O(g) defined in Eqs. 6.50 and

6.51 are single particle sums, meaning that its is only 2Nterms

which need to be calculated instead of N2. This results in a tremen-

dous speed up of the simulation and allows larger systems sizes

and/or particle numbers.

Finally, the fact that the Fourier contributions are essentially single

particle sums allows to use simple and straight forward paralleliza-

tion methods like OpenMP when using the programming languages

C or C++. Especially on quad-, hex- or octa-core computer systems

with shared memory such methods are easy to apply and show sig-

nificant reduction in simulation time.

126

Bibliography

[1] Glotzer Sharon C. and Solomon Michael J. Anisotropy of

building blocks and their assembly into complex structures.

Nat Mater,6(7):557–562, aug 2007.10.1038/nmat1949.

[2] Jelena Jordanovic, Sebastian J¨

ager, and Sabine H. L. Klapp.

Crossover from normal to anomalous diffusion in systems of

field-aligned dipolar particles. Phys. Rev. Lett.,106:038301,

Jan 2011.

[3] Stefano Scanna, Laura Rossi, and David J. Pine. Magnetic click

colloidal assembly. Journal of the American Chemical Society,

134(14):6112–6115,2012.

[4] Yufeng Wang, Yu Wang, Dana R. Breed, Vinothan N. Manoha-

ran, Lang Feng, Andrew D. Hollingsworth, Marcus Weck, and

David J. Pine. Colloids with valence and specific directional

bonding. Nature,491(7422):51–55,2012.

[5] Randall M. Erb, Nathan J. Jenness, Robert L. Clark, and Ben-

jamin B. Yellen. Towards holonomic control of janus particles

in optomagnetic traps. Advanced Materials,21(47):4825–4829,

2009.

[6] Yan Jing, Bloom Moses, Bae Sung Chul, Luijten Erik, and

Granick Steve. Linking synchronization to self-assembly us-

ing magnetic Janus colloids. Nature,491(7425):578–581, nov

2012.10.1038/nature11619.

[7] J. K. G. Dhont. An introduction to dynamics of colloids. Else-

vier, (Amsterdam), 2003.

[8] Emanuela Bianchi, Gerhard Kahl, Christos N Likos, and

Francesco Sciortino. Patchy particles. Journal of Physics: Con-

densed Matter,27(23):230301,2015.

[9] Woltman Scott J., Jay Gregory D., and Crawford Gregory

P. Liquid-crystal materials find a new order in biomed-

ical applications. Nat Mater,6(12):929–938, dec 2007.

10.1038/nmat2010.

[10] Joe W. Tavacoli, Pierre Bauer, Marc Fermigier, Denis Bartolo,

Julien Heuvingh, and Olivia du Roure. The fabrication and di-

Bibliography

rected self-assembly of micron-sized superparamagnetic non-

spherical particles. Soft Matter,9:9103–9110,2013.

[11] Pietro Tierno. Recent advances in anisotropic magnetic col-

loids: realization, assembly and applications. Phys. Chem.

Chem. Phys.,16:23515–23528,2014.

[12] C. Wyatt Shields IV, Shan Zhu, Ye Yang, Bhuvnesh Bharti,

Jonathan Liu, Benjamin B. Yellen, Orlin D. Velev, and

Gabriel P. Lopez. Field-directed assembly of patchy

anisotropic microparticles with defined shape. Soft Matter,

9:9219–9229,2013.

[13] Denis Rodr´

ıguez-Fern´

andez and Luis M. Liz-Marz´

an. Metallic

janus and patchy particles. Particle and Particle Systems Char-

acterization,30(1):46–60,2013.

[14] Andreas Walther and Axel H. E. M ¨

uller. Janus particles. Soft

Matter,4(5):663–668,2008.

[15] Sumit Gangwal, Olivier J. Cayre, and Orlin D. Velev. Dielec-

trophoretic assembly of metallodielectric janus particles in ac

electric fields. Langmuir,24(23):13312–13320,2008.

[16] Ivo Buttinoni, Julian Bialk´

e, Felix K ¨

ummel, Hartmut L¨

owen,

Clemens Bechinger, and Thomas Speck. Dynamical cluster-

ing and phase separation in suspensions of self-propelled col-

loidal particles. Phys. Rev. Lett.,110:238301, Jun 2013.

[17] Gerald Rosenthal, Keith E. Gubbins, and Sabine H. L. Klapp.

Self-assembly of model amphiphilic janus particles. The Jour-

nal of Chemical Physics,136(17):174901,2012.

[18] Ankit Kumar, Bum Jun Park, Fuquan Tu, and Daeyeon Lee.

Amphiphilic janus particles at fluid interfaces. Soft Matter,

9:6604–6617,2013.

[19] Yusei Kobayashi and Noriyoshi Arai. Self-assembly of janus

nanoparticles with a hydrophobic hemisphere in nanotubes.

Soft Matter, pages –, 2015.

[20] Pietro Tierno, Sathavaram V. Reddy, Michael G. Roper, Tom H.

Johansen, , and Thomas M. Fischer. Transport and separa-

tion of biomolecular cargo on paramagnetic colloidal parti-

cles in a magnetic ratchet. The Journal of Physical Chemistry B,

112(12):3833–3837,2008. PMID: 18318526.

[21] Jones Matthew R. and Mirkin Chad A. Materials science: Self-

assembly gets new direction. Nature,491(7422):42–43, nov

2012.10.1038/491042a.

[22] J.-J. Weis. Simulation of quasi-two-dimensional dipolar sys-

tems. Journal of Physics: Condensed Matter,55(15):1471,2003.

128

Bibliography

[23] Fuduo Ma, David T. Wu, and Ning Wu. Formation of

colloidal molecules induced by alternating-current electric

fields. Journal of the American Chemical Society,135(21):7839–

7842,2013.

[24] S. O. Lumsdon, E. W. Kaler, and O. D. Velev. Two-dimensional

crystallization of microspheres by a coplanar ac electric field.

Langmuir,20(6):2108–2116,2004.

[25] Bhuvnesh Bharti and Orlin D. Velev. Assembly of reconfig-

urable colloidal structures by multidirectional field-induced

interactions. Langmuir,31(29):7897–7908,2015.

[26] Sumit Gangwal, Amar Pawar, Ilona Kretzschmar, and Orlin D.

Velev. Programmed assembly of metallodielectric patchy par-

ticles in external ac electric fields. Soft Matter,6(6):1413–

1418,2010.

[27] Marco Klinkigt, Rudolf Weeber, Sofia Kantorovich, and Chris-

tian Holm. Cluster formation in systems of shifted-dipole par-

ticles. Soft Matter,9:3535–3546,2013.

[28] Aleksey Ruditskiy, Bin Ren, and Ilona Kretzschmar. Be-

haviour of iron oxide (fe3o4) janus particles in overlapping

external ac electric and static magnetic fields. Soft Matter,

9(38):9174–9181,2013.

[29] Hartmut L¨

owen. Colloidal dispersions in external fields: re-

cent developments. J. Phys.: Condens. Matter,20(20):404201,

2008.

[30] Tassilo Ott, Hartmut L¨

owen, and M. Bonitz. Dynamics of two-

dimensional one-component and binary yukawa systems in a

magnetic field. Phys. Rev. E,89:013105, Jan 2014.

[31] Eric M. Furst. Directed self-assembly. Soft Matter,9:9039–

9045,2013.

[32] S. Elborai, D.-K. Kim, X. He, S.-H. Lee, S. Rhodes, and

M. Zahn. Self-forming, quasi-two-dimensional, magnetic-

fluid patterns with applied in-plane-rotating and dc-axial

magnetic fields. Journal of Applied Physics,97(10):10Q303,

2005.

[33] Anit Dawar and Amita Chandra. Electric field driven frac-

tal growth dynamics in polymeric medium. Physics Letters A,

378(40):2951–2958,2014.

[34] Michael J. Solomon. Directions for targeted self-assembly of

anisotropic colloids from statistical thermodynamics. Current

Opinion in Colloid & Interface Science,16(2):158–167,2011.

[35] G. De Magistris and D. Marenduzzo. An introduction to the

physics of active matter. Physica A: Statistical Mechanics and

129

Bibliography

its Applications,418:65 –77,2015. Proceedings of the 13th In-

ternational Summer School on Fundamental Problems in Sta-

tistical Physics.

[36] Yashodhan Hatwalne, Sriram Ramaswamy, Madan Rao, and

R. Aditi Simha. Rheology of active-particle suspensions. Phys.

Rev. Lett.,92:118101, Mar 2004.

[37] Browne Wesley R. and Feringa Ben L. Making molec-

ular machines work. Nat Nano,1(1):25–35, oct 2006.

10.1038/nnano.2006.45.

[38] Andrew Cressman, Yuichi Togashi, Alexander S. Mikhailov,

and Raymond Kapral. Mesoscale modeling of molecular ma-

chines: Cyclic dynamics and hydrodynamical fluctuations.

Phys. Rev. E,77:050901, May 2008.

[39] M. E. Cates and J. Tailleur. When are active brownian particles

and run-and-tumble particles equivalent? consequences for

motility induced phase separation. Eur. Phys. Lett.,101:20010,

2013.

[40] Igor S. Aranson. Collective behavior in out-of-equilibrium col-

loidal suspensions. Comptes Rendus Physique,14(6):518 –527,

2013. Living fluids / Fluides vivants.

[41] S. J¨

ager and S. H. L. Klapp. Pattern formation of dipolar col-

loids in rotating fields: layering and synchronization. Soft

Matter,7:6606,2011.

[42] K. Lichtner and S. H. L. Klapp. Novel structure formation of

a phase separating colloidal fluid in a ratchet potential. Euro-

phys. Lett.,106:56004,2014.

[43] Dmitry Borin, Andrey Zubarev, Dmitry Chirikov, Robert

M¨

uller, and Stefan Odenbach. Ferrofluid with clustered iron

nanoparticles: Slow relaxation of rheological properties un-

der joint action of shear flow and magnetic field. Journal of

Magnetism and Magnetic Materials,323(10):1273 –1277,2011.

Proceedings of 12th International Conference on Magnetic

Fluid.

[44] Jason J. Benkoski, Jennifer L. Breidenich, O. Manuel Uy,

Allen T. Hayes, Ryan M. Deacon, H. Bruce Land, Jane M.

Spicer, Pei Yuin Keng, and Jeffrey Pyun. Dipolar organiza-

tion and magnetic actuation of flagella-like nanoparticle as-

semblies. J. Mater. Chem.,21:7314–7325,2011.

[45] A. Snezhko, M. Belkin, I. S. Aranson, and W.-K. Kwok.

Self-assembled magnetic surface swimmers. Phys. Rev. Lett.,

102:118103, Mar 2009.

130

Bibliography

[46] T. Glanz and H. L¨

owen. The nature of the laning transi-

tion in two dimensions. Journal of Physics: Condensed Matter,

24(46):464114,2012.

[47] Stephen J. Ebbens and Jonathan R. Howse. In pursuit of

propulsion at the nanoscale. Soft Matter,6:726–738,2010.

[48] Dreyfus Remi, Baudry Jean, Roper Marcus L., Fermigier Marc,

Stone Howard A., and Bibette Jerome. Microscopic artificial

swimmers. Nature,437(7060):862–865, oct 2005.10.1038/na-

ture04090.

[49] Shradha Mishra and Sriram Ramaswamy. Active nematics are

intrinsically phase separated. Phys. Rev. Lett.,97:090602, Aug

2006.

[50] Xiaogang Yang and Qi Wang. Role of the active viscosity and

self-propelling speed in channel flows of active polar liquid

crystals. Soft Matter, pages –, 2016.

[51] Jonathan R. Howse, Richard A. L. Jones, Anthony J. Ryan, Tim

Gough, Reza Vafabakhsh, and Ramin Golestanian. Self-motile

colloidal particles: From directed propulsion to random walk.

Phys. Rev. Lett.,99:048102, Jul 2007.

[52] Ramin Golestanian. Collective behavior of thermally active

colloids. Phys. Rev. Lett.,108:038303, Jan 2012.

[53] Hong-Ren Jiang, Natsuhiko Yoshinaga, and Masaki Sano. Ac-

tive motion of a janus particle by self-thermophoresis in a de-

focused laser beam. Phys. Rev. Lett.,105:268302, Dec 2010.

[54] Alicia M. Boymelgreen and Touvia Miloh. Alternating current

induced-charge electrophoresis of leaky dielectric janus parti-

cles. Physics of Fluids,24(8):082003–082003,2012.

[55] Sumit Gangwal, Olivier J. Cayre, Martin Z. Bazant, and Or-

lin D. Velev. Induced-charge electrophoresis of metallodielec-

tric particles. Phys. Rev. Lett.,100:058302, Feb 2008.

[56] J. Elgeti, R. G. Winkler, and G. Gompper. Physics of mi-

croswimmers—single particle motion and collective behavior:

a review. Reports on Progress in Physics,78(5):056601,2015.

[57] Tam´

as Vicsek and Anna Zafeiris. Collective motion. Physics

Reports,517(3–4):71 –140,2012. Collective motion.

[58] Julian Bialk´

e, Thomas Speck, and Hartmut L˙ Active colloidal

suspensions: Clustering and phase behavior. Journal of Non-

Crystalline Solids,407:367 –375,2015.7th IDMRCS: Relax-

ation in Complex Systems.

[59] Tam´

as Vicsek, Andr´

as Czir ´

ok, Eshel Ben-Jacob, Inon Cohen,

and Ofer Shochet. Novel type of phase transition in a system

131

Bibliography

of self-driven particles. Phys. Rev. Lett.,75:1226–1229, Aug

1995.

[60] Udo Erdmann, Werner Ebeling, and Alexander S. Mikhailov.

Noise-induced transition from translational to rotational mo-

tion of swarms. Phys. Rev. E,71:051904, May 2005.

[61] Romanczuk, P., B¨

ar, M., Ebeling, W., Lindner, B., and

Schimansky-Geier, L. Active brownian particles. Eur. Phys.

J. Special Topics,202:1–162,2012.

[62] H. Chat´

e, F. Ginelli, G. Gr´

egoire, F. Peruani, and F. Raynaud.

Modeling collective motion: variations on the vicsek model.

The European Physical Journal B,64(3-4):451–456,2008.

[63] Gabriel Baglietto and Ezequiel V. Albano. Nature of the order-

disorder transition in the vicsek model for the collective mo-

tion of self-propelled particles. Phys. Rev. E,80:050103, Nov

2009.

[64] Julian Bialk´

e, Thomas Speck, and Hartmut L¨

owen. Crys-

tallization in a dense suspension of self-propelled particles.

Phys. Rev. Lett.,108:168301, Apr 2012.

[65] J. Dzubiella and H. L¨

owen. Pattern formation in driven col-

loidal mixtures: tilted driving forces and re-entrant crystal

freezing. J. Phys.: Condens. Matter,14(40):9383,2002.

[66] Dirk Helbing, Ill´

es J. Farkas, and Tam´

as Vicsek. Freezing

by heating in a driven mesoscopic system. Phys. Rev. Lett.,

84:1240–1243, Feb 2000.

[67] Robert Großmann, Pawel Romanczuk, Markus B¨

ar, and Lutz

Schimansky-Geier. Vortex arrays and mesoscale turbulence

of self-propelled particles. Phys. Rev. Lett.,113:258104, Dec

2014.

[68] I. Theurkauff, C. Cottin-Bizonne, J. Palacci, C. Ybert, and

L. Bocquet. Dynamic clustering in active colloidal suspen-

sions with chemical signaling. Phys. Rev. Lett.,108:268303,

Jun 2012.

[69] A. Cacciuto M. E. Cates D. Marenduzzo A. N. Morozov

J. Schwarz-Linek, C. Valeriani and W. C. K. Poon. Phase sep-

aration and rotor self-assembly in active particle suspensions.

PNAS,109(11):4052–4057,2012.

[70] Gabriel S. Redner, Michael F. Hagan, and Aparna Baskaran.

Structure and dynamics of a phase-separating active colloidal

fluid. Phys. Rev. Lett.,110:055701, Jan 2013.

[71] Gabriel S. Redner, Aparna Baskaran, and Michael F. Hagan.

Reentrant phase behavior in active colloids with attraction.

Phys. Rev. E,88:012305, Jul 2013.

132

Bibliography

[72] Ethayaraja Mani and Hartmut L¨

owen. Effect of self-

propulsion on equilibrium clustering. Phys. Rev. E,92:032301,

Sep 2015.

[73] Yaouen Fily and M. Cristina Marchetti. Athermal phase sepa-

ration of self-propelled particles with no alignment. Phys. Rev.

Lett.,108:235702, Jun 2012.

[74] T. F. F. Farage, P. Krinninger, and J. M. Brader. Effective

interactions in active brownian suspensions. Phys. Rev. E,

91:042310, Apr 2015.

[75] Thomas Speck, Julian Bialk´

e, Andreas M. Menzel, and Hart-

mut L¨

owen. Effective cahn-hilliard equation for the phase

separation of active brownian particles. Phys. Rev. Lett.,

112:218304, May 2014.

[76] F´

elix Ginot, Isaac Theurkauff, Demian Levis, Christophe

Ybert, Lyd´

eric Bocquet, Ludovic Berthier, and C´

ecile Cottin-

Bizonne. Nonequilibrium equation of state in suspensions of

active colloids. Phys. Rev. X,5:011004, Jan 2015.

[77] S. C. Takatori and J. F. Brady. Towards a thermodynamics of

active matter. Phys. Rev. E,91:032117, Mar 2015.

[78] Julian Bialk´

e, Jonathan T. Siebert, Hartmut L¨

owen, and

Thomas Speck. Negative interfacial tension in phase-

separated active brownian particles. Phys. Rev. Lett.,

115:098301, Aug 2015.

[79] Andreas Kaiser, Katarina Popowa, and Hartmut L¨

owen. Ac-

tive dipole clusters: From helical motion to fission. Phys. Rev.

E,92:012301, Jul 2015.

[80] J. Dzubiella, G. P. Hoffmann, and H. L¨

owen. Lane formation

in colloidal mixtures driven by an external field. Phys. Rev. E,

65:021402, Jan 2002.

[81] Leunissen Mirjam E., Christova Christina G., Hynninen Antti-

Pekka, Royall C. Patrick, Campbell Andrew I., Imhof Arnout,

Dijkstra Marjolein, van Roij Rene, and van Blaaderen Alfons.

Ionic colloidal crystals of oppositely charged particles. Nature,

437(7056):235–240, sep 2005.10.1038/nature03946.

[82] Teun Vissers, Adam Wysocki, Martin Rex, Hartmut Lowen,

C. Patrick Royall, Arnout Imhof, and Alfons van Blaaderen.

Lane formation in driven mixtures of oppositely charged col-

loids. Soft Matter,7:2352–2356,2011.

[83] Roland. R. Netz. Conduction and diffusion in two-

dimensional electrolytes. Europhys. Lett.,63:616,2003.

[84] K R S ¨

utterlin, A Wysocki, C R¨

ath, A V Ivlev, H M Thomas,

S Khrapak, S Zhdanov, M Rubin-Zuzic, W J Goedheer, V E

133

Bibliography

Fortov, A M Lipaev, V I Molotkov, O F Petrov, G E Morfill,

and H L¨

owen. Non-equilibrium phase transitions in complex

plasma. Plasma Physics and Controlled Fusion,52(12):124042,

2010.

[85] C.-R. Du, K. R. S ¨

utterlin, A. V. Ivlev, H. M. Thomas, and G. E.

Morfill. Model experiment for studying lane formation in bi-

nary complex plasmas. Europhys. Lett.,99(4):45001,2012.

[86] K. R. S ¨

utterlin, A. Wysocki, A. V. Ivlev, C. R¨

ath, H. M. Thomas,

M. Rubin-Zuzic, W. J. Goedheer, V. E. Fortov, A. M. Lipaev,

V. I. Molotkov, O. F. Petrov, G. E. Morfill, and H. L¨

owen. Dy-

namics of lane formation in driven binary complex plasmas.

Phys. Rev. Lett.,102:085003, Feb 2009.

[87] K. Jiang, C.-R. Du, K. R. S ¨

utterlin, A. V. Ivlev, and G. E. Mor-

fill. Lane formation in binary complex plasmas: Role of non-

additive interactions and initial configurations. EPL (Euro-

physics Letters),92(6):65002,2010.

[88] V. E. Fortov and G. E. Morfill. Strongly coupled dusty plas-

mas on iss: experimental results and theoretical explanation.

Plasma Physics and Controlled Fusion,54(12):124040,2012.

[89] A. M. Menzel. Unidirectional laning and migrating cluster

crystals in confined self-propelled particle systems. Journal of

Physics: Condensed Matter,25(50):505103,2013.

[90] I. D. Couzin and N. R. Franks. Self-organized lane forma-

tion and optimized traffic flow in army ants. Proceedings of the

Royal Society of London B: Biological Sciences,270(1511):139–

146,2003.

[91] Dzubiella J. Chakrabarti J. and L¨

owen H. Reentrance effect

in the lane formation of driven colloids. Physical Review E,

70(1):012401,2004.

[92] M. Rex and H. L¨

owen. Influence of hydrodynamic interactions

on lane formation in oppositely charged driven colloids. The

European Physical Journal E,26(1-2):143–150,2008.

[93] Masahiro Ikeda, Hirofumi Wada, and Hisao Hayakawa. Insta-

bilities and turbulence-like dynamics in an oppositely driven

binary particle mixture. Europhys. Lett.,99(6):68005,2012.

[94] H. Ohta. Lane formation in a lattice model for oppo-

sitely driven binary particles. EPL (Europhysics Letters),

99(4):40006,2012.

[95] Heiko Schmidle, Sebastian J¨

ager, Carol K. Hall, Orlin D. Velev,

and Sabine H. L. Klapp. Two-dimensional colloidal networks

induced by a uni-axial external field. Soft Matter,9(9):2518–

2524,2013.

134

Bibliography

[96] Martin Z. Bazant and Todd M. Squires. Induced-charge elec-

trokinetic phenomena: Theory and microfluidic applications.

Phys. Rev. Lett.,92:066101, Feb 2004.

[97] Sciortino Francesco. Disordered materials: One liq-

uid, two glasses. Nat Mater,1(3):145–146, nov 2002.

10.1038/nmat752.

[98] Julie Byrom and Sibani Lisa Biswal. Magnetic field directed

assembly of two-dimensional fractal colloidal aggregates. Soft

Matter,9(9):9167–9173,2013.

[99] Rajesh Ganapathy, Mark R. Buckley, Sharon J. Gerbode, and

Itai Cohen. Direct measurements of island growth and step-

edge barriers in colloidal epitaxy. Science,327(5964):445–448,

2010.

[100] Bommel S., Kleppmann N., Weber C., Spranger H., Sch¨

afer P.,

Novak J., Roth S.V., Schreiber F., Klapp S.H.L., and Kowarik S.

Unravelling the multilayer growth of the fullerene C60 in real

time. Nat Commun,5, nov 2014.

[101] E. Zaccarelli. Colloidal gels: equilibrium and non-equilibrium

routes. J. Phys.: Condens. Matter.,19(32):323101,2007.

[102] Amar B. Pawar and Ilona Kretzschmar. Fabrication, assem-

bly, and application of patchy particles. Macromolecular Rapid

Communications,31(2):150–168,2010.

[103] E. Bianchi, J. Largo, P. Tartaglia, E. Zaccarelli, and F. Sciortino.

Phase diagram of patchy colloids: Towards empty liquids.

Phys. Rev. Lett.,97(16):168301,2006.

[104] T. Tlusty and S. A. Safran. Defect-induced phase separation in

dipolar fluids. Science,290(5495):1328–1331,2000.

[105] J. Russo, J. M. Tavares, P. I. C. Teixeira, M. M. Telo da Gama,

and F. Sciortino. Reentrant phase diagram of network fluids.

Phys. Rev. Lett.,106(106):085703,2011.

[106] Mark A. Miller, Ronald Blaak, Craig N. Lumb, and Jean-Pierre

Hansen. Dynamical arrest in low density dipolar colloidal

gels. The Journal of Chemical Physics,130(11):–, 2009.

[107] R. Blaak, M. A. Miller, and J.-P. Hansen. Reversible gelation

and dynamical arrest of dipolar colloids. EPL (Europhysics Let-

ters),78(2):26002,2007.

[108] H. Schmidle, C. K. Hall, O. D. Velev, and S. H. L. Klapp. Phase

diagram of two-dimensional systems of dipole-like colloids.

Soft Matter,8(8):1521–1531,2012.

[109] Lorenzo Rovigatti, John Russo, and Francesco Sciortino. No

135

Bibliography

evidence of gas-liquid coexistence in dipolar hard spheres.

Phys. Rev. Lett.,107:237801, Nov 2011.

[110] Zhan-Wei Li, You-Liang Zhu, Zhong-Yuan Lu, and Zhao-Yan

Sun. A versatile model for soft patchy particles with various

patch arrangements. Soft Matter, pages –, 2015.

[111] Florian Kogler, Orlin D. Velev, Carol K. Hall, and Sabine H. L.

Klapp. Generic model for tunable colloidal aggregation in

multidirectional fields. Soft Matter,11:7356–7366,2015.

[112] Peter J. Lu, Jacinta C. Conrad, Hans M. Wyss, Andrew B.

Schofield, and David A. Weitz. Fluids of clusters in attractive

colloids. Phys. Rev. Lett.,96:028306, Jan 2006.

[113] Rui Zhang, Prateek K. Jhab, and Monica Olvera de la

Cruz. Non-equilibrium ionic assemblies of oppositely charged

nanoparticle. Soft Matter,9(20):5042–5051,2013.

[114] Lisa J. Teece, Malcolm A. Faers, and Paul Bartlett. Ageing

and collapse in gels with long-range attractions. Soft Matter,

7:1341–1351,2011.

[115] Florian Johannes Maier and Thomas Martin Fischer. Criti-

cal nucleation mesh-size of coarsening transient colloidal net-

works. Soft Matter, pages –, 2015.

[116] Clair R. Seager and Thomas G. Mason. Slippery diffusion-

limited aggregation. Physical Review E,75(1):011406,2007.

[117] Isadora R. Nogueira, Sidiney G. Alves, and Silvio C. Ferreira.

Scaling laws in the diffusion limited aggregation of persistent

random walkers. Physica A: Statistical Mechanics and its Appli-

cations,390(23–24):4087 –4094,2011.

[118] S. G. Alves and S. C. Ferreira. Aggregation in a mixture

of brownian and ballistic wandering particles. Phys. Rev. E,

73:051401, May 2006.

[119] Susan Tolman and Paul Meakin. Off-lattice and hypercubic-

lattice models for diffusion-limited aggregation in dimension-

alities 2-8.Phys. Rev. A,40:428–437, Jul 1989.

[120] Paul Meakin. Progress in dla research. Physica D: Nonlinear

Phenomena,86(2):104–112,1995.

[121] S. Corezzi, C. De Michele, E. Zaccarelli, P. Tartaglia, and

F. Sciortino. Connecting irreversible to reversible aggregation:

Time and temperature. The Journal of Physical Chemistry B,

113(5):1233–1236,2009.

[122] M. D. Haw, M. Sievwright, W. C. K. Poon, and P. N. Pusey.

Cluster-cluster gelation with finite bond energy. Advances in

Colloid and Interface Science,62(1):1–16,1995.

136

Bibliography

[123] S. Babu, J. C. Gimel, and T. Nicolai. Diffusion limited cluster

aggregation with irreversible slippery bonds. Eur. Phys. J. E,

27(3):297–308,2008.

[124] T. A. Witten and L. M. Sander. Diffusion-limited aggregation:

a kinetic critical phenomenon. Phys. Rev. Lett.,47:1400–1403,

Nov 1981.

[125] Silvia Corezzi, Cristiano De Michele, Emanuela Zaccarelli,

Daniele Fioretto, and Francesco Sciortino. A molecular dy-

namics study of chemical gelation in a patchy particle model.

Soft Matter,4(4):1173–1177,2008.

[126] Silvia Corezzi, Daniele Fioretto, and Francesco Sciortino.

Chemical and physical aggregation of small-functionality par-

ticles. Soft Matter,8(8):11207–11216,2012.

[127] G. Helgesen, A. T. Skjeltorp, P. M. Mors, R. Botet, and R. Jul-

lien. Aggregation of magnetic microspheres: Experiments and

simulations. Phys. Rev. Lett.,61:1736–1739, Oct 1988.

[128] N. Yoshioka, Imre Varga, Ferenc Kun, Satoshi Yukawa, and

Nobuyasu Ito. Attraction-limited cluster-cluster aggregation

of ising dipolar particles. Phys. Rev. E,72:061403, Dec 2005.

[129] Masaru Suzuki, Ferenc Kun, and Nobuyasu Ito. Cluster-

cluster aggregation of ising dipolar particles under thermal

noise. Phys. Rev. E,80:021402, Aug 2009.

[130] M. N. Popescu, H. G. E. Hentschel, and F. Family. Anisotropic

diffusion-limited aggregation. Phys. Rev. E,69:061403, Jun

2004.

[131] R. C. Ball. Diffusion limited aggregation and its response to

anisotropy. Physica A: Statistical Mechanics and its Applications,

140(1):62–69,1986.

[132] J.-M. Jin, K. Parbhakar, L. H. Dao, and K. H. Lee. Gel for-

mation by reversible cluster-cluster aggregation. Phys. Rev. E,

54:997–1000, Jul 1996.

[133] B. Bharti, F. Kogler, S. H. L. Klapp, C. K. Hall, and O. D. Velev.

preprint.

[134] M. P. Allen and D. J. Tildesley. Computer simulation of liq-

uids. Oxford University Press, (New York), 2009.

[135] D. L. Ermak. A computer simulation of charged particles in

solution. i. technique and equilibrium properties. J. Chem.

Phys.,62(10):4189,1975.

[136] J. P. Hansen and I. R. McDonald. Theory of simple liquids

(third edition). Academic Press, (Burlington), 2006.

137

Bibliography

[137] M. S. Wertheim. Exact solution of the percus-yevick integral

equation for hard spheres. Phys. Rev. Lett.,10:321–323,1963.

[138] M. Baus and J. L. Colot. Thermodynamics and structure of

a fluid of hard rods, disks, spheres, or hyperspheres from

rescaled virial expansions. Phys. Rev. A,36:3912–3925, Oct

1987.

[139] X. Guo and U. Riebel. Theoretical direct correlation function

for two-dimensional fluids of monodisperse hard spheres.

Journal of Chemical Physics,125(14):144504–144508,2006.

[140] Franz Schwabel. Statistische mechanik. Springer,1st edition,

2000.

[141] Gabriel M. Range and Sabine H. L. Klapp. Demixing in simple

dipolar mixtures: Integral equation versus density functional

results. Phys. Rev. E,70:031201, Sep 2004.

[142] David J. Griffiths. Introduction to electrodynamics (3rd ed.).

Addison-Wesley,1999.

[143] Andreas Engel and Peter Reimann. Thermal ratchet effects in

ferrofluids. Phys. Rev. E,70:051107, Nov 2004.

[144] Martial Mazars. Long ranged interactions in computer simu-

lations and for quasi-2d systems. Physics Reports,500(2):43–

116,2011.

[145] David Griffiths. Introduction to electrodynamics. (Addison-

Wesley), 2012.4th ed.

[146] James E. Martin, Robert A. Anderson, and Rodney L.

Williamson. Generating strange magnetic and dielectric in-

teractions: Classical molecules and particle foams. Journal of

Chemical Physics,118(3):1557–1570,2003.

[147] N. Osterman, I. Poberaj, J. Dobnikar, D. Frenkel, P. Ziherl, and

D. Babi´

c. Field-induced self-assembly of suspended colloidal

membranes. Phys. Rev. Lett.,103:228301, Nov 2009.

[148] K. M ¨

uller, N. Osterman, D. Babi`

c, C. N. Likos, J. Dobnikar,

and A. Nikoubashman. Pattern formation and coarse-graining

in 2d colloids driven by multiaxial magnetic fields. Langmuir,

30(18):5088–5096,2014.

[149] Maarten M. van Oene, Laura E. Dickinson, Francesco Pedaci,

Mariana K¨

ober, David Dulin, Jan Lipfert, and Nynke H.

Dekker. Biological magnetometry: Torque on superparam-

agnetic beads in magnetic fields. Phys. Rev. Lett.,114:218301,

May 2015.

138

Bibliography

[150] Sabine H. L. Klapp and Martin Schoen. Reviews in computa-

tional chemistry volume 24.John Wiley & Sons, (New Jersey),

2007.

[151] Francesco Sciortino and Emanuela Zaccarelli. Reversible gels

of patchy particles. Current Opinion in Solid State and Materials

Science,15(15):246–253,2011.

[152] Amit Goyal, Carol K. Hall, and Orlin D. Velev. Phase diagram

for stimulus-responsive materials containing dipolar colloidal

particles. Phys. Rev. E,77:031401, Mar 2008.

[153] F. Kogler and S. H. L. Klapp. Lane formation in a sys-

tem of dipolar microswimmers. EPL (Europhysics Letters),

110(1):10004,2015.

[154] James M. Phillips, L. W. Bruch, and R. D. Murphy. The two-

dimensional lennard-jones system: Sublimation, vaporization

and melting. J. Chem. Phys.,75(0):5907,1981.

[155] B. Smit and D. Frenkel. Vapor-liquid equilibria of the tw-

dimensional lennard-jones fluid(s). J. Chem. Phys.,94(0):5663,

1991.

[156] John G. Kirkwood and Frank P. Buff. The statistical me-

chanical theory of solutions. The Journal of Chemical Physics,

19(6):774–777,1951.

[157] F. L. Braga and M. S. Ribeiro. Diffusion limited aggregation:

Algorithm optimization revisited. Computer Physics Commu-

nications,182(8):1602–1605,2011.

[158] Mark Newman. Networks: An introduction. (Oxford Univer-

sity Press), 2010.4th ed.

[159] A. Wysocki and H. L¨

owen. Instability of a fluid–fluid inter-

face in driven colloidal mixtures. J. Phys.: Condens. Matter,

16(14):7209,2004.

[160] Christopher W¨

achtler, Florian Kogler, and Sabine H. L. Klapp.

Lane formation of the two-dimensional lennard-jones fluid.

2016. In preparation.

[161] Bhuvnesh Bharti, Florian Kogler, Carol K. Hall, Sabine H. L.

Klapp, and Orlin D. Velev. Multidirectional colloidal assem-

bly in concurrent electric and magnetic fields. 2016. In prepa-

ration.

[162] Peter J. Krommenhoek and Joseph B. Tracy. Magnetic field-

directed self-assembly of magnetic nanoparticle chains in

bulk polymers. Particle & Particle Systems Characterization,

30(9):759–763,2013.

139

Bibliography

[163] Mao Xiaoming, Chen Qian, and Granick Steve. Entropy

favours open colloidal lattices. Nat Mater,12(3):217–222, mar

2013.10.1038/nmat3496.

[164] Marie-Pierre Valignat, Olivier Theodoly, John C. Crocker,

William B. Russel, and Paul M. Chaikin. Reversible self-

assembly and directed assembly of dna-linked micrometer-

sized colloids. PNAS,102(12):4225–4229,2005.

[165] Andrew I. Campbell, Valerie J. Anderson, Jeroen S. van Dui-

jneveldt, and Paul Bartlett. Dynamical arrest in attractive

colloids: The effect of long-range repulsion. Phys. Rev. Lett.,

94:208301, May 2005.

[166] Oliver Pohl and Holger Stark. Dynamic clustering and chemo-

tactic collapse of self-phoretic active particles. Phys. Rev. Lett.,

112:238303, Jun 2014.

[167] Jonathan K. Withmer and Erik Luijten. Influence of hydrody-

namics on cluster formation in colloid-polymer mixtures. The

Journal of Physical Chemistry B,115(22):7294–7300,2011.

140