Active Matter and Choreography at the Colloidal Scale Joseph

Active Matter and Choreography at the Colloidal Scale

Joseph Harder

Submitted in partial fulﬁllment of the requirements for the degree of Doctor of Philosophy in the Graduate School of Arts and Sciences

COLUMBIA UNIVERSITY

Active Matter and Choreography at the Colloidal Scale

Joseph Harder

In this thesis, I present numerical simulations that explore the applications of self-propelled particles to the field of self-assembly and to the design of ‘smart’ micromachines. Self-propelled particles, as conceived of here, are colloidal particles that take some energy from their surroundings and turn it into directed motion. These non-equilibrium particles can move persistently for long times in the same direction, a fact that makes the behavior of dense and semi-dilute systems of these particles very different from that of their passive counterparts. The first section of this thesis deals with the interactions between passive components and baths of hard, isotropic self-propelled particles. First, I present simulations showing how the depletion attraction can be made into a short ranged repulsive, or long ranged attractive interaction for passive components with different geometries in a bath of self-propelled particles, and show how the form of these interactions is consistent with how active particles move near fixed walls. In the next chapter, a rigid filament acts as a flexible wall that engages in a feedback loop with an active bath to undergo repeated folding and unfolding events, behavior which would not occur for a filament in a passive environment. The subsequent chapters deal with self-propelled particles that have long ranged and anisotropic interactions. When the orientations of active particles are coupled, they can undergo remarkable collective motion. While the first chapter in this section begins with a discussion of how active disks interacting via an isotropic potential consisting of a long ranged repulsion and short ranged attraction self-assemble into living clusters of controllable size, I show how replacing the disks with anisotropic dumbbells causes these clusters to rotate coherently. In the last chapter, I show that weakly screened active dipoles form lines and clusters that move coherently. These particles can become anchored to the surface of a passive charged colloid in various ways that lead to two different kinds of active motion: rotations of a corona of dipoles around the colloid, and active translation of the colloid, pushed by a tail of dipoles. Finally, a mixture of many charged colloids and dipoles can reproduce the swarming behavior of the pure dipoles at a larger length scale with coherent motion of the colloids. These are all examples of how activity is a useful tool for controlling motion at the micro-scale. Contents

List of Figures iii

Acknowledgements xii

I The Physics of self-Propelled Particles 1

0.1 Introduction ...... 2

0.2 Generating Motion at the Microscale ...... 3

0.3 Colloidal Phoretic Motion ...... 5

0.4 Non-equilibrium Behavior of self-Propelled Particles ...... 9

II Purely Repulsive Active Brownian Particles 13

1 Introduction 14

2 Active Depletion 19

2.1 Introduction ...... 19

2.2 Model ...... 21

2.3 Results and Discussion ...... 23

2.4 Conclusions ...... 35

3 A Polymer in an Active Bath 36

3.1 Introduction ...... 36

3.2 Model ...... 37

i 3.3 Results and Discussion ...... 39

3.4 Conclusions ...... 45

III Long Ranged Interactions Between Active Brownian Particles 46

4 Introduction 47

5 Active Living Clusters 51

5.1 Introduction ...... 51

5.2 Model ...... 53

5.3 Results and Discussion ...... 55

5.4 Conclusions ...... 64

6 Active Dipoles 66

6.1 Introduction ...... 66

6.2 Model ...... 68

6.3 Results and Discussion ...... 70

6.4 Conclusions ...... 85

Conclusion 87

Bibliography 89

ii List of Figures

0.1 Schematic showing two different mechanisms of phoretic motion. A: A concentration gradient of solute molecules tangent to a surface induces fluid flow. B: A similar gradient, but with charge density can also lead to flows. (Image taken from [45]) ...... 5

0.2 Top Left: Diagram of the electrophoresis powering a self-propelled Pt-Au

rod immersed in a bath of H2O2. [8] Top Right: A: Light activated active particle made up of a hematite cube embedded in a polystyrene bead. B: Living clusters formed when the particles are active. C: When particles are

passive, the clusters break up. [14] Bottom: A: Pt-SiO2 bead in hydrogen peroxide as it approaches a wall. B: Model of the reactant concentration near the particle surface [50]...... 7

0.3 Trajectories of Pt-silica bead microswimmers at diﬀerent fuel concentrations over the same elapsed time. [60]...... 9

1.1 An idealized active particle as it collides with a wall (∆t1), slides along the wall (∆t ), and then, when time is run backwards ( ∆t ), does not 2 − 1 return to its original position...... 15

1.2 A: Series of funnels takes uniformly distributed motile bacteria and concentrates them in the rightmost reservoir [102]. B: Ramps with steeper inside or outside faces concentrate colloids pushed by motile bacterial baths [116]. C: Motile E. Coli are trapped in the teeth of a ratchet motor and cause it to spin in one direction [106]...... 16

iii 1.3 Two explanations for the depletion attraction. A: Non-uniform distribution around the large colloids leads to a pressure that pushes the particles together. B: Overlap between excluded volume regions increases available area of the small particles, increasing their entropy. [118] ...... 17

2.1 Schematic representation of two colloidal disks in a bath of active particles. The smaller, active components move according to Eq.1. The persistent

force Fa acts along a deﬁned axis (as shown by the arrow as well as the colors, where red corresponds to the back of the particle and yellow the front)...... 23

2.2 Radial distribution function g(r) of the large colloids for three diﬀerent

values of self-propulsion βFaσ. In a passive bath (blue, solid) the expected peak signiﬁes a short ranged attraction between colloids. In an active bath (dashed/dotted lines), g(r) takes values which are less than 1, suggesting a repulsion between the colloids. This repulsion increases with the bath’s activity...... 25

2.3 Eﬀective rescaled forces F/Fa experienced by two colloidal disks as a function of separation for diﬀerent values of depletant’s activity. (a) Shows

the result for σc = 5σ and (b) for σc = 10σ. For both sets of simulations φ = 0.1. Rescaling is only applied as long as βF σ = 0. Positive values b a 6 correspond to a repulsion, which clearly dominates any depletion driven interaction when the bath is active. The larger the active force and the larger the colloid-to-depletant size ratio is, the stronger the repulsion. . . 26

iv 2.4 (a) Sketch showing the eﬀective forces exerted on the disks by the active particles in the two diﬀerent regions. (b) Snapshot from a simulation of

two colloidal disks of diameter σc = 10σ, at βFaσ = 50. The two large disks experience a net repulsion due to the trapping of active bath particles. As in (a), the yellow portions of the particles indicate the direction along which the propulsive force is applied.[134] ...... 27

2.5 Time averaged density maps of active particles for βFaσ = 50, φb = 0.1

around two colloidal disks of diameter σc = 10σ at a center-to-center distance of 10σ (a), and 15σ (b). In both cases, it is clear that active particles aggregate on the colloid surfaces. When the colloids are in contact, a region of high active particle density can be seen near the eﬀectively concave surface between the colloids. The scale bar corresponds to the density of the bath particles, and goes from a minimum of zero in the space occupied by the colloids to a maximum value near the point where the colloids meet in (a)...... 28

2.6 Eﬀective force between two disks in contact and in the presence of active

depletants as a function of self-propulsion βFaσ for disks of diameter σc =

10σ and 20σ, at φb = 0.1 and box side length L = 150σ. The inset shows that for moderate active forces, an enhanced attraction is observed. . . 29

2.7 Measured force (scaled by the active force) felt by two rods in a bath of active particles as a function of their separation for diﬀerent strengths of self-propulsion. (a) Shows the behavior for small separations, (r σ) ∼ while (b) shows the curve for large separations (r σ). The inset in (b)

shows how the decay length `p of the ﬁtted exponential curves in the limit of large separations (a measure of the interaction range) grows linearly

with the particle persistence length d/σ = βFaσ/3...... 30

v 2.8 Time averaged density maps of active particles around larger rods. (a)

βFaσ = 0 at rod separation r = 4σ, (b) βFaσ = 50 at rod separation

r = 2σ, (c) βFaσ = 50 at rod separation r = 4σ, and (d) βFaσ = 50 at rod separation r = 6σ. In the passive case (a), the bath particle density is uniform across the simulation box, resulting in no long range interaction between the rods. (b) shows the bath particle density proﬁle around two rods in contact. (c) shows particles getting trapped in between the rods and giving rise to a net repulsion. In (d), the density of bath particles on the outside of the rods is greater than that inside, leading to the observed long-ranged attraction. As before, the scale bar shows the particle density, and goes from a minimum in the regions excluded by the rods to a maximum at points near the rod surfaces...... 31

2.9 Eﬀective force between two rods in contact with each other and in the

presence of active depletants as a function of self-propulsion βFaσ for

rods of length ` = 10σ, at φb = 0.1 and box side length L = 77σ. The dashed line is a linear ﬁt to the net force at high bath activity and shows that our simulation results are consistent with Eq. 2.4...... 33

2.10 Eﬀective force between two rods in contact with each other and in the

presence of active depletants as a function of rod length for βFaσ = 50,

at φb = 0.1 and box side length L = 200σ. A linear ﬁt is also plotted, showing that our simulation results are consistent with Eq. 2.5...... 33

3.1 Time averaged radius of gyration of a ﬁlament, normalized by its value cal-

culated in the absence of activity (Fa = 0), as a function of the propelling

force Fa for diﬀerent ﬁlament rigidities, κ...... 39

vi 3.2 Snapshots from our simulations depicting the filament at various stages of folding and unfolding (a) (f). The propelling force F acts along the → a axis connecting the poles of the two hemispheres we used to depict the active particles in the blue-to-red direction. Panel (a) shows explicitly our convention for the definition of inner and outer regions of a bent filament. For the sake of clarity, active particles away from the filament have been rendered with a semi-transparent filter.[134] ...... 42

3.3 Probability distribution of the radius of gyration P (Rg) for increasing

values of the propelling forces Fa. Panel (a) shows the results for a fully ﬂexible chain, panel (b) those for a polymer with rigidity βκ = 20, and panel (c) those for the stiﬀest polymer considered in this work, βκ = 90. 43

3.4 Panels (a)-(c). Time averaged probability distribution of ﬁnding a rigid

ﬁlament, βκ = 90, with a given radius of gyration Rg and asphericity A for

diﬀerent values of Fa. When βFaσ = 0, panel (a), the ﬁlament is almost exclusively extended (A 1, R 22σ). At βF σ = 20, panel (b), two ∼ g ∼ a distinct peaks can be seen at large values of the asphericity, where one

has roughly half the Rg of a fully extended chain. These peaks correspond to the folded (hairpin) and unfolded (extended) states. At larger values

of Fa, panel (c), the distribution smears out and the filament can more freely explore a wider variety of configurations. Note how the distribution in panel (c) is very similar to that in panel (d) corresponding to a fully flexible filament, βκ = 0, in an active fluid having the same propelling

force βFaσ = 80. All distributions have been divided by their respective maximum values so that they can be represented with the same color scale. 44

vii 4.1 A-D Snapshots from simulations performed with the Vicsek model [11]. A: Initial motion at low density and high noise. B: Finite sized ﬂocks at low density and low noise. C: Little coherent motion at high noise and density. D: Ordered motion at high density and low noise. E: In one step of the Vicsek model, the red particle adopts the average direction of its nearby neighbors...... 48

4.2 Phase diagram of active rods. Dilute (D), Swarming (S), Laning (L), Bio- nematic (B), Turbulent (T) and Jammed (J) structures emerge depending on the density and rod aspect ratios. [150] ...... 49

4.3 A disordered ﬂuid, collective swarming, lines and phase separated clusters form in a system of self-propelled particles with dipole-like interactions. [68] 49

5.1 A summary of all the structures observed in our simulations of active spheres as a function of P´ecletnumber Pe and area fraction φ...... 56

5.2 Self-propelled spheres. Top: snapshots at φ = 0.24, bottom: φ = 0.55 (See Fig. 1 as a reference). From left to right: Pe = 0, 10, 20. At Pe = 10, meso-phases of living clusters are observed at lower densities, and living stripes are seen at higher densities...... 56

5.3 Cluster size distributions as a function of Pe are shown for spheres at φ = 0.24. At Pe 5, ﬂuids of living clusters are seen, for which particle ' exchange occurs. At Pe 15, there is no longer a stable cluster size and ' the system is a ﬂuid. The inset shows the average cluster size for increasing Pe...... 57

5.4 Cluster-to-cluster bond order parameter Ψ6 as a function of Pe for φ = 0.24 of the mesophase formed by active spherical particles...... 58

viii 5.5 A summary of all the structures observed in our simulations of active dumbbells as a function of P´ecletnumber Pe and area fraction φ. In contrast to the spheres, at low values of Pe, clusters rotate, each with a speciﬁc handedness ,and translate...... 59

5.6 Dumbbells at φ = 0.24 showing the variety of phases formed. From left to right: Pe = 0 (static clusters), Pe =5 (rotating clusters), Pe =15 (living clusters), Pe =40 (ﬂuid phase)...... 60

5.7 Cluster size distributions for dumbbells at φ = 0.24 and ε = 50kBT with increasing Pe. The cluster distribution shifts towards larger clusters and broadens with increasing activity until the system becomes a ﬂuid. The inset shows the average cluster size for increasing values of Pe...... 61

5.8 The angular velocities for several rotating dumbbellar clusters containing

roughly 20-30 dumbbells from a system at φ = 0.24, ε = 50kBT , and Pe = 20 are shown. The clusters show unidirectional rotation over the duration of the simulation. Clusters shown here have been selected to have velocities around the average and at the tail of the distribution. . . 62

5.9 Average rotational speed ω of a cluster of dumbells of size n, as measured h i in our simulations, for Pe=20. The solid line is the expected theoretical dependence ω 1/n ...... 63 h i ∼

6.1 Structural self-assembly diagram showing typical conﬁgurations acquired by active dipolar janus particles for a range of activities and particle area fractions...... 70

ix 6.2 The average ADJP chain length, normalized by the length of the simu-

lation box L, as a function of the self-propelling force Fa, for diﬀerent values of the particles’ area fraction φ . n is the average size of clusters b h i which are constructed by considering particles that are in contact with each other and for which their mutual orientation is oﬀ-set by an angle that is not larger than π/6...... 72

6.3 A: Snapshot from simulations of a rotating brush of ADJPs chains over the surface of a PCC (see also supplementary movie S1). B: Idealized sketch showing how lateral interactions between ADJPs can lead to a tilt of the chain axis with respect to the surface of the PCC. This is responsible for the tangential active forces that lead to the rotation of the brush. C: Long tails of ADJPs developing over a PCC at low are fractions and intermediate activities. (supplementary movie S2) D: Short chains developing over a PCC at low are fractions and large activities...... 73

6.4 Anisotropy of PCC-ADJP complexes, expresses in terms of ∆com (see text

for a deﬁnition of ∆com), as a function of the strength of the active forces

Fa and for diﬀerent values of the ADJP area fraction φb...... 75

6.5 Structural phase diagram showing where a charged colloid is fully and partially coated. As the area fraction increases at a given bath activity, the charged colloid is more readily fully coated. At the border between the two regions, the charged colloid can alternate between being fully and

partially coated, especially at higher ADJP area fractions, φb. The line separating the two regimes is a guide to the eye...... 76

6.6 Total angular displacement of the ADJP brush (corona) around a charged

colloid as a function of time for φb = 0.03. The inset shows the average time t spent by the corona rotating with a given handedness as a function h i

of the self-propelling force Fa...... 77

x 6.7 Typical trajectories of a PCC partially coated by a bundle of ADJPs for

four diﬀerent values of Fa over the same elapsed time. As the bath particle activity increases, the trajectories become rougher. This trajectories were

takes at a ADJP area fraction φb = 0.01...... 78 6.8 A: Orientational correlation functions for the normalized displacement vec-

tor u(t) of a partially coated PCC-ADJP complex at φb = 0.01. B: Effec- tive rotational diffusion coefficients of a PCC-ADJP complex measured by fitting an exponential to the orientational correlation functions presented in A. C: Rotational diffusion coefficients of the PCC-ADJP complex as a function of the chain length ∆ . The dotted line is a power-law fit to ∝ com the data...... 79 6.9 Top: Snapshots of steady state configurations taken from simulations for

diﬀerent values of PCC area fraction φc at a ﬁxed value of βFaσ = 20

and number of ADJP per PCC, χ = 20. φc increases when going from A to C (supplementary movie S3). Bottom: Snapshots of steady state conﬁgurations taken from simulations for diﬀerent values of χ at constant

PCC area fraction φc = 0.2 and a fixed value of βFaσ = 20. χ increases when going from D to F...... 81 6.10 Matrix of figures showing the order parameter ψ associated with the motion of the PCC-ADJP complexes for three different values of the number of ADJP per PCC, χ; χ = 20, χ = 40 and χ = 60. Each panel shows the

same data for diﬀerent values of PCC area fractions φc and strength of

the active forces Fa...... 83 6.11 Distribution of the number of ADJP chains bound to the surface of the

PCCs, P (λ), for a passive and an active (βFaσ = 10) suspension at different values of number of ADJPs per PCC, χ...... 84

xi Acknowledgements

Throughout my time in his lab, Professor Angelo Cacciuto has created an environment that was rich with creativity, collaboration, and coﬀee, all things for which I extend my deepest gratitude. Most importantly, he was endlessly curious, and had the remarkable ability of picking out the most important thread of whatever knotted problem I showed him. I also extend my thanks to my other committee members, Profs. Bruce Berne, Sanat Kumar, David Reichman, and Xavier Roy, both for agree- ing to serve on my Defense committee, as well as for teaching me through various courses, seminars and conversations.

My co-workers, Andela, Clarion, and Stewart are nothing short of amazing. They were simultaneously supportive and challenging in a way that made them a joy to work with. In addition to being great Pokemon Go players, they are all exemplary scientists who I’ve tried to emulate. The largest group conﬂicts in my 5 years have been over the thermostat setting and the deﬁnition of the word ‘truce’, which speaks to both my group-mates’ characters, as well as Angelo’s care in building a group.

They say your friends are the family you choose, and while I was not on the admis- sions committee and so did not technically choose the students that were admitted between 2009 and 2016, I would like to thank whoever did. My friends’ successes in the laboratory have only been matched by their generosity, humor, and sense of fun outside of it. I would like to thank all of the members of every sports team that I’ve been on (good and bad), and most of the group chats that I’ve been in (good and bad). Particular thanks go to Dat, Clarion, Stew and all the members of my year,

xii Andrew, Trevor, Nate, Tim, Glen, Tim, and Andela. My wonderful girlfriend, Mimi, has been amazingly supportive, understanding, and has even laughed at some of my jokes! I feel so lucky to have met her, and look forward to learning more from her than I already have, and experiencing New York, and the world together. Finally, I would like to thank my parents for giving me the freedom to playfully explore the world and allowing (even encouraging, sometimes) some of the more py- rotechnical expressions of this exploration that eventually led me to study chemistry. My grandparents’ lives are testaments to the joy of living through and for loved ones, and my brother’s compassion and work ethic inspire me constantly. Every good quality that I possess, I owe to my family.

xiii For my family, in every sense of the word.

xiv Part I

The Physics of self-Propelled Particles

1 0.1 Introduction

The benefits of cooperation are taught to children at a very young age, often using a school of fish as an example, where a large group of small fish moves in concert to scare a single large fish. Lessons from the behavior of fish provide an example of what choices can be made to enhance cooperation, and which make achieving a desired outcome easier. Unlike humans, fish do not make the conscious decision that swimming in more or less complex, concerted patterns leads to material benefits (protection from predators [1–3], foraging advantages [4], and possibly even more efficient swimming [5]), instead, rules for how to move in schools emerged over the course of the fishes’ evolution. Whatever mental capabilities you chose to ascribe to fish, however, the fact that even simpler swimming bacteria also exhibit collective behavior [6–9] suggests that there are simple rules that govern the motion of these kinds of ‘particles’ that move persistently in one direction (that I will refer to as self-propelled, or active particles in this thesis). These simple rules were first enu- merated in 1987 in the context of computer generated graphics of flocking animals [10], and then studied further in 1995 with the development of the Vicsek model [11]. Fundamentally, these models are concerned with the steady state, non-equilibrium behavior of particles that are made to move along an axis given by some internal rather than externally imposed broken symmetry. These pioneering works led to a rapid development of many methods, both coarse and fine grained [12, 13], to study self-propelled particles, a class of particles with a growing number of experimental realizations and applications [14–20].

This thesis focuses on the behavior of a recently developed type of these inherently non-equilibrium particles called artiﬁcial microswimmers. While a single one of these inherently non-equilibrium swimmers has complex and interesting behavior, by combining large numbers of these swimmers with passive components we can design systems that behave completely diﬀerently from their passive, equilibrium equiva-

2 lents. In this thesis, I study these self-propelled particles computationally as coarse grained Active Brownian Particles (ABP) with the goal of understanding their interactions with each other and with other passive components. The ultimate goal is to understand how particle activity can be used to control self-assembly and to design ‘smart’ micromachines that are powered by ‘dumb’ microswimmers.

0.2 Generating Motion at the Microscale

Nature has evolved a huge range of self-propulsion mechanisms, from the self-propelled motor proteins that power muscle contractions [21, 22] to the organisms to which those muscles give the ability to move. Broadly defined self-propelled particles exist at every biologically relevant length scale suggesting that there are some fairly universal benefits of particles being able to generate their own motion [23–25]. Absent this semi-autonomous motion, a living entity must rely on external forces to provide fuel sources, via either fluid flows that bring fuel to a stationary entity, or flows that move the entity itself. These forces come from many sources depending on the size and nature of the organism in question. Large, immobile organisms depend on macroscopic fluid flows, but if an organism is small enough (σ 1µm), it can also take advantage ∼ of diffusion [26]. While there are certainly organisms that rely on passive transport, the transport of these particles has significant disadvantages when compared to that of active particles. The first disadvantage is that diffusion scales with the inverse of particle size, so it can be very slow for large particles. The second is that diffusion is random, making it inefficient in a world where biological processes demand efficiency [27, 28]. Self-propelled particles move orders of magnitude faster than their passive counterparts (as shown in Table 0.1) solving the problem of slow diffusion. There are also propulsion mechanisms that couple a particle’s orientation to its environment, thus solving the problem of undirected motion [28, 29].

3 Microswimmer Swimming Mechanism Swim Speed

1 E. Coli Bundle of Flagella 20-30µs− 1 Pt-Au Janus rod Self-electrophoresis 10 µs− Hematite cube embedded Self-diﬀusiophoresis when 1 in polystyrene bead exposed to light 15 µs−

Table 0.1: Swim speeds of some common self-propelled particles.

How self-propelled particles move depends in large part on the particle size and the properties of the fluid through which they are moving, two quantities that can be combined in a dimensionless hydrodynamic parameter called the Reynolds number. The Reynolds number is defined as the ratio of the inertial forces to viscous forces (Re = uL/ν) in a given fluid. Humans typically live at high Reynolds number (Re 106 in water) where the inertial motion of immersed bodies dominates over ∼ the viscosity of the fluid. These are the fluid dynamics that we are generally familiar with, where objects put into motion in water will continue to move for some significant time after any propelling force is removed. At small length scales, the velocity of the particles is small compared to the viscosity of the fluid. E. coli, a classic example

5 of a biological self-propelled particle has Re 10− in water, and therefore exists in ∼ an entirely different regime. A person swimming in honey is an illustrative (though still inadequate) example of the hydrodynamic forces that govern the motion of such a bacterium or similarly sized colloid. In this laminar flow regime, particles that are put into motion stop almost instantaneously when any force is removed, a fact that has significant bearing on how particles can self-propel. In order to swim at low Reynolds number, particles cannot rely on time-symmetric (or reciprocal) swimming modes to generate net motion. The requirement for time-asymmetric motion in high viscosity fluids is known as the Scallop Theorem [30], a reference to the fact that scallops, which can move at high Reynold number by repeatedly slowly opening and quickly closing their shells, would not be able to swim at low Reynolds number.

4 Figure 0.1: Schematic showing two different mechanisms of phoretic motion. A: A concentration gradient of solute molecules tangent to a surface induces fluid flow. B: A similar gradient, but with charge density can also lead to flows. (Image taken from [45])

Self-propelled particles use several diﬀerent non-reciprocal propulsion mechanisms to generate motion ranging from rotating helical tails [31–35], to cilia [36–39] in the case of biological systems, and self-generated concentration gradients [14, 19, 40–44] in the case of colloidal particles.

0.3 Colloidal Phoretic Motion

In this thesis, we are principally concerned with the design of systems that include synthetic microswimmers, so a brief description of some of these particles’ common propulsion mechanisms follows. Taking advantage of active motion in non-biological systems requires using controllable synthetic self-propelled particles, and preferably many such types of particles that can be used in many diﬀerent environments. Fortu- nately, in the past 15 years, numerous synthetic microswimmers have been developed [46]. Most of these are micron sized, and rely on some sort of local self-induced

5 gradient to generate motion. Diffusiophoretic motion, a long known phenomenon explaining the spontaneous motion of particles or flow of fluid due to solvent concentration gradients [47–49], can be explained by accounting for the forces in a fluid that emerge when there is a tangential concentration gradient of solute molecules along a flat surface to which they adsorb [45]. Forces from attractive interactions between the solute and the surface must be balanced against the viscous stress due to the different solute concentrations along the tangent to the surface. Assuming that there is no slip velocity at the surface, and that the velocity of the solute is zero in the bulk solution, a flow emerges along the surface from high solute concentration to low. If we take the surface to be a particle that is large compared to the solute and allow it to move, in the reference frame of the bulk fluid, the particle moves opposite the fluid flow from low solute concentration to high. The direction of motion can be reversed when the solute-surface interaction is energetically unfavorable [43], but a relatively strong interaction is required to generate any motion at all. Self-phoretic motion occurs when a particle induces a solute concentration gradient tangentially around its own surface, spontaneously breaking the local symmetry of the fluid around the particle. This concentration gradient can be created in many ways, so active particles can be used in many different experimental systems.

The earliest example of a synthetic microscopic self-propelled particle is a Pt-Au Janus rods immersed in a solution of H O [8]. These 2 micron rods swim by self- 2 2 ∼ 1 electrophoresis, with the Pt coated end facing forward, at roughly 10µm s− where the swim speed can be controlled with the concentration of hydrogen peroxide. An electrochemical redox reaction occurs along the length of the rods with oxidation of hydrogen peroxide into oxygen gas and protons occurring at the platinum end and reduction to water taking place at the gold patch. The protons that are produced by this reaction ﬂow in solution in the same direction as the current inside the rod (from Pt to Au) which induces an “electrophoretic” motion of the particle in the

6 Figure 0.2: Top Left: Diagram of the electrophoresis powering a self-propelled Pt-Au rod immersed in a bath of H2O2. [8] Top Right: A: Light activated active particle made up of a hematite cube embedded in a polystyrene bead. B: Living clusters formed when the particles are active. C: When particles are passive, the clusters break up. [14] Bottom: A: Pt-SiO2 bead in hydrogen peroxide as it approaches a wall. B: Model of the reactant concentration near the particle surface [50]. opposite direction (Fig. 0.2A). A more recent example of a synthetic active particle is a polystyrene bead coated with a Pt patch immersed in a bath of hydrogen peroxide. These were thought to move by neutral diffusiophoresis (where colloids are driven by the catalytic decomposition of H2O2 at the Pt end, and the resulting fluid flow due to a hydrogen peroxide concentration gradient along the particle surface), but more recent experiments suggest that self-electrophoresis causes these particles to move, with a redox reaction occurring along the platinum surface due to differences in thickness or roughness within the patch [45, 51]. In a particularly exciting example of self-propelled motion, a cube of hematite embedded in a polymer sphere immersed in hydrogen peroxide can be controlled with light [52]. Under normal il- lumination, these particles are passive and diffuse accordingly, but when they are illuminated with blue light, catalytic decomposition of hydrogen peroxide takes place and the particles begin to self-propel, giving rise to a system with switchable activity. The concentration gradient around all of these particles is established by the energy

7 contained in chemical reactions, but the reactants are depleted over time, and the particles eventually slow down and become passive. One solution to this problem has been the development of ﬂow chambers that replenish the supply of fuel, but these add experimental complexity to these systems, and make these kinds of particles less applicable outside of the lab.

There are, however, particles that can be made active without the need for an internal energy source. A gold-polystyrene Janus particle illuminated by a laser generates its own local temperature gradient in the surrounding fluid leading to thermophoresis. The gold half of the particle heats the fluid more rapidly than the polystyrene half, and the resulting thermal gradient induces motion in the fluid from high to low temperature, and motion in the particle in the opposite direction [40]. Temperature gradients can also be used to induce diffusiophoretic motion in gold capped colloids immersed in a critical mixture of lutidine and water just below the lower critical solution temperature [43]. As in the previous example, when the particles are illuminated with a laser, the gold patch heats up the surrounding solution more than the polystyrene, but here, the solution demixes in the neighborhood of the gold patch creating a concentration gradient in the fluid. The gold patch can be functionalized with hydrophilic or hydrophobic molecules and so the interactions between the solute and surface are attractive or repulsive, allowing for control over the direction that the particles move.

There are many other ways to engineer self-propelled motion, which are not based on phoretic mechanisms, or even limited to use with microscopic particles. A generic non-reciprocal motion of three connected beads at low Reynolds number has been used to construct a swimmer that moves only because of some external forcing (optical tweezers in one realization) and hydrodynamics [18, 53], leading to a swimmer that could be used in any low Reynolds number environment. Artificial flagella made of DNA linked magnetic colloids can be controlled with external fields to propel larger

8 Figure 0.3: Trajectories of Pt-silica bead microswimmers at different fuel concentrations over the same elapsed time. [60]. objects [35, 54]. A strong applied electric field can cause colloids to spontaneously begin to roll in a process known as Quincke rotation [55], which has been used to study swarms of millions of self-propelled colloids [56]. At larger length scales, bubbles generated by chemical reactions have been used to propel large self-propelled particles [20]. These kinds of particles have been used to swim upstream in blood vessels to deliver clotting agents which help stop bleeding of a wound [57]. Finally, macroscopic particles with asymmetric mass distributions become active granular particles on a shake table, and have been used to model flocking behavior [58, 59].

0.4 Non-equilibrium Behavior of self-Propelled

Particles

Self-propelled particles are often referred to as being “out-of-equilibrium at the single particle level” [44, 61–64]. This is fundamentally due to the fact that each particle individually takes energy from some external source and uses that energy to generate persistent motion in a direction given by an internal axis [8, 15, 17, 40, 41, 43, 45, 63– 69]. Most self-propelled particles are mesoscale objects that reorient randomly, due to internal biological processes in the case of bacteria (run-and-tumble motion) [70–74], or Brownian rotational diﬀusion in the case of synthetic microswimmers [13, 75, 76]. When viewed at a large enough length scale and when the particles’ swim speeds

9 and reorientation dynamics are independent of the swimming direction, the diffusion properties of run-and-tumble and synthetic microswimmers are identical [77, 78]. A simple explanation for the equivalence of these two apparently different swimming mechanisms comes from the fact that both the bacteria and the microswimmers are executing persistent random walks [75], i.e. a random walk with larger step sizes than would be expected for passive particles at the same temperature. Figure 0.3 shows experimental trajectories of Pt-silica janus colloids at different H2O2 fuel concentrations over the same elapsed time interval. The active particles move orders of magnitude faster than their passive counterparts, moreover, as the fuel concentration increases and the particles becomes more active, the persistence length of the trajectories increases even though the particles have the same Brownian rotational dynamics in each case [60]. In fact, one of the main practical differences between active and passive particles is that the persistence length lp of self-propelled particle motion, can be many times larger than a particle diameter and can be controlled with the degree of activity of the particle.

One of the most common models for simulating and analyzing self-propelled particles takes advantage of the fact that self-propelled particles undergo the same rotational dynamics as Brownian particles. Their translational dynamics are also identical apart from an additional active forcing term. We call self-propelled particles simu- lated with this model Active Brownian Particles (ABPs) [13, 75]. The translational motion of these particles is given by:

1 r˙(t) = U(r) + v n + √2D ξ(t) (1) −γ ∇ p where U(r) is the potential that a particle feels, γ is the friction coefficient, D = kBT/γ is the translational diffusion coefficient, vp is an added active term accounting for the self-propulsion velocity of the particle, and n is the orientation of the propelling

10 force that evolves according to equilibrium Brownian rotational dynamics given by:

p n˙ (t) = 2D ξ(t) n (2) r ×

2 where Dr = 3D/σ . For both translations and rotations, solvent eﬀects are given by

ξ(t), a Gaussian white noise term with the properties: ξ (t) = 0 and ξ (t)ξ (t0) = h i i h i j i δ δ(t t0). i,j − Computing the mean squared displacement ( r2(t) ) illustrates some of the dif- h i ferences between passive and self-propelled particles. For an active particle in two dimensions (a constraint that will persist throughout this thesis), a term accounting for the active motion is added to the well known equilibrium expression, r2(t) = 4Dt h i which becomes [75]:

2 2 Drt r (t) = 4Dt + 2 l [D t + 1 e− ] (3) h i p r −

At times that are long compared to the particle reorientation time, a self-propelled

2 particle moves diffusively with an enhanced diffusion coefficient Deff = D + Pe /9 where Pe is the dimensionless Péclet number defined as the ratio between active and diffusive transport (Pe = vpσ/D). An active particle moves ballistically at times that are short compared to the rotational diffusion (but long compared to the timescales of ballistic motion of passive particles). We will see throughout this thesis that the extension of the timescale of ballistic motion has profound consequences on the interactions between baths of self-propelled particles and passive objects. The fact that self-propelled particles are out of equilibrium does not mean that we can’t sometimes use equilibrium thermodynamic and statistical mechanical concepts to describe their behavior. There are certain cases where an effective temperature and diffusion coefficient can be defined for active particles. The density profile of sedimenting self-propelled particles, for example, has been measured to have the same

∆mgz/kBT functional form as that for equilibrium particles, ρ(z) = ρ(0)e− where the temperature for self-propelled particles is given by an eﬀective temperature, Teﬀ =

11 1 + Pe [62, 79]. In these experiments, Pe 5, and so the effective temperature is 9 ∼ roughly 2000K. One of the most fascinating phenomena of active matter, and another key to understanding their behavior, is that they undergo large density fluctuations at the steady state [59, 80–88], even to the point of spontaneously phase separating into a fluid and gas above a critical packing fraction (φ 0.4 for hard, repulsive ABPs). c ≈ These particles can form a coexisting fluid-vapor phase with no attractive or aligning interactions. Self-propelled particles swim more slowly in high density regions and can become trapped, further increasing the local density, initiating a feedback loop where more and more particles particles are trapped until phase separation occurs. An active chemical potential that accounts for this dependence of the swim speed on the local density can predict this unusual phase separation [80, 85, 86, 89, 90]. This theory suggests that there may not be a useful definition of the temperature of a system of self-propelled particles for the context of motility-induced phase separation, but shows how a mechanical analysis of the motion of these particles leads to useful results. Practically, this self-trapping of active particles can be used for switchable self-assembly of macroscopic structures [66]. This thesis can be divided into two sections: In the first, I investigate interactions between purely repulsive isotropic self-propelled particles and passive objects, and how these interactions can be used to direct self-assembly [91], as well as design ’smart’ micromachines [92]. In the second part, I explore the dynamics of self-propelled particles with anisotropic and long ranged interactions [93] which have complex swarming and collective behavior.

12 Part II

Purely Repulsive Active Brownian Particles

13 Chapter 1

Introduction

The first part of this thesis concerns the behavior of purely repulsive, hard Active Brownian Particles and their interactions with other, passive components. Active particles’ long lived ballistic motion can profoundly alter the nature of the collisions between these active and passive components. The translational motion of an active particle is in a sense ‘decoupled’ from its diffusive rotational motion and so when a highly active particle collides with a wall, instead of diffusing away, it can apply a large force on the wall for a long time (until the particle slides off, or rotates so that the propulsion axis no longer points towards the wall). Another way to put this is to say that the motion of active particles near walls is not time symmetric [94–

104]. In Figure 1.1, a sketch of an idealized active particle (with speed vp >> kbT ) colliding with a hard wall shows how as time runs forward, the particle moves from the bulk, into contact with a wall, then applies a force on the wall as it slides along the surface. If time is run backwards after the particle has slid against the wall, it does not return to its original position. This is an example of how self-propelled particles violate detailed balance and gives a sense of the potential practical applications of self-propelled particles when they interact with dynamic environments [105–107].

The most basic consequence of this time asymmetric interaction between active particles and walls is that purely repulsive active particles in a container will not have a uniform spatial distribution [94, 97, 99]. Instead, these particles aggregate at the container edge to a degree controlled by the rotational diﬀusion (via the temperature) and the curvature of the container [94–96, 108].Particle trapping at walls is useful both

14 ∆t ∆t 1 − 1

∆t2

Figure 1.1: An idealized active particle as it collides with a wall (∆t1), slides along the wall (∆t2), and then, when time is run backwards ( ∆t1), does not return to its original position. − for theoretical studies into the physics of active particles, and as a concept for the rational design of systems containing active particles.

Theoretically, the concept of constraining active particles in some region of space has been useful for the deﬁnition of active analogies to an equilibrium pressure [94– 96, 99] (and also probing the limits of when this is can be done). Practically, wall aggregation of self-propelled particles is a principle for the design of microstructures that control, or are controlled by active particles. The shape of obstacles can be chosen so that they induce ﬂows of active particles, and immersed passive objects can be designed so that they are made active by a bath of self-propelled particles. For example, asymmetric saw tooth patterns in the form of funnels, ramps, wedges, and similar shapes have been used to rectify the motion of active particles [102, 109– 115]. In Figure 1.2A, a series of funnels guides motile bacteria in a controllable direction from one reservoir to another. In another example (Fig. 1.2B), a series of concentric ramps with a steeper slope on either the inside or outside face is used to rectify the motion of colloids pushed by a bath of bacteria. The bacteria can push a colloid up the ramp face with a shallower slope more easily, so the colloids can be either expelled from or corralled towards the center of the ramps [116]. In an example of mobile passive structures (Fig. 1.2C), a saw-tooth patterned disk is immersed in a bath of motile E. Coli where the teeth of the disk are steeper on one face than

15 Figure 1.2: A: Series of funnels takes uniformly distributed motile bacteria and concentrates them in the rightmost reservoir [102]. B: Ramps with steeper inside or outside faces concentrate colloids pushed by motile bacterial baths [116]. C: Motile E. Coli are trapped in the teeth of a ratchet motor and cause it to spin in one direction [106].

the other. Because the bacteria moving towards the steeper face of the teeth become trapped, the disk rotates in one direction, becoming a motor powered by bacteria [106].

In this section, I present numerical simulations that explore how active particles interact with static and flexible passive components. In the first example, I show how self-propelled particles modify the entropic depletion attraction between colloids. The depletion attraction occurs whenever there is a mixture of large and small volume excluding particles (with diameters σl and σs respectively) [117, 118]. Large particles have an effective volume which is excluded from the small particles where the closest a small particle can get to a large particle is (σ + σ ) 0.5. When two large particles l s · are brought together the two large particles’ excluded volume, two circles of radius

16 Figure 1.3: Two explanations for the depletion attraction. A: Non-uniform distribution around the large colloids leads to a pressure that pushes the particles together. B: Overlap between excluded volume regions increases available area of the small particles, increasing their entropy. [118]

(σ +σ ) 0.5 separated by σ , overlap, becoming redundant (Fig. 1.3). This reduction l s · l of the excluded volume of the large particles corresponds with an increase of accessible volume for the small particles, and is therefore entropically favorable. This is a short ranged attraction that does not depend on the nature of the large and small particles (only their relative sizes, and overlap geometries) and so can be used for self-assembly in many environments.

The depletion attraction depends on interactions at particle surfaces, so it is not surprising that using self-propelled particles (that aggregate at surfaces) as the small depletants can profoundly alter the nature of this attraction. In fact, the self-propelled particle mediated ‘active’ depletion interaction between two large disks becomes repulsive, whereas for two rods, the interaction has short ranged repulsive and long ranged attractive components. These changes can be explained by how self-propelled particles interact with ﬁxed walls.

In the second chapter of this section, I present simulations of a semi-ﬂexible pas-

17 sive filament immersed in a bath of self-propelled particles. A feedback loop between spontaneous shape fluctuations in the filament and the local bath particle density can lead to cooperative structural changes of the filament. When the filament is rigid, it is repeatedly folded and unfolded by active particles aggregating on its surface, suggesting the development of an actuator powered by active particles. These simulation studies present some of the constantly growing body of research into how self-propelled particles can be used to provide exquisite control over motion at the microscale, with applications ranging from self-assembly to the rational design of micromachines.

18 Chapter 2

The Depletion Interaction in an Active Bath1

2.1 Introduction

Complex ﬂuids and colloidal mixtures are some of the most ubiquitous substances on our planet. Aerosols, foams, emulsions, and gels have countless applications, and are the subject of intense scientiﬁc research across all disciplines. Recently, self-propelled or active colloidal systems have garnered considerable interest because of their exciting rheological properties and unusual phenomenological behavior. The combination of their unique non-equilibrium driving force and the inherent stochastic nature of microscopic processes have endowed active systems with remarkable collective behavior. Self-propulsion is typically achieved by conversion of chemical or ambient free energy into consistent, directed motion. There are numerous examples of biological and synthetic active systems at the nanoscale, including the cytoskeleton of eukaryotic cells [21], bacterial suspensions, and catalytically activated colloidal particles [18, 41, 119, 120].

Although signiﬁcant work has been carried out to understand the phenomenological behavior of self-propelled systems (for a recent review of the subject we refer the reader to [16]), understanding of how immersion into an active environment can aﬀect the dynamic self-assembly pathways of large non-active bodies is still an active area of research. How introducing activity changes well-understood equilibrium systems

19 is a very important question in colloidal science where eﬀective interactions (i.e. solvent mediated interactions) play a crucial role in stabilizing or driving self-assembly of colloidal particles.

One of the simplest ways of inducing a short range attraction among colloids is by taking advantage of the depletion effect which is an effective interaction achieved by the addition of numerous small, non-adsorbing components such as polymers (colloid- polymer mixtures) or colloids (asymmetric binary mixtures). The strength of this attraction increases linearly with the depletants’ concentration (the small particles) and the range is comparable to the depletants’ diameter. This attractive force is purely entropic and is due to an osmotic pressure difference when depletants are expelled from the region between two colloids [117]. In the simplest case where ideal polymers are used as depletants, this attraction takes the general form F (r) = πρk TR2(1 (r/2R)2), where r is the center-to-center distance between two colloids, B − R is the colloidal radius, ρ the density of depletant, and T is the system temperature. If σ is the diameter of the depletant, then the force between two colloids is present as long as r (2R + σ). For sufficiently large attractions, usually controlled by ≤ the depletant’s concentration, phase separation will occur [88, 121]. The overall phase behavior as a function of polymer size and concentration has been thoroughly studied within the Oosawa-Asakura approximation [117, 122–124]. More recently there has also been an effort to characterize this force when the system is no longer in equilibrium [125], and a few studies have considered the phase behavior of active particles in a system of passive depletants [126, 127].

In previous work [94], our group has studied the thermomechanical properties of an active gas, and found that the force acting on two rods kept at a constant separation in the presence of active depletants has an anomalous, non-monotonic dependence on the temperature a notable deviation from the typical behavior of equilibrium − systems. Additional studies [128–130] have further revealed that using active particles

20 as depleting agents can give rise to behavior that is drastically different from that induced by passive depletants. In these works the forces induced by an active bath on two plates of a given length were measured, and layering effects and mid-to-long range interactions between plates were reported to develop when increasing the self- propulsion. Additionally, Angelani et al. [131] have recently shown that the depletion attraction alone cannot describe the effective interactions between passive colloids in a bath of active particles. In a way, it is therefore inaccurate to refer to these forces as active depletion, but we will nevertheless carry on with this nomenclature throughout this chapter to keep the analogy with the parent equilibrium system.

In this chapter, we show how the strength, the sign and the range of this effective interaction can be controlled by tuning the geometry of the passive bodies in a way that is very different from what would be expected of their equilibrium counterparts. Specifically, we characterize how the effective interaction between two colloidal particles varies as a function of the magnitude of the self-propelling force of the depletant and the depletant-to-colloid size ratios. In addition, we highlight the strikingly different nature of the induced interaction when the colloids are rods or disks.

2.2 Model

We consider a two dimensional system of large, passive, colloidal particles of diameter

σc immersed in a bath of smaller active particles of diameter σ and unit mass m at a volume fraction φb. Each active particle undergoes Langevin dynamics at a constant temperature, T , while self-propulsion is introduced through a directional force which has a constant magnitude Fa, along a predeﬁned orientation vector, n = [sin(θ), cos(θ)]. The equations of motion of a bath particle are given by the coupled Langevin equations

21 p 2 p mr¨ = γr˙ ∂rV (r) + 2γ Dξ(t) + F n and θ˙ = 2D ξ (t) (2.1) − − a r r where γ is the friction coeﬃcient, V the total conservative potential acting between any pair of particles, D and Dr are the translational and rotational diﬀusion constants,

2 respectively (with Dr = 3D/σ ). The typical solvent induced Gaussian white noise terms for both the translational and rotational motion are characterized by ξ (t) = 0 h i i and ξ (t) ξ (t0) = δ δ(t t0) and ξ (t) = 0 and ξ (t) ξ (t0) = δ(t t0), respectively. h i · j i ij − h r i h r · r i − Bath particles are disks with diameter σ which interact with each other via the Weeks Chandler Andersen (WCA) potential " 12 6 # σij σij 1 V (rij) = 4 + (2.2) rij − rij 4

1/6 with a range of interaction extending out to rij = 2 σ. Here rij is the center to center distance between any two particles i and j, and is their interaction energy. Suspended colloids are either rods or disks. The large colloidal disks interact with the bath particles through the same WCA potential deﬁned above, with

σij = (σ + σc)/2, where σc is the colloidal diameter. The rods are modeled as rect- angular regions of width σw = 2.5σ and vertical length `, and also repel the particles according to Eq. 6.3, where the separation rij is the smallest distance between the particle the wall. Figure 2.1 shows a sketch of the model for disks. The strength of interaction for both the depletant-depletant interaction and the depletant-colloid interaction is chosen to be ε = 10 kBT . The simulation box is a square with periodic

1 boundary conditions, the Langevin damping parameter is set to γ = 10τ0− (here τ0

3 is the dimensionless time), and the timestep to ∆t = 10− τ0. Each simulation is run for a minimum of 3 107 iterations. All simulations were carried out using the × numerical package LAMMPS [132], and throughout this chapter we use the default dimensionless Lennard Jones units as deﬁned in LAMMPS, for which the fundamental quantities mass m0, length σ0, epsilon 0, and the Boltzmann constant kB are set

22 to 1, and all of the speciﬁed masses, distances, and energies are multiples of these fundamental values. In our simulations we have T = T0 = 0/kB, m = m0, σ = σ0, q 2 and τ = m0σ0 . 0 0

σc σ

Figure 2.1: Schematic representation of two colloidal disks in a bath of active particles. The smaller, active components move according to Eq.1. The persistent force Fa acts along a deﬁned axis (as shown by the arrow as well as the colors, where red corresponds to the back of the particle and yellow the front).

2.3 Results and Discussion

We ﬁrst present the results for disk-shaped colloids. To understand the eﬀective interactions induced by active bath particles on the suspended colloids, we proceed in two ways: (1) We measure the radial distribution function g(r) for a suspension of passive colloidal disks in the presence of the active depletants. In this case the large colloids move according to the Langevin dynamics in Eq.1, but with βFaσ = 0

1 (β (k T )− ), and without the rotational component. (2) We calculate the eﬀective ≡ B force between two colloids by directly measuring the mean force acting on the particles when they are frozen in place as a function of the bath activity, colloidal shape (disks and rods), and colloid separation r. For non-active equilibrium systems, the reversible work theorem provides a simple relationship between the potential of mean force and the radial distribution function,

23 namely U(r) = k T log[g(r)] [133]. Unfortunately, such a relation does not necessar- − B ily hold in the presence of an inherently out-of-equilibrium active bath. Nevertheless, from the g(r) it is possible to extract qualitative information about the sign, strength and range of the interaction. To determine the g(r), simulations were carried out with 100 colloidal disks of diameter σc = 5σ immersed in an active bath at a volume fraction φb = 0.1, and the simulation box is a periodic cube with box length 150. The resulting radial distribution functions are shown in Fig. 2.2. Each simulation was run for over 108 time-steps.

In the passive system with βFaσ = 0, the g(r) presents a large peak at the colloid contact separation as expected for this system which is characterized by a strong depletion attraction. In other words, this peak indicates a significant likelihood of finding two colloids in contact with each other. When the bath is active, however, the radial distribution function is smaller than 1 for small colloid separations, which strongly suggests that there is an effective repulsion between the colloids.

To provide a more quantitative measurement of this repulsion and to better understand its nature, we proceed by performing simulations where two colloids are frozen in place and the force between them is measured directly from their interactions with the active bath particles. All results presented below are obtained at a constant volume fraction φb = 0.1. The net force exerted on the two disks by the bath along the inter-colloidal axis was evaluated for two diﬀerent colloidal sizes σc = 5σ and σc = 10σ. The results are shown in Fig. 2.3.

In a passive bath, the interaction between two large colloids is well understood and is given by the depletion attraction previously discussed. Surprisingly, as the bath becomes increasingly active, the eﬀective interaction between the colloids becomes purely repulsive. This result is consistent with the observed behavior of the g(r). The introduction of activity results in a repulsive force much larger than the depletion attraction observed in passive systems, and grows with the extent of the activity.

ÌÓÔ ÌiØÐe

½º4

½º¾ ½

) ¼º8 r (

¼º6 ¼º4 βFaσ = 0

¼º¾ βFaσ = 10

βFaσ = 20

¼º8 ½ ½º¾ ½º4 ½º6 ½º8 ¾ ¾º¾ ¾º4

r/σc (= 5σ) Figure 2.2: Radial distribution function g(r) of the large colloids for three diﬀerent values of self-propulsion βFaσ. In a passive bath (blue, solid) the expected peak signiﬁes a short ranged attraction between colloids. In an active bath (dashed/dotted lines), g(r) takes values which are less than 1, suggesting a repulsion between the colloids. This repulsion increases with the bath’s activity.

Notice however, that the range of the interaction is rather insensitive to the propulsion strength, and extends to a distance of roughly half the colloidal diameter.

To better understand this phenomenon, we examine the duration of collisions between bath particles and the colloidal disks as well as where along the colloids’ surface these collisions take place. Here, we deﬁne the inner surface of a colloid (Region II Fig. 2.4(a)) as the half circle which lies closer to the center of the other colloid, and the outer surface (Region I Fig. 2.4(a)) as the half circle which is further away from the other colloid. When a bath particle strikes the outer surface of either of the large disks a force is generated with a net component, FI , which pushes the disks toward each other. When a particle strikes the inner surface of either disk it generates a force with components, FII , which pushes the disks away from each other (see Fig. 2.4(a) for a sketch of these forces). The eﬀective force experienced by the two disks is determined by the number of particles at the surface of each region as well as by the average duration of a collision event.

Unlike equilibrium systems for which one expects a particle to bounce oﬀ a wall upon collision, the collision of an active particle with a wall is similar to that of a car

25 6

βFaσ = 0 5

βFaσ = 5 4 βFaσ = 10

¿ βFaσ = 25 a

¾ βFaσ = 50 ½

F/F

¹½

¹¾

½ ½º¾ ½º4 ½º6 ½º8 ¾ ¾º¾ ¾º4

r/σc (= 5σ) ½¾

βFaσ = 0 ½¼ βFaσ = 5

8 βFaσ = 10 βFaσ = 25

a 6

βFaσ = 50 4

F/F

¹¾

½ ½º¾ ½º4 ½º6 ½º8 ¾ ¾º¾ ¾º4

r/σc (= 10σ)

Figure 2.3: Effective rescaled forces F/Fa experienced by two colloidal disks as a function of separation for different values of depletant’s activity. (a) Shows the result for σc = 5σ and (b) for σc = 10σ. For both sets of simulations φb = 0.1. Rescaling is only applied as long as βFaσ = 0. Positive values correspond to a repulsion, which clearly dominates any depletion6 driven interaction when the bath is active. The larger the active force and the larger the colloid-to-depletant size ratio is, the stronger the repulsion. driving into a wall. The active particle will continue to exert a force into a barrier until its propulsion axis begins to rotate, upon which the particle will slide along the wall. For a given strength of self-propulsion, the duration of a collision is controlled by the rotational diffusion, which is governed by thermal fluctuations, and has a strong dependence on the local environment. During collisions, active particles remain in contact with the surface of the larger colloids for some amount of time before sliding off or rotating away. The duration of contact is in large part determined by the geometry of the colloids.

When the colloids are far apart, active particles have equal probability of striking either their inner or outer surfaces, leading to a zero net eﬀective force between

26 Region IRegion II Region I

(a) (b)

Figure 2.4: (a) Sketch showing the effective forces exerted on the disks by the active particles in the two different regions. (b) Snapshot from a simulation of two colloidal disks of diameter σc = 10σ, at βFaσ = 50. The two large disks experience a net repulsion due to the trapping of active bath particles. As in (a), the yellow portions of the particles indicate the direction along which the propulsive force is applied.[134] them. When the colloids are in contact, they form an object characterized by regions of both positive and negative curvature. The outer surfaces have positive curvature, and colliding particles can slide off rather quickly. The inner surfaces have negative curvature and can create a trap [135] for the active particles, greatly increasing the duration of a collision. The result is a net gradient in particle concentration along the colloidal surface, leading to the effective repulsion reported in our simulations (See Fig. 2.4(b) for a snapshot from our numerical simulations).

To determine the surface concentration gradient of bath particles, we compute a density map of the active particles around the disks at large and small separations (Fig. 2.5). As expected, when the disks are sufficiently far apart, there is no significant difference between the particle density on the inner and outer region. When the surface-to-surface separation between the disks is of the order of 2σ however, the inner density is significantly larger than the outer one resulting in the observed repulsion.

We also measure the net force between two disks at contact as a function of the active force for σc = 10σ, this is shown in Fig. 2.6. In the absence of activity, we recover the expected depletion attraction for equilibrium systems, and in the limit of

27 Max (a) (b)

Min

Figure 2.5: Time averaged density maps of active particles for βFaσ = 50, φb = 0.1 around two colloidal disks of diameter σc = 10σ at a center-to-center distance of 10σ (a), and 15σ (b). In both cases, it is clear that active particles aggregate on the colloid surfaces. When the colloids are in contact, a region of high active particle density can be seen near the eﬀectively concave surface between the colloids. The scale bar corresponds to the density of the bath particles, and goes from a minimum of zero in the space occupied by the colloids to a maximum value near the point where the colloids meet in (a).

large active forces, we observe the repulsive behavior discussed above. However, for intermediate values of βF σ [1, 10], we observe a strengthening of the attraction a ∈ between the colloids as a function of Fa. These results suggest that as long as Fa is sufficiently small, the main effect of the propelling force is that of an effective higher temperature of the bath, leading to a strengthening of the depletion interactions. It should be noted that, for this range of active forces, the persistence length of the tra-

1 2 jectory traced by a single active particle, estimated as d (F /γ)D− = (βF σ )/3, ' a r a is signiﬁcantly smaller than the colloidal diameter used in these simulations σc = 10σ. One possible interpretation of this result is that as long as d σ the colloids always c experience an attractive interaction. To see whether this is true, we repeated our simulations for the same range of active forces for two larger colloids with twice the diameter σc = 20σ, placed in contact with each other. Surprisingly, the sign of the interaction switches over at approximately the same value of Fa as for the smaller colloids (see Fig. 2.6 inset). If the d σ argument were correct, the attraction c should persist to larger values of Fa for the larger colloids, but we ﬁnd that this is

28 not the case. One reason for this could be due to the fact that larger colloidal diameters also correspond to larger regions where particles can be trapped. This leads to an enhanced repulsion between the colloids that competes with the strengthened attraction. This enhanced repulsion is easily seen when comparing the two plots in Fig. 2.3 showing that larger colloids experience overall larger repulsive forces. Fi- nally, it should be noticed that the range of the interaction between disks is not very sensitive to the strength of the propelling force, and does not extend to separations

much further than a fraction of the colloidal diameter.

¾¼¼¼ ½¼

σc = ¾¼

σc =

½6¼¼ ½¾¼¼

¾¼

8¼¼ ½¼ βF σ i

h ¼

βF σ 4¼¼

¹½¼

¹¾¼

¼ 5 ½¼ ½5 ¼

βFaσ

¹4¼¼

¼ ½¼¼ ¾¼¼ ¿¼¼

βFaσ Figure 2.6: Eﬀective force between two disks in contact and in the presence of active depletants as a function of self-propulsion βFaσ for disks of diameter σc = 10σ and 20σ, at φb = 0.1 and box side length L = 150σ. The inset shows that for moderate active forces, an enhanced attraction is observed.

We now turn our attention to the case of two colloidal rods. A system composed of two such rods was one of the earliest to be studied in the context of the depletion attraction. As was the case for disks, two rods in a bath of smaller particles experience an entropic attractive force which depends on the size of the excluded area, as well as the size and density of the depletants, and the temperature. This force can be large when compared to that between two suspended disks due to the relatively larger excluded area when rods are in contact. Unlike colloidal disks, rods have no curved

29 surfaces, so active bath particles which come into contact with the surface of a rod are eﬀectively conﬁned to move along this surface until they rotate away or slide to the end of the rod.

When the separation between the rods is small, we observe an oscillating attractive and repulsive force. As also reported in [130], this behavior is due to a competition between the forces exerted by the active particles on the outer surfaces, and the buildup of ordered layer of particles between the rods. (see Fig. 2.7(a)). Surprisingly, at larger separations (Fig. 2.7(b)), a large long-ranged attraction is induced between the rods. In agreement with [130] this attractive effective interaction can be well fit to an exponential and the range of the interaction is controlled by the effective persistence length of the path traced by the active particles d/σ = (βFaσ)/3. The inset of Fig. 2.7(b) shows the linear dependence of the interaction decay length `p as

a function of the particle persistence length d/σ = βFaσ/3. ´bµ áµ 4 βFaσ = 0 0 3 βFaσ = 5 βFaσ = 10 2 Faβσ = 25 0.2 − βFaσ = 50 10 a a 1 0.4 − 8 F/F F/F 0 ℓp 0.6 6 1 − − 4 0.8 2 − 0 25 50 75100 − βFaσ 3 1 − 3 4 5 − 5 15 25 35 45 55 r/σ r/σ Figure 2.7: Measured force (scaled by the active force) felt by two rods in a bath of active particles as a function of their separation for different strengths of self- propulsion. (a) Shows the behavior for small separations, (r σ) while (b) shows the curve for large separations (r σ). The inset in (b) shows∼ how the decay length `p of the fitted exponential curves in the limit of large separations (a measure of the interaction range) grows linearly with the particle persistence length d/σ = βFaσ/3.

Further insight can be obtained by looking at the time-averaged density map of the depletants for diﬀerent rod separations (Fig. 2.8.) In the passive case, the density

30 of bath particles is uniform throughout the simulation box. However, when the bath particles are active, they aggregate on the surfaces of the rods and there is a marked difference in local density on the different rod surfaces. Specifically, once the rods are separated by more than 4σ, there are more bath particles on the outer surfaces ∼ than on the inner ones, resulting in an attractive force between the rods. When the rods are at a smaller separation, the situation is the opposite.

Max (a) (b)

Min Figure 2.8: Time averaged density maps of active particles around larger rods. (a) βFaσ = 0 at rod separation r = 4σ, (b) βFaσ = 50 at rod separation r = 2σ, (c) βFaσ = 50 at rod separation r = 4σ, and (d) βFaσ = 50 at rod separation r = 6σ. In the passive case (a), the bath particle density is uniform across the simulation box, resulting in no long range interaction between the rods. (b) shows the bath particle density proﬁle around two rods in contact. (c) shows particles getting trapped in between the rods and giving rise to a net repulsion. In (d), the density of bath particles on the outside of the rods is greater than that inside, leading to the observed long-ranged attraction. As before, the scale bar shows the particle density, and goes from a minimum in the regions excluded by the rods to a maximum at points near the rod surfaces.

An estimate of how the force exerted on two rods at contact scales with their length and with the strength of the bath activity can be obtained with the following simple

31 argument. In the diffusive limit (i.e. when the length of the rods is sufficiently large such that the particles can diffuse over their surface before sliding off) Fily et. al [96], have shown that for large self-propelling forces the typical time t1 a particle remains in

γ` 2 1 1 contact with a rod scales as t1 ( ) 3 ( ) 3 . During this time the particle will exert ∼ Fa Dr an average force on the rod, that to leading order scales like Fa. The time required for a particle in a container of lateral size L to ﬁnd the rods can be estimated as

L2 γ t2 , which accounts for the probability of ﬁnding the rod when moving at a ∼ ` Fa speed va = Fa/γ across the box. Alternatively, one can think of 1/t2 as the average

2 collision rate between an active particle and the rod. So that 1/t2 = (1/L )Cva, where C = ` is the cross section of the rod.

During this time the particle will exert no force on the rod. The net average force can then be estimated as F NFat1 ,(N is the number of active particles) and for h i ' t1+t2 sufficiently large systems, t t , it can be simplified to F NF t /t , leading to 2 1 h i ' a 1 2 the scaling behavior 1 F 4`5 3 F ρ a (2.3) h i ' γDr where ρ is the number density. In the non-diffusive limit, when the rods are short and the force is so large that an active particle slides off of the surface before any diffusion can occur, t `γ/F , the average force should scale as 1 ∼ a

F ρF `2 (2.4) h i ' a

For long rods or weak propelling forces, the residence time of the particles on the surface is simply controlled by the rotational diﬀusion t1 = 1/Dr. In fact, in these cases a particle leaves the surface as soon as its axis turns away from the surface’s normal, with θ2 = (π/2)2 = 2D tmax as the upper bound. In this limit the average h i r 1 force should scale as F 2` F ρ a (2.5) h i ' γDr

32 Finally, whenever t t (for suﬃciently high densities) one should expect to ﬁrst 1 2 t2 order F Fa(1 ).

h i ' − t1

¾¼¼

¹¾¼¼ ¹4¼¼

βF σ

¹6¼¼

¹8¼¼

¹½¼¼¼

¹½¾¼¼

¼ 5¼ ½¼¼ ½5¼ ¾¼¼ ¾5¼ ¿¼¼

βFaσ Figure 2.9: Eﬀective force between two rods in contact with each other and in the presence of active depletants as a function of self-propulsion βFaσ for rods of length ` = 10σ, at φb = 0.1 and box side length L = 77σ. The dashed line is a linear ﬁt to the net force at high bath activity and shows that our simulation results are

consistent with Eq. 2.4.

¾¼¼

¹¾¼¼

¹4¼¼

¹6¼¼ ¹8¼¼

βF σ

¹½¼¼¼

¹½¾¼¼

¹½4¼¼

¹½6¼¼

¹½8¼¼

¼ ½¼ ¾¼ ¿¼ 4¼ 5¼ 6¼ 7¼ 8¼ ℓ/σ Figure 2.10: Eﬀective force between two rods in contact with each other and in the presence of active depletants as a function of rod length for βFaσ = 50, at φb = 0.1 and box side length L = 200σ. A linear ﬁt is also plotted, showing that our simulation results are consistent with Eq. 2.5.

Deviations from this simple scaling are also expected at moderate and large densities due to the excluded volume interactions between particles. Figures 2.9 and 2.10 show how the force between the rods scales with the strength of the activity and with the length of the rods in our simulations. Our numerical data have been taken at volume fraction φb = 0.1, which is suﬃciently low to prevent any bulk phase separation

33 or aggregation of the active particles, yet, it is large enough to give non-negligible excluded volume effects. The relatively short length of the rods, ` = 10σ, in these simulations implies that Eq. 2.4 should give the most appropriate description for the effective force. This is consistent with Fig. 2.9, that shows a linear dependence of the force with F in the large propulsion limit. When F is small, we expect F a a h i to depend quadratically on Fa. In fact, in this case the rotational diffusion is fast enough to limit the persistence-time of the particles on the colloidal surface, thus Fa will contribute to the average force a quadratic (thermal-like) term corresponding to an enhanced velocity of the particles [108].

The analysis for the dependence on the length of the rods is a bit more complicated. The problem is that the short ` limit is characterized by a short residence time that is inversely proportional to the self-propulsion, it grows linearly with `, and for which clearly t t . As ` becomes larger (up to ` = 80σ), t should scale as 2 1 1 γ` 2 1 1 t1 ( ) 3 ( ) 3 (as long as t1 < 1/Dr), however, as the particles’ residence time ∼ Fa Dr becomes longer, the average number of particles at contact becomes larger, and for moderate volume fractions, the assumption that t t becomes less adequate. To 2 1 complicate matters even further, for large `, excluded volume interactions begin to matter, and local self-trapping of the particles may effectively increase of t1 to values larger than 1/Dr, so that the time to rotationally diffuse away from the surface may become faster than the time required to slide off the surface edge; making Eq. 2.5 more appropriate in this regime. This phenomenon is quite visible in our simulations for the longest rods (` = 80σ), where diffusive correlated motion of linear clusters of active particles over the rod surfaces takes place. At significantly lower densities than the ones considered in this chapter, we would have expected Eq. 2.3 to hold, but at φb = 0.1, all these effects become relevant. We therefore expect a combination of Eq. 2.4 and Eq. 2.5 to provide a good approximation to our data. Indeed, in the long rod limit the average force seems to be well fitted by a linear dependence on `.

34 2.4 Conclusions

In this chapter, we have studied the effective interactions induced by small active components on large passive colloidal particles as a function of the strength of the propelling force of the active bath and of the geometry of the colloids. Our results indicate that the induced colloidal interactions are crucially dependent on their shape, and that while a long ranged, predominantly attractive interaction is induced between rods, disks undergo a purely short range repulsion that grows in strength with the size ratio between the colloid and the active component. Crucial to this difference is the role of curvature, which determines whether passive bodies act as traps or as efficient scatterers of active particles. For instance, we have recently shown how curvature can be exploited to activate C-shaped passive bodies, by creating density gradients across the colloids [108]. Although our study has been performed in two dimensions, the essence of our results should be easily extendable to three dimensions when considering colloidal rods and spheres. This work further highlights the many differences between the effective forces induced by small active components and those produced by the corresponding equilibrium system, and suggests that active depletion can have dramatic consequences on both the phase behavior and the self-assembly of differently shaped colloids, with possible applications in material engineering and particle sorting.

35 Chapter 3

Folding Dynamics of a Polymer in an Active Bath1

3.1 Introduction

When a micron-sized object is immersed in a thermally equilibrated solution, it will undergo Brownian motion due to random collisions with the surrounding fluid. The situation is vastly different when this object is suspended in a bath of active (self- propelled) particles, especially when its shape is not spherical. Indeed, it is well known that self-propelled particles tend to accumulate around highly curved regions of suspended obstacles, thus generating local density gradients which can lead to effective propulsive forces, and anomalous rotational and translational diffusion. Recent work has explored the trapping of active particles on boundaries and microstructures of different shapes and how passive obstacles can become activated in the presence of self-propelled particles [94, 96, 97, 101, 109, 112, 113, 128, 136–142]. More recently, we have explicitly characterized how the curvature of passive tracers is related to the extent of their activation [108]. Needless to say, having a way of selectively activating passive microstructures and their aggregates based on their shape is the first step towards the fabrication of microscopic engines, where the random active motion of one component can be converted into the persistent motion of another. A beautiful example of this process was recently shown in [105, 106].

In this chapter, we consider the case of interactions between a passive structure

36 with a flexible shape and self-propelled particles. Specifically, we study a semi-flexible polymer (or filament) confined in two-dimensions in the presence of a low concentration of active spherical particles. We analyze the behavior of the filament for different values of the strength of the bath activity (characterized by the particles’ self-propelling force) and of the filament rigidity. Recent work on active flexible filaments (i.e. chains formed by active Brownian particles) has shown very interesting phenomenological behavior, including periodic beating and rotational motion [34]. Kaiser et al. [143] have also considered a system similar to ours, but in the limit of a fully flexible polymer. In their study they show how a filament undergoes anomalous swelling upon increasing activity of the bath. Here, we confirm the results reported in [143], and show how adding rigidity to the filament leads to an even more complex non-monotonic behavior, where the chain softens for moderate activities, can collapse into a metastable hairpin for intermediate activities, and behaves as fully flexible chain does in the large activity limit. Our data also indicate that at intermediate activities a rigid filament behaves as a two state system that dynamically ‘breathes’ between a tight hairpin and an ensemble of extended conformations because of a positive feedback loop between the polymer shape and active particle density distribution.

3.2 Model

We perform numerical simulations of a self-avoiding polymer conﬁned in a two- dimensional plane and immersed in a bath of spherical self-propelled particles of diameter σ, whose motion is described via the coupled Langevin equations

p 2 mr¨ = γr˙ + F n(θ) ∂rV + 2γ Dξ(t) (3.1) − a − p θ˙ = 2Dr ξr(t) (3.2)

Here, γ is the friction coeﬃcient, V the pairwise interaction potential between the particles, D and Dr are the translational and rotational diﬀusion constants respec-

37 2 tively, satisfying the relation Dr = 3D/σ . n(θ) is a unit vector pointing along the propelling axis of the particles, and its orientation, tracked by the angular vari- able θ, and is controlled by the simple rotational diﬀusion in Eq 2. F 0 is the a ≥ magnitude of the self-propelling force acting on each particle. The solvent-induced Gaussian white noise terms for both the translational and rotational motion are characterized by the usual relations ξ (t) = 0, ξ (t) ξ (t0) = δ δ(t t0) and h i i h i · j i ij − ξ (t) = 0, ξ (t) ξ (t0) = δ(t t0). The polymer consists of N = 70 monomers h r i h r · r i − of diameter σ, linearly connected via harmonic bonds according to the potential U = ε (r r )2 where r is the center-to-center distance between linked monomers, b b − 0 3 2 1/6 εb = 10 kBT/σ , and r0 = 2 σ is the equilibrium distance. An additional harmonic potential, U = κ (φ φ )2, is used to introduce angular rigidity to the polymer, a − 0 where φ is the angle between adjacent bonds, and φ0 = π is the equilibrium angle.

Values of κ considered in this work range from 0 to 90kBT . The equation of motion of the ﬁlament monomers is given by Eq 1., with Fa = 0.

The excluded volume between any two particles in the system (including the monomers in the chain) is enforced with a purely repulsive Weeks-Chandler-Andersen potential σ 12 σ 6 1 V (r) = 4 + (3.3) r − r 4 with range extending up to r = 21/6σ. Here r is the center-to-center distance between any two particles, and ε = 10 kBT . The simulation box is a square with periodic boundary conditions. All simulations have been performed using LAMMPS[132] at an active particle volume fraction φ = 0.1 (corresponding to 318 active particles), which is sufficiently small to prevent spontaneous phase separation of the active particles [88, 121]. Throughout this work we use the default dimensionless Lennard Jones units as defined in LAMMPS, for which the fundamental quantities mass m0, length σ0, epsilon 0, and the Boltzmann constant kB are set to 1, and all of the specified masses, distances, and energies are multiples of these fundamental values. In our simulations

38 1 we have T = T0 = 0/kB, m = m0, σ = σ0. Finally γ = 10τ0− where τ0 is the q 2 dimensionless unit time deﬁned as τ = m0σ0 . 0 0

3.3 Results and Discussion

2 We begin our analysis by measuring the radius of gyration of the filament, Rg = 1 P 2 2 (ri rj) as a function of the strength of the propelling force of the surround- 2N i,j − ing particles in the bath, Fa, for different values of the chain stiffness κ normalized by the radius of gyration of the same polymer in a passive bath. The results are shown in Fig. 3.1.

1.4 1.3

i 1.2

(0) 1.1 g βκ = 0 R

h 1

/ βκ = 10 i

) βκ = 20

a 0.9

F βκ = 90 (

g 0.8 R h 0.7 0.6 0.5 0 20 40 60 80 100

βFaσ Figure 3.1: Time averaged radius of gyration of a filament, normalized by its value calculated in the absence of activity (Fa = 0), as a function of the propelling force Fa for different filament rigidities, κ.

Consistent with the work by Kaiser et al. [143], a fully flexible self-avoiding filament swells when immersed in an active bath. This is due to the forces exerted on the filament by the active particles as they cluster along different regions of the filament and stretch it by pushing it in different directions. In other words, active particles which slide along the polymer with some potentially very long run length can stretch the polymer by smoothing out kinks. Interestingly, when the filament is somewhat

39 rigid, the opposite behavior is observed and the radius of gyration decreases for small values of Fa, reaching a minimum for intermediate activities before increasing for large propelling forces. This non-monotonic behavior of the radius of gyration with

Fa is reminiscent of that of polyelectrolytes in the presence of multivalent salt [144] as a function of salt concentration.

When an active particle strikes a fully flexible filament, the filament can be stretched from a relatively compact configuration to a more extended one. How- ever, when the filament is stiff, it is mostly stretched to begin with, and any local asymmetries in the number of active particles that cluster on either side of the filament can provide sufficiently large forces to bend it. As the filament begins to bend (say in the middle), different regions of its surface will have different exposure to the active particles. Specifically, the outer region (see Fig. 2(a)) will experience a larger number of collisions than the inner region as the former offers a larger cross section to the active particles. Simultaneously, active particles in the inner region tend to accumulate around the areas with highest curvature [96, 108] in a feedback loop that reinforces the bent structure. The combination of the forces acting on the inner and outer regions of the filament creates a pivot around which the filament can fold into a tight hairpin.

Because there are no direct attractive interactions between different parts of the filament or between the filament and the active particles, the hairpin eventually un- folds. This process is initiated by the active particles located in the inner region whose forces cause the location of the hairpin’s head to slide along the filament, effectively shortening one arm and lengthening the other until full unfolding is achieved. Fluctuations in the number of active particles applying a force on the outer arms of the filament allow it to open slightly, thus letting active particles either out of the inner region or into it which stabilizes or de-stabilizes the filament, respectively. For a range of intermediate bath activities, the filament will therefore breath dynami-

40 cally between a tightly folded and an ensemble of bent but extended configurations. Fig. 3.2 shows several snapshots from our simulations at each stage of the the folding and unfolding process of the filament. Finally, it is worth noting how the position of the minimum in Fig. 3.1 shifts towards larger values of Fa upon increasing the rigidity of the polymer. This is not surprising as larger active forces are required to bend more rigid polymers. How exactly this minimum shifts is unclear because of the nontrivial dependence of the number of particles accumulating on the filament as a function of Fa. In general, an increase in Fa also comes with an increase in the average number of particles pushing on the filament, resulting in a net force which increases faster than linearly with Fa. This has been explicitly shown in our recent work [91] where we computed the force induced on two parallel rods by a bath of active spherical particles. The result is a complex dependence of the force between the rods on Fa, on the active particles density, and on the length on the rods.

Given these complex dynamics, the average value of the ﬁlament radius of gyration provides only a limited characterization of the system. Crucial information about the conformation of the ﬁlament can be extracted from the probability distribution,

P (Rg), of the radius of gyration, for diﬀerent values of κ and Fa. For a fully ﬂexible chain, Fig. 3.3(a), we observe that upon increasing the bath activity, the distribution,

P (Rg), begins to broaden towards larger values of Rg, indicating that the filament is able to reach configurations that are more extended when compared to those expected in the absence of active particles, which is consistent with recent results by Kaiser et al. For rigid filaments, Fig. 3.3(b)-(c), we observe the opposite behavior as a function of the bath activity. Namely, the sharply peaked distribution for large values of Rg that one would expect in the absence of particle activity and that corresponds to a fully stretched filament, broadens and shifts towards lower values of Rg as Fa is increased in the systems. This is indicative of the softening of the chain. Upon further increase of Fa, distinct peaks begin to develop (signature of the formation

41 (a) (b) (c) Outer Region Inner Region

(d)(e) (f)

Figure 3.2: Snapshots from our simulations depicting the filament at various stages of folding and unfolding (a) (f). The propelling force Fa acts along the axis connecting the poles of the two hemispheres→ we used to depict the active particles in the blue- to-red direction. Panel (a) shows explicitly our convention for the definition of inner and outer regions of a bent filament. For the sake of clarity, active particles away from the filament have been rendered with a semi-transparent filter.[134]

of folded conformations). Finally, for even larger values of Fa, we observe a broad distribution covering a wide range of Rg. Strikingly, the distributions in the large activity limit are independent of κ, suggesting that, as we will discuss later on, a rigid filament in a bath at large activity behaves akin to a fully flexible filament in the same bath.

Further evidence for the formation of hairpins and their relative stability with respect to more extended conﬁgurations can be obtained by combining calculations of Rg with the corresponding information about the asphericity of the ﬁlament. Fol-

✵ ✵ ✵

☞ ✔ ✾ ☞ ✔ ✷ ☞ ✔

✵ ✵ ✵ ✼

✿✷ ✿✻ ✿

✵ ✵

☞ ❋ ✛ ✺ ☞ ❋ ✛ ☞ ❋ ✛

❛ ❛ ❛

✵ ✽

✿✶

✵

☞ ❋ ✛ ✶✺ ☞ ❋ ✛ ✺ ☞ ❋ ✛ ✷

❛ ❛ ❛

✵

✿✻

✵

✿✺

✵ ✵ ✹✵

☞ ❋ ✛ ✷ ☞ ❋ ✛ ✶ ☞ ❋ ✛

❛ ❛ ❛

✵

✿✶✻

✹✵ ✹✵ ✽✵

☞ ❋ ✛ ☞ ❋ ✛ ☞ ❋ ✛

❛ ❛ ❛

✵

✿✺

✵ ✹

✿✶

✽✵

☞ ❋ ✛

❛

✵ ✹

✿

✵

✿✶✷

✮

✵ ✹

✿

✂

✵ ✵

✿✶ ✿✸

✁

✭

✵

✿✸

✵ ✵✽

✿

✵

✿✷

✵ ✵

✿ ✻

✵

✿✷

✵ ✵✹

✿

✵

✿✶

✵

✿✶

✵ ✵

✿ ✷

✵ ✵ ✵

✽ ✵ ✹ ✽ ✵ ✹ ✽ ✵ ✹ ✽ ✵ ✹ ✽ ✵ ✹ ✽ ✵ ✹

✻ ✶ ✶✷ ✶ ✶✻ ✶ ✷ ✷✷ ✷ ✻ ✶ ✶✷ ✶ ✶✻ ✶ ✷ ✷✷ ✷ ✻ ✶ ✶✷ ✶ ✶✻ ✶ ✷ ✷✷ ✷

❘ ❂✛ ❘ ❂✛ ❘ ❂✛

❣ ❣ ❣

Figure 3.3: Probability distribution of the radius of gyration P (Rg) for increasing values of the propelling forces Fa. Panel (a) shows the results for a fully flexible chain, panel (b) those for a polymer with rigidity βκ = 20, and panel (c) those for the stiffest polymer considered in this work, βκ = 90. lowing Rudnick et al. [145] we define asphericity as: P 2 (λi λj) A = i>jh − i (3.4) h i (P λ )2 h i i i th where λi is the i eigenvalue of the shape tensor. In two dimensions, this parameter is equal to 0 for a circular or isotropic object, and 1 for an object which extends along one dimension. In our system, small values represent configurations where the filament is relatively symmetric about its center of mass, whereas large values indicate a fully stretched, rigid filament. We show the result of this analysis for βκ = 90 in Fig. 3.4, where we plot the time averaged probability distribution of configurations with a given Rg and asphericity,

A. When βFaσ = 0 the plot shows, as expected, a distribution centered around large values of Rg and A close to 1, corresponding to a stiff filament. As the bath activity increases, at βFaσ = 20 two distinct high probability regions emerge. Both have large asphericities, indicating elongated configurations, but with well separated values of R . The folded configuration has a value of R 11σ and the unfolded corresponds g g ∼ to R 18σ. This scenario, consistent with our previous analysis, suggests a two- g ∼ state system (folded-unfolded) where the two states have comparable probabilities.

43 As Fa is further increased, the distribution smears out as local folding and unfolding of filament segments begins to occur at every length-scale along the chain and the filament’s statistics begin to resemble that of a fully flexible polymer in an active bath. For comparison we also show in the same figure (bottom-right) the probability

distribution for a fully ﬂexible ﬁlament immersed in a bath with the same active force. βFaσ =0 βFaσ = 20

βκ = 90 βκ = 90

/σ g

R βFaσ = 80 βFaσ = 80

βκ = 90 βκ =0

Figure 3.4: Panels (a)-(c). Time averaged probability distribution of finding a rigid filament, βκ = 90, with a given radius of gyration Rg and asphericity A for different values of Fa. When βFaσ = 0, panel (a), the filament is almost exclusively extended (A 1, R 22σ). At βF σ = 20, panel (b), two distinct peaks can be seen at large ∼ g ∼ a values of the asphericity, where one has roughly half the Rg of a fully extended chain. These peaks correspond to the folded (hairpin) and unfolded (extended) states. At larger values of Fa, panel (c), the distribution smears out and the filament can more freely explore a wider variety of configurations. Note how the distribution in panel (c) is very similar to that in panel (d) corresponding to a fully flexible filament, βκ = 0, in an active fluid having the same propelling force βFaσ = 80. All distributions have been divided by their respective maximum values so that they can be represented with the same color scale.

44 3.4 Conclusions

In this chapter we studied the behavior of a rigid filament immersed in a low volume fraction suspension of active particles confined in two dimensions as a function of filament rigidity and bath activity. Using shape-sensitive observables, we have characterized the conformational properties of the filament. Our findings indicate that fully flexible filaments swell monotonically with bath activity, but rigid filaments present a more peculiar non-monotonic behavior. For small activities, rigid filaments soften and become more flexible. As the bath activity is further increased, we observe the formation of a two state system characterized by extended and hairpin configurations, with complex local dynamics leading to their formation and de-stabilization. Finally, in the large bath activity limit the behavior of fully flexible and rigid filaments become nearly identical with a large distribution of accessible filament conformations. It should be stressed that this behavior is peculiar to two-dimensional or quasi-two dimensional confinement. In fact, simulations of the same system in three dimensions (not presented) do not show traces of the remarkable phenomenology observed in two dimensions. This is because of the reduced probability of a collision between active particles and filaments, but more importantly because of the inability of a one- dimensional filament to effectively trap active particles in three dimensions. Other work done in our group [146] has shown that two-dimensional membranes embedded in a three-dimensional bath of self-propelled particles exhibit a similar kind of dynamic breathing between collapsed and open states.

45 Part III

Long Ranged Interactions Between Active Brownian Particles

46 Chapter 4

Introduction

In this section, our focus shifts from isotropic purely repulsive self-propelled particles to particles that have long ranged and anisotropic interactions. The study of active matter began in earnest with the development of the Vicsek model in 1995 [11]. This minimal model for swarming particles that have some built in ‘awareness’ of their surroundings was able to reproduce some of the complex behavior of swarming bacteria [8], ﬂocks of birds [147], schools of ﬁsh [148], and even crowds of people in some cases [149]. The Vicsek model consists of three simple rules for how particles interact:

1. Particles with given orientations and positions move at a constant speed. 2. At each time step, all particles adopt the average direction of all their neighbors within a certain cutoﬀ distance. 3. Some random noise term is added to the direction of each particle.

The noise term effectively represents how strongly particles interact with each other. At low densities and high noise, particles move more or less randomly. When the noise term and particle density are small, finite sized groups of particles begin to swarm, with no large scale coherent motion occurring. Finally, as the density increases past a certain point for low enough noise, all the particles begin moving coherently in the same direction (Fig. 4.1). The particles in this model undergo a so- called first order ‘kinetic phase transition’ between disordered and collective motion as the density increases and the noise term decreases. The Vicsek model can also be modified to account for different types of noise which can change its physics. In

47 Figure 4.1: A-D Snapshots from simulations performed with the Vicsek model [11]. A: Initial motion at low density and high noise. B: Finite sized ﬂocks at low density and low noise. C: Little coherent motion at high noise and density. D: Ordered motion at high density and low noise. E: In one step of the Vicsek model, the red particle adopts the average direction of its nearby neighbors. general, however, mobile particles that align undergo some transition from disordered to ordered motion when inter-particle interactions are suﬃciently important compared to their random motion.

The success of this model inspired the development of both hydrodynamic [12] and particle based methods [13, 75] to study the motion of self-propelled particles with more or less complex directional interactions. A key characteristic of these swarming particles is that they have some aligning interaction which can come in the form of specialized interactions like in the Vicsek model where particles have some explicitly deﬁned spatial awareness, or asymmetries in particle shape and interactions in the case of colloids.

Both long and short ranged interactions have been used to induce complex swarming behavior in colloidal microswimmers, a remarkable fact, given that the only two

48 Figure 4.2: Phase diagram of active rods. Dilute (D), Swarming (S), Figure 4.3: A disordered fluid, collec- Laning (L), Bio-nematic (B), Turbu- tive swarming, lines and phase sepa- lent (T) and Jammed (J) structures rated clusters form in a system of self- emerge depending on the density and propelled particles with dipole-like in- rod aspect ratios. [150] teractions. [68] ingredients that give rise to this life-like collective behavior are anisotropic interactions and active motion. For self-propelled microswimmers, long ranged anisotropic interactions can come from hydrodynamics, charges, or dipoles, and short ranged anisotropy can be introduced through the particle shape. In Figure 4.2, simulations of zero-temperature self-propelled rods [150, 151] show a rich phase behavior, including coherent behavior like swarming, laning, and a mobile nematic phase, that depend on the aspect ratio and density of the rods. These particles have relatively short ranged interactions, and so the collective behavior occurs at high densities. As seen in the images from experiments in Figure 4.3, colloids with longer ranged dipole or magnetic interactions can form low density swarms that move collectively, lines of colloids that move together, and large clusters. The different types of active behavior seen in these experiments of the same system are accessible through the manipulation of the charge-charge interactions between the hemispheres of janus colloids with an external electric field. [68]

In this section, I present two stories about how tunable long ranged interactions can lead to exceptional control of self-propelled particle systems. In the ﬁrst chapter,

49 disks with isotropic long ranged interactions form structures like cluster crystals and lines, reminiscent of the self-assembly of block co-polymers. These structures assemble due to an inter-particle interaction composed of a long ranged repulsion and short ranged attraction. When activity is added to this system, the structures become ‘living’ in that they exchange particles between themselves in a self-regulatory process. Dumbbells with the same interaction potential are an anisotropic version of this system, and the self-assembled structures begin to rotate and translate coherently due to a coupling between the particles via their activity and shape [93]. The ﬁnal chapter concerns the collective behavior of self-propelled dipoles with long ranged anisotropic interactions. These particles swarm and move collectively like particles in the Vicsek model. When a charged, passive colloid is introduced, dipoles can bind to its surface in diﬀerent ways leading to remarkable hierarchical self-assembly and a diverse array of induced active motion. Finally, in an example of hierarchical collective motion, a mixture of many dipoles and colloids can be made to swarm much like a system of pure dipoles does. New developments in functionalizing active particles with anisotropic or long ranged interactions promise to open up new avenues for controllable self-assembly, just as the development of janus particles did for colloids.

50 Chapter 5

Living Clusters of Active Particles1

5.1 Introduction

Spontaneous pattern formation is a ubiquitous phenomenon in nature and arises in both equilibrium and out-of-equilibrium systems. Apart from the many biological examples [21, 152], important synthetic systems such as colloids and block copolymers have been shown to exhibit complex spatial equilibrium patterns upon self-assembly. Control of patterns at the micro and nanoscale is integral to the development of materials with novel optical, electrical, and rheological properties [153].

One route to spontaneous pattern formation in equilibrium systems is achieved via micro-phase separation [154], a phenomenon that typically occurs when geometrical or chemical constraints prevent a system from fully phase separating. Block copolymers, for instance, exhibit a wide variety of patterns upon micro-phase separation that can be controlled by tuning the relative length of the two blocks; from lamellae to cylinders to networks [155–157]. An alternative route is achieved with competing interactions. In colloidal suspensions, for instance, the interplay between a short range attraction and a long range repulsion has been shown to lead to micro-phase separation into a variety of patterns [158] with symmetry dependent on the relative weight of the two interactions (see for instance [159–163] and references therein). In these cases, the short range attraction is usually induced by depletion, hydrophobic or van der

1Based on work published in Soft Matter, 2016, 12, 555-561. Copyright The Royal Society of Chemistry 2016

51 Waals forces, while the long range repulsion may come from dipolar forces or screened electrostatics [164–167].

Recent experiments in two dimensions have shown that dilute suspensions of self- propelled colloidal particles can self-assemble into ‘dynamic’, ‘living’ crystals [14, 66, 168], where finite-sized aggregates continually join, break apart, dissolve, and reform. In this case, what limits the growth of the crystal to a macroscopic size a scenario − that would be favored because of inter-particle attractive interactions is the self- − propulsion of the active particles. This behavior seems to be specific to spherical particles, as these are able to freely re-orient within the developing crystallites under the influence of thermal forces, thus creating large stresses within the crystal. The formation of living clusters has also been recently observed in computer simulations of a three dimensional diluted suspension of self-propelled attractive spheres [169, 170]; however, in three dimensions clusters did not present any crystalline order. Experiments with swimming bacteria in a three dimensional container and in the presence of small polymer depletants [126] have resulted in the formation of micro- clusters with net rotational velocity dependent on the size and shape of the clusters (micro-rotors). This was also observed in simulations of two dimensional of dumbbells with different motilities [83]. Both living crystals and micro-rotors are beautiful examples of active finite-size structures. However, it is not clear whether a fluid with living crystals represents a truly stable phase [171]; furthermore, the micro-rotors observed experimentally are short-lived, and over time they merge to phase separate into a macroscopic structures.

The aims of the simulations presented in this chapter are two-fold. On the one hand, we want to understand whether explicitly introducing a micro-phase separation- inducing long range potential between self-propelled attractive particles can be exploited to control size and activity of living crystals and micro-rotors. On the other, hand we seek to understand how self-propulsion of colloidal particles can be used as a

52 means to alter the morphological features of the micro-patterns formed by the passive particles. With this in mind, we explore the behavior of two dimensional diluted suspensions of active Brownian spheres and dumbbells interacting with competing short and long range potentials for diﬀerent degrees of particle activity. We report a rich and complex range of physical behaviors that includes the formation of meso-phases of spinning and living crystals.

5.2 Model

We study systems of N active colloidal spheres and dumbbells in a two dimensional box of size L with periodic boundary conditions. Spherical particles are characterized by a diameter σ, and dumbbells are modeled as two spheres, each of diameter σ, rigidly connected to each other at contact distance, with length l = 2σ. Both spheres and dumbbells undergo Brownian motion, with an additional constant propelling force Fa acting along an orientation vector n = (cos θ, sin θ); for dumbbells this axis coincide with the particle long axis. The equations of motion of these systems are described using Brownian dynamics simulations:

D r˙i = ( iV (r) + Fa ) ni + √2D ξi (5.1) kBT −∇ | | Dr p n˙ i = ni V (r) + 2Dr ξri ni, (5.2) −kBT ∇ × where the ﬁrst equation describes the translational and second the rotational motion of the particles. V (r) is the interparticle potential. Notice that the ﬁrst term on the right side of Eq. 2 accounts for the external torques that develop exclusively for dumbbellar interactions. The relation between the translational, D = kBT/γ, and the

2 rotational diffusion coefficient, Dr, for spherical particles is set to Dr = (3D)/σ . Here γ is the friction coefficient. Following [172], the translational dynamics of dumbbells is solved independently for the parallel (ri //) and the perpendicular (ri⊥) components

53 of the particles’ coordinates with respect to the active axis ni. Correspondingly, there are two different diffusion coefficients D (D//,D ) with D = D///2 and → ⊥ ⊥ 2 Dr = (6D//)/l [173]. The solvent induced random fluctuations for the translational (ξ (t)) and rotational (ξ (t)) degrees of freedom obey the relations ξ (t) = 0 and i ri h i i ξ (t) ξ (t) = δ δ(t t0) and ξ (t) = 0 and ξ (t) ξ (t) = δ δ(t t0), respectively. h i · j i ij − h ri i h ri · rj i ij −

Following Imperio et al. [162], we set the interaction between any two non- connected spheres in the system to be

n 2 2 σ aσ r rσ r V (r) = ε A 2 exp + 2 exp (5.3) r − Ra −Ra Rr −Rr

The interaction parameters are n = 12, A = 0.018, Ra = σ, a = 1, Rr = 2σ, and

r = 1, and ε ranging from 25 to 50 kBT . This set of parameters has been chosen to guarantee micro-phase separation of the passive system as reported in [162]. This potential consists of a Lennard-Jones repulsion at short distances to enforce excluded area (the ﬁrst term); a short range attractive well to induce clustering of the particles (the second term); a long range soft repulsion to guarantee micro-phase separation(the last term). The potential is cut oﬀ and truncated at r = 10σ.

Throughout this study, we used dimensionless units, where lengths, energies and

2 2 times are expressed in terms of σ, kBT and τ = σ /D (for spheres), τ = σ /D// (for dumbbells), respectively. We use the dimensionless P´ecletnumber, deﬁned as

Fa D τ Fa D// τ Pe= | | for spheres and Pe= | | for dumbbells, as a measure of the degree kBT σ kBT σ 5 of self-propulsion of the particles. The time step was set to ∆t = 5 10− τ and all × simulations were run for a minimum of 107 time steps, with N = 1000 3000 spheres, − and N = 500 2000 dumbbells, at area fractions φ = π Nσ2/L2 ranging from 0.05 to − 4 0.5, and P´ecletnumbers from Pe = 0 to Pe = 40.

54 5.3 Results and Discussion

For the chosen parameters of the interaction potential, in the absence of active forces (Pe=0) and at small area fractions (typically smaller than φ 0.5), both spheres ' and dumbbells micro-phase separate to form cluster crystals (see Fig. 2 and 6), i.e. isotropic finite-sized crystallites of particles acting as lattice sites of a larger mesoscopic phase with overall hexagonal packing apart from the dislocations that − develop as a result of the size polydispersity of the small crystalline clusters. As the area fraction increases, so does the average size of the crystallites, until they eventually merge to form linear aggregates of finite width. This phase behavior is consistent with what reported in previous studies [162]. Because spheres can rotate freely within a crystallite at no energy cost without affecting its underlying structure, but dumbbells cannot, aggregates made of spheres and dumbbells behave quite differently when particles’ self-propulsion is switched on. We shall therefore discuss the two cases separately.

Spheres

All simulations with the spheres have been performed at ε = 25kBT . This corresponds to a pairwise binding energy of ε 5k T ; this particular value was selected to co- 0 ' − B incide with previous numerical studies with this potential [162]. A structural diagram in terms of the area fraction and P´ecletnumber is shown in Fig. 5.1, while Fig. 5.2 shows snapshots of the corresponding morphologies taken from our simulations.

At low area fractions and small P´ecletnumbers, we observe the expected mesoscopic crystal of static clusters. As soon as Pe becomes larger than ε /k T , not only | 0| B does particle exchange between clusters begin to occur, resembling the behavior observed in cluster crystals of soft particles at high densities [174, 175], but also, clusters begin to deform, split, and merge with neighboring clusters due to the stresses im-

55 static clusters living stripes living clusters ﬂuid static stripes 0.8 0.7 0.6 0.5

φ 0.4 0.3 0.2 0.1 0.0 0 10 20 30 40 Pe

Figure 5.1: A summary of all the structures observed in our simulations of active spheres as a function of P´ecletnumber Pe and area fraction φ.

Figure 5.2: Self-propelled spheres. Top: snapshots at φ = 0.24, bottom: φ = 0.55 (See Fig. 1 as a reference). From left to right: Pe = 0, 10, 20. At Pe = 10, meso- phases of living clusters are observed at lower densities, and living stripes are seen at higher densities. posed by the active forces of the constituent particles. The net result is the formation of a new phase consisting of a ﬂuid of living clusters. In this phase, the motion of the

56 clusters away from the lattice sites of the mesophase is driven by the ‘living’ character of the clusters themselves. In fact, as soon as a cluster splits and one or both of its parts are incorporated into the neighboring clusters, an imbalance in the long range forces holding together the mesophase takes place resulting in a global rearrangement of the clusters or in their further repartitioning to re-establish the overall balance of the forces.

Figure 5.3: Cluster size distributions as a function of Pe are shown for spheres at φ = 0.24. At Pe 5, ﬂuids of living clusters are seen, for which particle exchange occurs. At Pe 15,' there is no longer a stable cluster size and the system is a ﬂuid. The inset shows' the average cluster size for increasing Pe.

While, for small values of Pe, the mesophase is able to relax to accommodate the vacancies or density inhomogeneities created by the living clusters, at large values of Pe, the rate of cluster subdivision becomes too fast, the meso-phase cannot keep up, and it finally melts, resulting in a system of smaller clusters in a gas of dispersed active particles (fluid). This behavior can be easily tracked in the steady state cluster size distributions, P (n), shown in Fig. 5.3, where we study how P (n) changes when the system moves from the static cluster, to the living cluster phase and all the way to the fluid phase (changing the value of Pe). The well defined distributions centered

57 around large values of n for small Pécletnumbers give rise to a large peak at n = 1 (isolated particles) and a very wide distribution of clusters sizes as the meso-phase melts. Also notice that the average size of the clusters n (see inset of Fig. 5.3) h i tends to increase upon activating the self-propulsion of the particles, but n shows h i an overall non-monotonic behavior with the strength of the active forces. Our data indicate that the decrease of the average size for large activities is followed by a significant broadening of the cluster’s polydisperisty, and melting of the mesophase. To quantify the degree of order of the mesophase as a function of Pe, we also tracked the cluster-to-cluster bond order parameter Ψ6 [176]. In our simulations, cluster connectivity is obtained using a simple Delaunay triangulation of the clusters’ centers of mass, and Ψ6 is defined as:

N Nb(i) 1 Xc 1 X Ψ = ei6θj , 6 N N (i) c i=1 b j=1 where Nc is the total number of clusters in a conﬁguration, Nb(i) is the number of clusters connected to cluster i, and θj is the angle between a neighboring cluster and a reference axis. As expected (see Fig. 5.4), Ψ6 shows the deterioration of order of the meso-phase with increasing Pe.

0.55 0.50 0.45 0.40

6 0.35 Ψ 0.30 0.25 0.20 0.15 0 2 4 6 8 10 12 14 16 Pe

Figure 5.4: Cluster-to-cluster bond order parameter Ψ6 as a function of Pe for φ = 0.24 of the mesophase formed by active spherical particles.

An analogous behavior occurs at higher densities, where the passive phase is char-

58 Static Clusters Static Stripes Rotating Clusters Living Stripes Living Clusters Fluid 0.8 0.7 0.6 0.5

φ 0.4 0.3 0.2 0.1 0 10 20 30 40 Pe

Figure 5.5: A summary of all the structures observed in our simulations of active dumbbells as a function of Pécletnumber Pe and area fraction φ. In contrast to the spheres, at low values of Pe, clusters rotate, each with a specific handedness ,and translate. acterized by linear aggregates, or stripes, spanning the system size. In this case, intermediate propelling forces lead to the continuous recombination of the maze formed by the particles via formation and breakage of junctions between the stripes. At even larger Pe the stripes begin to loose connectivity, breaking up to give rise to large-scale density fluctuations.

Dumbbells

We also study the behavior of dumbbells for ε = 25kBT and over the same range of activities and area fractions as the spheres. Figures 5.5 and 5.6 show the structural diagram of the different regions observed in our simulations in terms of the area fraction and the Pécletnumber, and the corresponding snapshots from simulations at different conditions, respectively. The phase behavior of the passive system is analogous to that observed for spherical particles, where mesocrystals of hexagonally

59 Figure 5.6: Dumbbells at φ = 0.24 showing the variety of phases formed. From left to right: Pe = 0 (static clusters), Pe =5 (rotating clusters), Pe =15 (living clusters), Pe =40 (fluid phase). packing finite size clusters are observed at low area fractions, and stripes develop for sufficiently large densities. In contrast to the spheres, however, when self-propulsion is switched on at low values of Pe and at low densities, clusters acquire a well defined rotational and translational motion (roto-translator) that develops as a result of the nonzero internal average net force acting on the clusters of dumbbells, as discussed in [83, 126]. Thus, the first phase that one encounters at low area fractions and at small activities is a fluid of rotating and translating compact clusters.

As observed in our simulations with spherical particles, in this regime, an increase of the particle activity leads to an overall increase of the average size of the clusters. Further increase of Pe takes the system into a different state. Here, the rotational and translational velocity of the clusters increases. However, this state is characterized by a fluid of drifting, rotating, and living clusters. In fact, shear forces within the clusters are now able to break them apart, and their net translational velocities are strong enough to make whole clusters collide and merge. For this range of values of Pe, single dumbbells rarely leave the clusters individually, but mostly re-arrange over the outer surface of the individual clusters leading to a less well defined rotational velocity. Finally, for even larger forces, single dumbbells are able to fly off the clusters which melt and reduce in size as observed in the case of the spheres.

The linear, stripe-like aggregates found at high area fractions are similar to those

60 35 Pe =0

30 Pe =4

0.20 n 25 Pe =10 20 0 15 30 45 Pe =20 0.15 Pe Pe =40 ) n (

P 0.10

0.05

0.00 0 10 20 30 40 50 60 n

Figure 5.7: Cluster size distributions for dumbbells at φ = 0.24 and ε = 50kBT with increasing Pe. The cluster distribution shifts towards larger clusters and broadens with increasing activity until the system becomes a ﬂuid. The inset shows the average cluster size for increasing values of Pe.

observed for spheres, and living stripes are also seen at high activity. In contrast to the dynamic maze-like patterns observed for spheres, the morphology of the living stripes of dumbbells changes when large portions of stripes break off coherently and fuse with neighboring stripes. This is due to large net internal forces that continually break up the stripes. At even larger activities (not shown), clusters can easily completely melt and reform resulting in a phase characterized by large-scale density fluctuations as observed in active spheres at large densities and activities. Figure 5.7 (and its inset) show the cluster size distributions at low densities for different values of Pe, and the non monotonic behavior of the average cluster size with degree of activity.

Figure. 5.8 shows the angular velocity of a few selected clusters containing roughly thirty dumbbells in the phase of rotating clusters at low density/low Pe over time. This indicates that the rotational motion in this regime persists over time, but the speciﬁc direction and velocity of this rotational motion varies from cluster to cluster and depends on the speciﬁc arrangement of the dumbbells constituting each crystallite. The simulations to measure the angular velocity of the rotors are performed with

61 200 150 100

) 50 1 −

τ 0 (

ω 50 − 100 − 150 − 200 − 0 25 50 75 100 125 150 t (τ)

Figure 5.8: The angular velocities for several rotating dumbbellar clusters containing roughly 20-30 dumbbells from a system at φ = 0.24, ε = 50kBT , and Pe = 20 are shown. The clusters show unidirectional rotation over the duration of the simulation. Clusters shown here have been selected to have velocities around the average and at the tail of the distribution.

ε = 50kBT to extend the stability of this phase against the active forces and better observe its phenomenology. Overall, the angular velocity is expected to decrease with the size of the clusters as previously reported in [83, 126]. A rough estimate of the angular velocity dependence on the cluster sizes can be obtained by assuming that the axes of n dumbbells forming a cluster of radius R, are randomly oriented. The sum of these random forces will add up to a net force imbalance f F √n F R. ' a ' a Thus, one can expect an average torque acting on the cluster for any given random conﬁguration of dumbbells that scales as τ = fR F R2. Balancing this torque with ' a the rotational friction that we take to be proportional to R4ω, where ω is the angular velocity (this is the result for typical simulations without hydrodynamics [83]), one expects ω F /R2 F /n. Figure 5.9 shows the dependence of the average angu- ∼ a ' a lar velocity ω on cluster size for our dumbbell rotors. The data appears compatible with our analysis, in that the average angular velocity is inversely proportional to the cluster size.

One of the most exciting new phases for dumbbells is the ﬂuid of rotating clusters

62 50

40 )

1 30 − τ ( i

ω 20 h

0 10 20 30 40 50 60 n

Figure 5.9: Average rotational speed ω of a cluster of dumbells of size n, as measured in our simulations, for Pe=20. The solidh i line is the expected theoretical dependence ω 1/n h i ∼ observed at low densities and small Pécletnumbers. These cannot form with spherical particles as spheres can freely rotate within the finite size clusters, but dumbbells become sterically locked in place by the attractive forces. As a result, once a cluster of dumbbells is formed, that cluster will acquire a constant angular velocity over time. As mentioned above, the stability of the clusters can be enhanced by increasing the strength of the potential ε, however, the overall order of the meso-phase is disrupted because of their active translational motion. This occurs because, as explained above, for any configuration of n randomly assembled dumbbellar clusters, a net imbalance of forces proportional to Fa√n will on average develop in a random direction. One could, however, envision that increasing the repulsive part of the interaction between the particles would create a sufficiently large energy barrier between the clusters to limit the effect of their active translation (at low Pe), while keeping their rotational motion unaffected. This would lead to the formation of yet another phase characterized by a cluster-crystal with spinning sites. Our simulations show that this is indeed the case, and can be achieved by setting, for instance, εr = 2.5εa in the potential V (r). A similar analysis for the spheres leads to better defined clusters, but in this case the

63 splitting/recombination of the clusters signature of their inner activity which now − − occurs at larger P´ecletnumbers, leads again to mass transfer to the neighboring sites, and subsequent relaxation of the whole meso-phase. Movies from our simulations for both spheres and dumbbells can be found in the SI.

5.4 Conclusions

In this chapter we studied the phase behavior of diluted suspensions of self-propelled dumbbells and spheres interacting with a micro-phase separation inducing potential. Speciﬁcally, we considered a pair potential composed of a short-range attraction and a long-range repulsion. This is a standard potential that has been used extensively in the literature of micro-phase separation of colloidal particles, and that mimics the interaction of weakly charged particles in the presence of depletants. Both forces can be easily tuned by either changing the salt concentration in solution or by increasing/decreasing the density of the depletants.

Our results indicate that for a range of parameters, it is possible to induce the formation of two previously unobserved states; a spinning cluster crystal, i.e. an ordered mesoscopic phase having finite size spinning crystallites as lattice sites, and a fluid of living clusters, i.e. a two dimensional fluid where each ”particle” is a finite size living crystallite. The first state develops from the self-assembly of dumbbells, whereas the second state occurs for spherical particles. We suggest ways to increase the stability of these states by appropriately selecting the relative weight of the two competing interactions with respect to the self-propulsion.

Several groups have observed phase separation of self-propelled hard particles at high activity and area fraction (see for instance [77, 88, 177]). To understand whether a suﬃciently large self-propulsion would overwhelm the role of the repulsive interactions and lead to a re-entrant behavior as observed in colloidal particles with

64 attractive interactions [178], we performed simulations for Pe up to 150 and area fractions up to 0.5, but haven’t observed a complete phase separation of the system, suggesting that at least within this range of parameters the long range repulsive tail of the potential prevents this transition. Another interesting result that emerges from our simulations concerns the size of the clusters that form at low densities. For both spherical and dumbbellar particles, we find that the average size of the crystallites has a non-monotonic behavior with the strength of the propelling forces. This very result was also reported in [179] for the case of spherical particles. Although our system is two dimensional, the combination of short-ranged attractive and long-ranged repulsive interactions also leads to micro- phase separation in three dimensions [159] We speculate that both the fluid of living clusters and crystal of micro-rotors will survive, but the morphological properties of the clusters will be different. Simulations of these systems in three dimensions will need to be performed to gain further insight on the role of dimensionality. It should be finally mentioned that our model of active Brownian particles ignores hydrodynamic effects from the solvent. Although there is experimental evidence that this does not qualitatively affect the formation of living clusters or micro-rotors, as both phases have been observed experimentally [14, 126], from a quantitative standpoint, the details of the hydrodynamic interactions could affect the cluster morphology. Also, long-range hydrodynamic interactions may develop between spinning clusters of dumbbells [180]. Possible consequences include cluster-cluster rotational velocity coupling, which would affect the magnitude of a clusters rotational velocity. Hydrodynamic interactions between fast-spinning clusters could also overwhelm the long-range repulsive interactions and destabilize the crystalline mesophase of rotors. However, we expect this could be prevented by increasing the strength of the long-range repulsive interaction. More work in this direction is underway.

65 Chapter 6

Hierarchical Motion of Active Dipoles and Passive

Charged Colloids

6.1 Introduction

Self-propelled particles convert energy from their surroundings into translational motion. This constant energy uptake results in a dynamics where each active particle, while still undergoing Brownian translational and rotational motion, can also move in the same direction with a persistence length of many times its own diameter [42, 46, 63]. Isotropic, hard self-propelled particles have been shown to have a remarkable collective behavior, ranging from the development of giant number ﬂuc- tuations to motility induced phase separation [14, 80, 86, 88, 181]. Furthermore, they present an unusual behavior near surfaces and passive obstacles [94, 95, 106, 111, 116, 130, 182, 183]. A key feature of systems of isotropic active particles is that there is no energy cost associated with the rotation of the particles within the self-assembled structures they form. While this rotational freedom is a crucial aspect of these particles’ unusual behavior, the isotropy of the interactions greatly limits the structural properties of the aggregates they can form. The idea of using anisotropic particles whose orientations are coupled to those of their neighbors is as old as the ﬁeld of active matter itself [10, 11], but it is only recently that long ranged aligning interactions have been considered in synthetic and colloidal systems, adding a phenomenological complexity to active particles ranging from turbulence to self-regulation and swarming behavior previously observed only in living systems (for recent reviews we refer

66 the reader to [16, 89, 184] and the references therein).

Much of the numerical work in the field of active spherical colloids has focused on particles with isotropic potentials [13], and less has been done to explore how anisotropy in inter-particle interactions can affect the large scale collective dynamics of active systems. Spherical Dipolar Janus Particles (DJP) are the simplest colloidal entity with controllable surface interaction asymmetry. DJPs are composed of two hemispheres each functionalized with charged molecules of opposite sign, and they can be made to be passive [185], or active [40, 43, 186–189]. Passive DJPs present a rich phase behavior which includes the formation of dipolar clusters, linear assemblies and ring-like aggregates depending on the range and strength of the inter-particle interactions and concentration [190–197]. Recently, numerical simulations and experiments [68, 184, 198, 199] have indicated that a new level of dynamic and reconfigurable self- assembly complexity can be achieved when self-propelling DJPs under the influence of external fields. Inspired by these results, and by recent work on the behavior of passive components in active fluids [108, 116, 200–202], we report numerical simulations of the complex behavior of a mixture of Active Dipolar Janus Particles (ADJPs) and Passive Charged Colloidal (PCC) tracer particles. We show how the interplay between the strength of the active forces and that of the dipolar interactions can lead to a remarkable multi-stage dynamic self-assembly of the PCCs.

We begin this chapter by introducing the models and the numerical methods used to gather our data. We then study the behavior of a suspension of pure ADJPs at different concentrations and activities. We next turn our attention to how a suspension of ADJPs interact with a single PCC. Finally, we consider the collective behavior of a mixture of multiple PCCs and ADJPs.

67 6.2 Model

Our simulations consist of a mixture of ADJPs, represented by disks of diameter σ with embedded point charges at area fractions φb, and PCCs represented as disks of diameter σc = 10σ with a point charge at the particle center and area fraction

φc. Both types of particles undergo over-damped Langevin dynamics, with the active dipoles having an additional term driving their self-propulsion with a force of strength

Fa according to the equations:

p 2 mr¨ = γr˙ rV (r) + 2γ Dξ(t) + F n (6.1) − − ∇ a p θ˙ = 2Drξr(t) (6.2)

Here n = [sin(θ), cos(θ)] represents the orientation vector along which the active force is directed, γ is the friction coeﬃcient, V (r) the total potential acting on a particle, and D and Dr are the translational and rotational diﬀusion constants, respectively

2 (with Dr = 3D/σ ). The solvent-induced Gaussian white noise terms are given by

ξ (t) = 0, ξ (t) ξ (t0) = δ δ(t t0) and by ξ (t) = 0, ξ (t) ξ (t0) = δ(t t0), h i i h i · j i ij − h r i h r · r i − for the translational and rotational motion respectively. All simulations are carried out using the numerical package LAMMPS [132], and throughout this work we use the default dimensionless Lennard-Jones units as deﬁned in LAMMPS, for which the fundamental quantities mass m0, length σ0, epsilon 0, and the Boltzmann constant kB are set to 1, and all of the speciﬁed masses, distances, and energies are multiples of these fundamental values corresponding to T = T0 = 0/kB, m = m0, σ = σ0, and q 2 τ = m0σ0 in our simulations. 0 0 The interaction potential between particles is the sum of a Weeks-Chandler-

Andersen potential VWCA, responsible for enforcing excluded volume between the disks, and electrostatic terms which depend on the particle type. The ﬁrst term is given as " 12 6 # σij σij 1 VWCA(rij) = 4 + , (6.3) rij − rij 4

68 where the indices refer to the particle type, σij = (σi + σj)/2, ε = 5ε0, and the

1 potential extends out to 2 6 σij. The typical interaction potential between the two charged hemispheres of DJPs can be quite complicated to describe accurately, in particular when interactions are strongly screened by the presence of a salt and a simple dipolar description becomes inadequate [196, 203]. In this study, however, we are not interested in the regime where particles are expected to self-assemble into isotropic dipole clusters, but in an intermediate regime where the dipolar character of the particles, although screened to some extent, still dominates their assembly, favoring the formation of elongated linear structures. We model this interaction by anchoring two explicit eﬀective charges of opposite sign inside of each dipole. These charges are placed along the vector n, at a distance 0.25σ from the center of the ± DJP, and rigidly translate and rotate with it. Similar models were used in [184, 192]. A single charge is placed at the center of each of the large passive colloids. Any two charges in the system interact according to:

qeﬀ qeﬀ i j κrij V (rij) = e− (6.4) εdrij

eff where the indices refer to the particle type, and qi is the effective charge they carry. The effective charges have been selected to provide typical interactions strengths observed in colloidal systems when at contact. Given our setup, we used qeff = 104 for the charged colloids, and qeff = 5 for the dipoles. Finally , and κ are set to | | d 1. With this setup the strongest attraction between a PCC and an ADJP occurs when the particles sit at their WCA minimum, and is on the order of 10kBT . The dipoles spontaneously assemble into chains and rings when taken in isolation and in the absence of active forces. All simulations are run for a minimum of 107 iterations

3 1 with a time step ∆t = 10− τ0 (where τ0 is the dimensionless time) and γ = 100τ0− . Simulation snapshots and movies are rendered using VMD [204].

69 6.3 Results and Discussion

We divide our results into sections where we first analyze the behavior of a suspension of ADJPs by themselves, we then study how these organize over the surface of a single passive charged colloid, and how different arrangements lead to different types of activated colloidal motion. Finally, we study the collective behavior of a mixture of multiple PCCs and ADJPs.

Suspensions of ADJPs

We begin our analysis by exploring the behavior of a suspension of ADJPs in the absence of charged colloids. Fig. 6.1 shows typical system configurations for different values of the active force, Fa and at different ADJP area fractions, φb.

Figure 6.1: Structural self-assembly diagram showing typical conﬁgurations acquired by active dipolar janus particles for a range of activities and particle area fractions.

Passive, weakly screened DJPs tend to self-assemble into closed loops or linear ag-

70 gregates. Beyond a percolation threshold, the passive DJPs organize into a disordered network whose branches become thicker, but more tortuous as the area fraction of the DJPs increases. The introduction of a small amount of activity along the dipolar axis in a low density suspension leads to the formation of structures morphologically analogous to those of passive DJPs: loops and lines that now actively spin or slither across the system as a result of the coherent alignment of the active forces via electrostatic interactions. Overall, the length of these aggregates decreases as the swimming speed of their constituent particles increases. This is because ﬂexible long chains are easily broken by the torques that develop from mis-aligned active forces within their backbone. In addition, swarming behavior similar to that seen in suspensions of active rods [151, 205, 206] is observed, albeit with smaller and smaller ADJP aggregates as Fa increases. The system is found in the ﬂuid state of mostly isolated

ADJPs as soon as Fa becomes suﬃciently large (not shown in the ﬁgure) to break apart all linear aggregates. Figure 6.2 shows how the average length of the ADJP chains decreases with Fa in low area fraction suspensions.

For the largest area fractions considered here φ 0.4, a small amount of activity b ≥ breaks the structure of the compressed network formed by the passive particles and generates ordered patterns of coherently moving linear aggregates that span the entire simulation box. Apart from the pattern for φc = 0.6 and βFaσ = 10 which seems to remain stable over the course of our simulations, the other patterns are transient as they form, break into smaller clusters, and then reform aperiodically over time. At these area fractions, larger activities tend to mostly compact the aggregates into ﬁngerprint-like structures similar to those seen in active nematics [207].

Single Charged Colloid in a Suspension of ADJPs

Next, we consider the behavior of a low density bath of ADJPs in the presence of a single PCC. When a negatively charged large colloid is added to a bath of

71 Figure 6.2: The average ADJP chain length, normalized by the length of the simulation box L, as a function of the self-propelling force Fa, for different values of the particles’ area fraction φb. n is the average size of clusters which are constructed by considering particles that areh i in contact with each other and for which their mutual orientation is off-set by an angle that is not larger than π/6. dipoles, DJP chains bind to the colloid forming radially extruding linear structures. In passive systems, we observe sparse coating of the colloid by a few long chains of dipoles, whereas ADJPs, moving in the direction of their minus-plus polar axis, cover the PCC surface more completely, albeit with shorter chains. Particle activity induces this change for two reasons: first, ADJP chains are shorter than passive ones so there are more chains in solution, and second, when the direction of the propelling axis is chosen so that the active force aligns with the direction of a favorable charge-dipole interaction (which is indeed the case in our simulations), the chains develop an effectively stronger binding attraction to the charged colloid than their bare electrostatic one, making the overall coverage of the colloidal surface by the chains more likely.

Two types of PCC-ADJP complexes form, depending on how much of the PCC surface is coated, with quite diﬀerent behaviors. At low activity and high φb, adsorbed ADJP chains fully coat the PCC to form a polymer brush-like structure around it (see

72 Figure 6.3: A: Snapshot from simulations of a rotating brush of ADJPs chains over the surface of a PCC (see also supplementary movie S1). B: Idealized sketch showing how lateral interactions between ADJPs can lead to a tilt of the chain axis with respect to the surface of the PCC. This is responsible for the tangential active forces that lead to the rotation of the brush. C: Long tails of ADJPs developing over a PCC at low are fractions and intermediate activities. (supplementary movie S2) D: Short chains developing over a PCC at low are fractions and large activities.

Fig. 6.3A), which we refer to as the ‘corona’. Neighboring ADJP chains that densely coat the PCC have attractive lateral electrostatic interactions which make bundling favorable. The lowest energy conﬁguration is one where all of the chains collapse around the PCC in the same direction and form a ‘swirl’ where the axis of ADJPs in the chains is tilted away from normal to the PCC surface (Fig. 6.3B). When the DJPs are active, non-zero tangential components of the ADJP forces result in coherent

73 rotations of the corona around the PCC. Thermal fluctuations and fluctuations in the chain lengths can occasionally change the handedness of the ADJP swirl, leading to a switching between clockwise and counterclockwise rotations. The persistence of these rotations in a specific handedness depends on the strength of the activity, the surface coverage, and the strength of the electrostatic interactions in a nontrivial manner. The PCC is rarely coated by chains of identical length, so in addition to the rotation of the corona, the PCC is propelled as an activated particle with force having strength and direction that depends on the fluctuating balance of forces induced by the adsorbed active chains. In this configuration, the PCC-ADJP complex tends to move in tight circles of varying radius and handedness similar to the motion recently observed in [208].

A rather different PCC-ADJP complex forms in the limit of a sufficiently large self-propelling force and low ADJP concentration. In this regime, the PCC is no longer evenly or fully coated. Any asymmetric coverage results in a clear imbalance of forces that translationally propels the colloid in a specific direction. Once a bare patch on the PCC surface is formed, especially at low φb, it is rare that the colloid becomes fully coated again. The partially coated PCC now effectively becomes an active particle whose source of propulsion is a tail of ADJP chains which are kept in a bundle by lateral dipole interactions (see Fig. 6.3C and D). Over time, these trailing chains grow away from the direction of motion through the addition of free ADJP chains from solution, or by the incorporation of ADJPs that initially bind to the bare face of the now mobile PCC, and subsequently slide around the colloid to merge with the pre-existing pushing chains. The trailing chains can grow to lengths that can be significantly larger than the diameter of the colloid.

A single parameter allows us to both diﬀerentiate fully coated from partially coated PCCs as well as explore the properties of the ADJP chains that are bound to partially coated colloids. We deﬁne ∆com as the distance between the center of the

74 Figure 6.4: Anisotropy of PCC-ADJP complexes, expresses in terms of ∆com (see text for a deﬁnition of ∆com), as a function of the strength of the active forces Fa and for diﬀerent values of the ADJP area fraction φb. colloid and the center of mass of all of the dipoles which are connected directly (via contact), or indirectly (through the ADJP chains) to the PCC. Figure 6.4 shows how

∆com depends on Fa at diﬀerent values of φb. At low activity, the center of mass of the corona is close to the center of the PCC, indicating that the PCC is fully coated by chains of roughly equal length (Fig. 6.3A). As the activity is increased, a sudden jump in ∆com, corresponding to the transition from a fully coated to a partially coated PCC (Fig. 6.3C), is observed. The location of this jump shifts to larger activities for higher φb, where ADJPs can fully coat the colloid more easily. As the activity increases further, the asymmetry decreases from its maximum value because, as in the system consisting of only ADJPs, chains break up more easily giving rise to shorter adsorbed tails (Fig. 6.3D).

We summarize these results with the structural phase diagram in Fig. 6.5 showing how for our speciﬁc set of electrostatic interactions, the extent to which the PCC is coated depends on the strength of the active forces and the area fraction of the ADJPs in solution. Above φ 0.05, and for propelling forces near the structural b ≈ phase diagram boundary, the system shows bistable behavior where a PCC alternates

75 Figure 6.5: Structural phase diagram showing where a charged colloid is fully and partially coated. As the area fraction increases at a given bath activity, the charged colloid is more readily fully coated. At the border between the two regions, the charged colloid can alternate between being fully and partially coated, especially at higher ADJP area fractions, φb. The line separating the two regimes is a guide to the eye.

repeatedly between being a fully coated colloid with a rotating corona, and a partially coated translational swimmer, somewhat reminiscent of the motion of run-and-tumble bacteria.

Starting simulations at small values of φb, low Fa and with all particles randomly distributed, the PCC is initially partially coated and can remain so for long times. Once the PCC becomes fully coated however, the ADJP swirl structure is very stable. Conversely when simulations are initialized in a conﬁguration where the PCC is fully and densely coated, the rotating corona phase persists to higher ADJP activities at a given φb, but once the activity is suﬃciently large the corona breaks up and the colloid becomes partially coated.

76 Figure 6.6: Total angular displacement of the ADJP brush (corona) around a charged colloid as a function of time for φb = 0.03. The inset shows the average time t spent by the corona rotating with a given handedness as a function of the self-propellingh i force Fa.

Rotational Dynamics of a Swirl of ADJPs

In this section, we consider the rotational dynamics of the fully coating corona of ADJPs over the colloid. In Fig. 6.6 we track the typical dependence of the global angular displacement of the ADJP swirl (θ(t) θ(0)) over time for different values − of the active force. At low activities, the sign of the rotational velocity of the swirl does not change often, indicating that the ADJP rotations persist with the same handedness for long times. As the activity increases, this persistence time decreases while the rotational speed of the corona increases. There is no bias in favor of rotation in any particular direction, so at very long times, the total rotation of the corona is diffusive. The inset in Fig. 6.6 shows the average persistence time of this rotational motion in a specific handedness t as a function of F . These data are h i a αβFaσ well characterized by a simple exponential function of the type e− with α 0.5. ≈ The corona changes direction through collisions with unbound chains in solution as well as through internal structural rearrangements. At low activities, electrostatic

77 chain-chain interactions dominate and adsorbed ADJPs chains are the longest, thus leading to the largest values of t . For larger values of F , the ADJP chains are h i a shorter and can rearrange more easily. As a result, the active motion of the ADJPs becomes more important relative to their electrostatic interactions, leading to overall smaller values of t . With increasing φ at a given F , ADJP chains grow longer and h i b a t generally increases. h i

Induced Active Motion of a PCC

Figure 6.7: Typical trajectories of a PCC partially coated by a bundle of ADJPs for four diﬀerent values of Fa over the same elapsed time. As the bath particle activity increases, the trajectories become rougher. This trajectories were takes at a ADJP area fraction φb = 0.01.

In this section, we explore the dynamics of a single partially coated activated

PCC. Typical two-dimensional trajectories of the PCC at low φb, and for values of Fa where the colloid is partially coated, plotted over the same amount of time (Fig. 6.7) show that the motion of the PCC-ADJP complex is strongly linked to the properties of the ADJP bath. In a bath with low activity, the motion of the aggregate persists in the same direction for very long times. As Fa increases, the net distance traveled by

78 the aggregate increases, but the persistence length of its trajectory becomes smaller.

Figure 6.8: A: Orientational correlation functions for the normalized displacement vector u(t) of a partially coated PCC-ADJP complex at φb = 0.01. B: Effective rotational diffusion coefficients of a PCC-ADJP complex measured by fitting an exponential to the orientational correlation functions presented in A. C: Rotational diffusion coefficients of the PCC-ADJP complex as a function of the chain length ∆ . The dotted line is a power-law fit to the data. ∝ com

The orientational correlation function of the PCC swimming direction,

2 Ruu = u(0)u(t) / u(0) , where u(t) is the normalized displacement vector of the h i h i

PCC between any two successive time steps of size 100τ0, reflects the roughening of the PCC-ADJP trajectory with increasing Fa (Fig. 6.8A), and helps to explain some of the dynamics of the activated tracer. The rotational dynamics of the PCC-ADJP complex is mostly diffusive when the dipoles do not organize into a corona, but rather into a linear bundle, therefore a simple exponential fit to the orientational correlation functions allows us to directly extract the effective rotational diffusion coefficient,

eﬀ Dr , of the complex at diﬀerent ADJP activities (shown Fig. 6.8B). The reason for

eﬀ the increase of Dr with Fa is not immediately obvious because, in principle, the rotational diﬀusion of an isolated active particle depends only on the temperature

79 and not on the strength of the active forces. In this case, however, the activated PCC is not an isolated particle, but is propelled by a tail of ADJPs of length, ` ∆ , ∝ com which is inversely dependent on Fa (see Fig. 6.4). In this case, it is useful to think of the PCC-ADJP complex as a rod-like object for which the rotational diﬀusion

eﬀ 3 coeﬃcient Dr is expected to scale as `− . This is consistent with our results, shown

eﬀ in Fig. 6.8C, where we plot Dr as a function of ∆com. A power-law ﬁt to the data

b with ax− at a value of φ = 0.01 leads to an exponent b = 3.1 0.2, indicating that b ± indeed, a PCC-ADJP complex with a long tail of ADJPs (low Fa) rotates more slowly than one with a short tail (high Fa). Interestingly, at higher values of φb the value of the ﬁt exponent remains fairly compatible with our model of the aggregate as a rotationally diﬀusing rod (b = 2.6 0.2 for φ = 0.05), though at these values of φ , ± b b more collisions between the aggregate and unbound ADJP chains occur, complicating the rotational dynamics of aggregate. Larger deviations are expected at even larger area fractions.

For completeness, we include the orientational correlation function of a fully coated PCC (βFaσ = 4) in Fig. 6.8A. The periodic behavior of the orientational correlation function suggests that the rotational motion of the ADJP swirl forces the colloid to move in rough circles, as we indeed observe in our simulations. The net active force on these fully coated colloids is much lower than that on a partially coated particle, so they move at a signiﬁcantly lower speed. Nevertheless, even the motion of the fully coated PCC can be considered as activated when compared to that of a colloid in a passive bath.

In summary, a bath of active dipoles can eﬀectively turn a passive charged colloid into an active colloid whose motion depends strongly on the assembly properties of the ADJPs on its surface. At low activities, the PCC moves aperiodically in circles as a swirl of active dipoles rotates around it. At intermediate bath activities, the PCC is pushed in the same direction for very long times by a bundle of long chains of

80 ADJPs that only partially coat the PCC surface. As the activity is increased further, the size of the ADJP tail decreases, inter-particle interactions become less important compared to the self-propelled motion of the ADJPs, and the PCC begins to move more like a tracer particle in an active bath, diffusively but with an enhanced diffusion coefficient [136, 141, 209].

Multiple Colloids

Figure 6.9: Top: Snapshots of steady state configurations taken from simulations for different values of PCC area fraction φc at a fixed value of βFaσ = 20 and number of ADJP per PCC, χ = 20. φc increases when going from A to C (supplementary movie S3). Bottom: Snapshots of steady state configurations taken from simulations for different values of χ at constant PCC area fraction φc = 0.2 and a fixed value of βFaσ = 20. χ increases when going from D to F.

In the previous sections we showed that by using ADJPs it is possible to turn a single isotropic, passive colloid into an anisotropic, active one. In this section we

81 investigate the collective behavior of these activated colloids. We perform simulations with Nc = 200 PCCs at different values of φc with different numbers of ADJPs per PCC, a quantity we define as χ, ranging from χ = 20 to χ = 60 (and several simulations with fewer colloids at higher values of χ). As a reference, with the electrostatic interaction parameters that we have chosen for this study, in the absence of ADJPs, a solution of PCCs forms a hexagonal structure above φ∗ 0.3. c ≈

In these simulations, the values of χ and φc can be used to create or destroy both structural order and coherent motion in PCCs. Simulation snapshots in the top row of Fig. 6.9(A-C) show typical configurations at a fixed βFaσ = 20 and χ = 20 for increasing colloidal densities. At the lowest densities, dynamic groups of PCC-ADJP complexes show swarming behavior, where the orientation of the complexes (and thus their motion) is correlated only between nearby complexes. As φc increases, most of the complexes have the same orientation, where the center of mass of the particles translates in one direction with little order within a fluid of partially coated PCCs. At sufficiently high φc, the PCC-ADJP complexes form an ordered crystal with center of mass that moves according to the average orientation of the complexes. A key requirement for coherent PCC motion is that χ is small enough so that when the ADJPs are partitioned evenly among the PCCs, none of the colloids are fully coated, and crucially, few or none are uncoated. In other words all of the PCCs must be activated. If this is the case, then electrostatic interactions can reorient neighboring particles so that they move in the same direction. While increasing φc increases the order in the system, simply increasing the extent by which ADJPs can coat PCCs can reduce, and even destroy the coherent motion of the complexes. This is illustrated in the bottom row of Fig. 6.9(D-F), where we show snapshots of typical configurations with increasing number of ADJP per colloid, χ, at a fixed density φc = 0.2 and

βFaσ = 20.

To quantitatively account for the coherency of the particle motion we measure

82 Figure 6.10: Matrix of figures showing the order parameter ψ associated with the motion of the PCC-ADJP complexes for three different values of the number of ADJP per PCC, χ; χ = 20, χ = 40 and χ = 60. Each panel shows the same data for different values of PCC area fractions φc and strength of the active forces Fa. the order parameter ψ defined as

N 1 Xc ψ = µ (6.5) N | i| c i=0 where µi is the unit vector along the axis connecting the center of mass of the colloid to the center of mass of the dipoles bound to that colloid (the direction associated with ∆com), and the sum runs over all PCC-ADJPs complexes in the system [11]. In Fig. 6.10, we plot the orientational order parameter of the PCC-ADJP complexes as a function of χ for diﬀerent values of Fa and φc. At low φc the motion of PCC-ADJP complexes is relatively uncorrelated regardless of the value of χ, albeit with a larger degree of order for small χs a trend that persists throughout the phase diagram. −

As φc increases at low Fa the small χ complexes develop a coherent, macroscopic ordered motion (χ = 20, shown in blue). For identical conditions, systems with larger surface coverage have signiﬁcantly less coherent motion, an eﬀect that becomes even more pronounced as the average PCC surface coverage increases beyond χ = 60

83 (not shown). Overall, as the activity increases, the orientational order in the system decreases. We also conﬁrm this for the case of χ = 20 in a system with φc = 0.4 and

βFaσ = 50, where we measure ψ = 0.27 (not shown in the ﬁgure). For intermediate values of χ, there is a regime, at low activity and high φc, where the colloids are pushed in the same direction and can swim in an orderly manner. However, as soon as the colloids can become fully coated, as is the case in systems with χ = 60, PCC-ADJPs complexes do not show any signiﬁcant coherent motion, as fully coated colloids are relatively passive. At these values of χ the system separates into fully coated and partially coated PCC-ADJPs complexes.

Figure 6.11: Distribution of the number of ADJP chains bound to the surface of the PCCs, P (λ), for a passive and an active (βFaσ = 10) suspension at diﬀerent values of number of ADJPs per PCC, χ.

The coherence of the PCC motion is related to the distribution of ADJPs bound to the surface of each PCC, P (λ), which has a nontrivial dependence on the value of χ. In this respect, it is instructive to look at the distribution of adsorbed ADJPs over the surface of the PCCs. Remarkably, at moderate activities, large φc and small χ = 20, i.e. when the colloids acquire a collective coherent motion, the width of the distribution of ADJPs across the PCCs is much narrower than it is in the parent passive system (Fig. 6.11, left). We believe that this is due to a self-regulation eﬀect where, the PCC-ADJPs complexes exchange ADJPs so that all of the colloids are

84 activated and move at roughly the same speed, so that particles can maintain a constant distance from each other, thus decreasing their mutual electrostatic repulsive energy. Upon increasing the number of ADJPs per colloid in solution to χ = 60, we find that the distribution becomes bimodal (see Fig. 6.11, middle). In this case we observe a mixture of fully coated and partially coated PCCs. A further increase of χ eventually leads to the full coating of all of the PCCs. This trend is already visible for χ = 100 (in Fig. 6.11, right) where most of the particles develop a full isotropic coating of ADJPs. This behavior is consistent with what observed in Fig. 6.5. As one moves at a constant propelling force and φc from small to large values of χ, one goes from partially coated, through mixed, to fully coated PCC-ADJPs complex configurations. Interestingly, the repulsive interactions between the fully coated PCCs are somewhat screened by the ADJP chains, and so the resulting PCC-ADJP complexes are less repulsive than the bare PCCs. This interaction is mediated by the chains of ADJPs coming from different particles that can align with each other forming transient attractive bonds between the particles. See supplementary material (S4) for an example showing the dynamics for this case.

6.4 Conclusions

In this chapter we have presented extensive simulations of a system of large, passive and charged colloids in a bath of small active dipolar janus particles. We believe this is one of the ﬁrst examples of activity driven multi-stage self-assembly, where the strength of the propelling forces not only controls how ADJPs self-assemble over the surface of passive colloids, but also determines and regulates the collective behavior of PCC-ADJP complexes via self-regulatory ADJPs exchange. The PCC-ADJP complexes that spontaneously develop in these systems as a result of the electrostatic interactions between the diﬀerent components acquire a net directional motion when-

85 ever the colloidal surface is not fully coated by the ADJPs, and a spiraling motion when it is fully coated. In the ﬁrst case, the ADJPs form long chains that bundle on one side of the charged colloid. In the second case, we observe the formation of a fully coating brush of ADJPs chains, which rotates coherently over the colloidal surface. In both cases, the persistence length of the activated colloidal trajectories is controlled by the strength of the active forces. We have also studied the collective behavior of PCC-ADJP complexes that develop upon immersing a number of PCCs in a bath of ADJPs. Remarkably, we observe that at suﬃciently large PCC densities and for a range of ADJP concentrations, a coherent swimming behavior of the complexes ensues due to a partitioning of ADJPs among the PCCs which narrows the natural (equilibrium) distribution of DJP chains per colloid. Interestingly, preliminary results also suggest that moderately dense systems of PCCs with a large number of ADJPs can develop attractive interactions due to chains interactions and interdigitation. Overall, we expect the size and charge of the PCCs to be important parameters that control the structural phase behavior of the complexes, and in this respect, it is worth noting that we observed the formation of a rotating corona also in simulations performed with a PCC half the size of the ones discussed in this chapter (σc = 5σ), and with an embedded point charge chosen to obtain comparable charge-dipole interactions at the colloidal surface. As is the case for all active systems, we expect hydrodynamics to play a non-trivial role, possibly leading to enhanced coupling of the motion of particle complexes and of the self-assembled ADJP chains. More work is needed in this direction.

86 Conclusion

In this thesis, I have presented numerical simulations that explore how self-propelled particles can be used for the design of ‘smart’ micromachines. Despite their apparent simplicity, isotropic, hard active particles can mediate novel controllable interactions between passive particles, and can engage in positive feedback loops that induce active motion in passive components. More complex interactions between active particles can improve control over their motion and opens up the possibility of using active particles that have more ‘awareness’ of their surroundings. Active particles with long ranged, directional interactions can imitate the swarming and ﬂocking of living systems, a phenomenon which can be reproduced hierarchically at several length scales.

The remarkable range of behaviors seen throughout this thesis have all been produced with a relatively simple model for active motion. There are many other ways to simulate self-propelled particles, and each method has its own advantages and drawbacks. The most important aspect of these particles’ motion that the Active Brownian Particle simulation omits is hydrodynamics. Self-propelled particles swim by producing flows and long ranged interactions between these flows (even for hard particles) can couple particle motion. Experiments and theory have shown that active particles can be trapped at walls due to hydrodynamic interactions. Additionally, external flows can be used to steer active particles, and in some cases, even to allow particles to swim upstream. These effects are not included in the results presented in this thesis, but I anticipate that including hydrodynamics would quantitatively, but

87 not qualitatively alter these results. Creatively combining self-propelled particles with passive components in ways that are informed by relatively straightforward principles of active motion can yield exceptional control of mesoscale systems. From enhanced and steerable cargo transporters, to biologically or chemically powered energy sources, to agents that can control and tune self-assembly, the applications of self-propelled particles are only beginning to be realized. On a more whimsical note, analogies between self-propelled particles and living creatures are made somewhat poetically, but not unjustiﬁably so. One could be forgiven for looking at movies of these bits of plastic and metal, or even simulations (of these bits on metal) and imagining that there was some intentionality to the particles’ motion. At this point, any intelligence in an active system comes from the design of an experiment, but the presence of self-propelled particles at all scales of life suggests that active motion should be a component of a truly living synthetic organism.

88 Bibliography

[1] T. J. Pitcher, A. E. Magurran, and I. J. Winﬁeld. Fish in larger shoals ﬁnd food faster. Behavioral Ecology and Sociobiology, 10(2):149–151, April 1982.

[2] D. J. Hoare, J. Krause, N. Peuhkuri, and J.-G. J. Godin. Body size and shoaling in ﬁsh. Journal of Fish Biology, 57(6):1351–1366, December 2000.

[3] Manfred Milinski and Rolf Heller. Inﬂuence of a predator on the opti- mal foraging behaviour of sticklebacks (Gasterosteus aculeatus L.). Nature, 275(5681):642–644, October 1978.

[4] Brian L. Partridge, Jonas Johansson, and John Kalish. The structure of schools of giant blueﬁn tuna in Cape Cod Bay. Environmental Biology of Fishes, 9(3- 4):253–262, December 1983.

[5] R. McNeill Alexander. Hitching a lift hydrodynamically - in swimming, ﬂying and cycling. Journal of Biology, 3:7, May 2004.

[6] Fernando Peruani, Jrn Starru, Vladimir Jakovljevic, Lotte Sgaard-Andersen, Andreas Deutsch, and Markus Br. Collective Motion and Nonequilibrium Cluster Formation in Colonies of Gliding Bacteria. Physical Review Letters, 108(9):098102, February 2012.

[7] Andrey Sokolov and Igor S. Aranson. Physical Properties of Collective Motion in Suspensions of Bacteria. Physical Review Letters, 109(24):248109, December 2012.

[8] H. P. Zhang, Avraham Beer, E.-L. Florin, and Harry L. Swinney. Collective motion and density ﬂuctuations in bacterial colonies. Proceedings of the National Academy of Sciences, 107(31):13626–13630, August 2010.

[9] Andrey Sokolov, Igor S. Aranson, John O. Kessler, and Raymond E. Goldstein. Concentration Dependence of the Collective Dynamics of Swimming Bacteria. Physical Review Letters, 98(15):158102, April 2007.

[10] Craig W. Reynolds. Flocks, Herds and Schools: A Distributed Behavioral Model. In Proceedings of the 14th Annual Conference on Computer Graphics and Interactive Techniques, SIGGRAPH ’87, pages 25–34, New York, NY, USA, 1987. ACM.

89 [11] TamásVicsek, AndràsCzirók,Eshel Ben-Jacob, Inon Cohen, and Ofer Shochet. Novel Type of Phase Transition in a System of Self-Driven Particles. Physical Review Letters, 75(6):1226–1229, August 1995.

[12] John Toner and Yuhai Tu. Long-Range Order in a Two-Dimensional Dynamical xy Model: How Birds Fly Together. Physical Review Letters, 75(23):4326–4329, December 1995.

[13] P. Romanczuk, M. Br, W. Ebeling, B. Lindner, and L. Schimansky-Geier. Active Brownian particles. The European Physical Journal Special Topics, 202(1):1–162, March 2012.

[14] Jeremie Palacci, Stefano Sacanna, Asher Preska Steinberg, David J. Pine, and Paul M. Chaikin. Living Crystals of Light-Activated Colloidal Surfers. Science, 339(6122):936–940, February 2013.

[15] J Palacci, S Sacanna, S-H Kim, G-R Yi, DJ Pine, and PM Chaikin. Light- activated self-propelled colloids. Philosophical Transactions of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, 372(2029):20130372, 2014.

[16] M. C. Marchetti, J. F. Joanny, S. Ramaswamy, T. B. Liverpool, J. Prost, Madan Rao, and R. Aditi Simha. Hydrodynamics of soft active matter. Reviews of Modern Physics, 85(3):1143–1189, July 2013.

[17] Walter F. Paxton, Kevin C. Kistler, Christine C. Olmeda, Ayusman Sen, Sarah K. St. Angelo, Yanyan Cao, Thomas E. Mallouk, Paul E. Lammert, and Vincent H. Crespi. Catalytic Nanomotors: Autonomous Movement of Striped Nanorods. Journal of the American Chemical Society, 126(41):13424–13431, October 2004.

[18] Ramin Golestanian, Tanniemola B. Liverpool, and Armand Ajdari. Propulsion of a Molecular Machine by Asymmetric Distribution of Reaction Products. Phys. Rev. Lett., 94(22):220801, June 2005.

[19] R. Golestanian, T. B. Liverpool, and A. Ajdari. Designing phoretic micro- and nano-swimmers. New Journal of Physics, 9(5):126, 2007.

[20] Rustem F. Ismagilov, Alexander Schwartz, Ned Bowden, and George M. White- sides. Autonomous Movement and Self-Assembly. Angew. Chem. Int. Ed., 41(4):652–654, 2002.

[21] B. Alberts, A. Johnson, J. Lewis, M. Raﬀ, K. Roberts, and P. Walter. Molecular Biology of the Cell. Garland Science, New York & Oxford, 5th edition, 2008.

[22] Anne E. Hafner, Ludger Santen, Heiko Rieger, and M. Reza Shaebani. Run-and- pause dynamics of cytoskeletal motor proteins. Scientiﬁc Reports, 6, November 2016.

90 [23] Gabriel Popkin. The physics of life. Nature News, 529(7584):16, January 2016.

[24] Reinhard Lipowsky and Stefan Klumpp. Life is motion: multiscale motility of molecular motors. Physica A: Statistical Mechanics and its Applications, 352(1):53–112, July 2005.

[25] Daniel A. Fletcher and Julie A. Theriot. An introduction to cell motility for the physical scientist. Physical Biology, 1(1):T1, 2004.

[26] A. S. Chuck, M. F. Clarke, and B. O. Palsson. Retroviral Infection Is Limited by Brownian Motion. Human Gene Therapy, 7(13):1527–1534, August 1996.

[27] Junghae Suh, Denis Wirtz, and Justin Hanes. Eﬃcient active transport of gene nanocarriers to the cell nucleus. Proceedings of the National Academy of Sciences, 100(7):3878–3882, April 2003.

[28] Anatoly B. Kolomeisky. Motor Proteins and Molecular Motors: How to Operate Machines at Nanoscale. Journal of physics. Condensed matter : an Institute of Physics journal, 25(46), November 2013.

[29] Kazushi Kinbara and Takuzo Aida. Toward Intelligent Molecular Machines: Di- rected Motions of Biological and Artiﬁcial Molecules and Assemblies. Chemical Reviews, 105(4):1377–1400, April 2005.

[30] E. M. Purcell. Life at low Reynolds number. American Journal of Physics, 45(1):3–11, January 1977.

[31] Howard C. Berg and Robert A. Anderson. Bacteria Swim by Rotating their Flagellar Filaments. Nature, 245(5425):380–382, October 1973.

[32] Michael Silverman and Melvin Simon. Flagellar rotation and the mechanism of bacterial motility. Nature, 249(5452):73–74, May 1974.

[33] Graeme Lowe, Markus Meister, and Howard C. Berg. Rapid rotation of ﬂagellar bundles in swimming bacteria. Nature, 325(6105):637–640, February 1987.

[34] Raghunath Chelakkot, Arvind Gopinath, L. Mahadevan, and Michael F. Hagan. Flagellar dynamics of a connected chain of active, polar, Brownian particles. Journal of The Royal Society Interface, 11(92):20130884, March 2014.

[35] RémiDreyfus, Jean Baudry, Marcus L. Roper, Marc Fermigier, Howard A. Stone, and Jérme Bibette. Microscopic artificial swimmers. Nature, 437(7060):862–865, October 2005.

[36] J. Elgeti, R. G. Winkler, and G. Gompper. Physics of microswimmerssingle particle motion and collective behavior: a review. Reports on Progress in Physics, 78(5):056601, 2015.

91 [37] S. L. Tamm and G. A. Horridge. The Relation between the Orientation of the Central Fibrils and the Direction of Beat in Cilia of Opalina. Proceedings of the Royal Society of London B: Biological Sciences, 175(1040):219–233, August 1970.

[38] Timothy Sanchez, David Welch, Daniela Nicastro, and Zvonimir Dogic. Cilia- Like Beating of Active Microtubule Bundles. Science, 333(6041):456–459, July 2011.

[39] C Brennen and and H. Winet. Fluid Mechanics of Propulsion by Cilia and Flagella. Annual Review of Fluid Mechanics, 9(1):339–398, 1977.

[40] Hong-Ren Jiang, Natsuhiko Yoshinaga, and Masaki Sano. Active Motion of a Janus Particle by Self-Thermophoresis in a Defocused Laser Beam. Physical Review Letters, 105(26):268302, December 2010.

[41] John Gibbs and Yiping Zhao. Catalytic nanomotors: fabrication, mechanism, and applications. Frontiers of Materials Science, 5(1):25–39, 2011.

[42] Wei Wang, Wentao Duan, Suzanne Ahmed, Thomas E. Mallouk, and Ayus- man Sen. Small power: Autonomous nano- and micromotors propelled by self-generated gradients. Nano Today, 8(5):531–554, October 2013.

[43] Ivo Buttinoni, Giovanni Volpe, Felix K¨ummel,Giorgio Volpe, and Clemens Bechinger. Active Brownian motion tunable by light. Journal of Physics: Condensed Matter, 24(28):284129, 2012.

[44] Stephen J. Ebbens and Jonathan R. Howse. In pursuit of propulsion at the nanoscale. Soft Matter, 6(4):726–738, 2010.

[45] Aidan Brown and Wilson Poon. Ionic eﬀects in self-propelled Pt-coated Janus swimmers. Soft Matter, 10(22):4016–4027, 2014.

[46] S. J. Ebbens. Active colloids: Progress and challenges towards realising autonomous applications. Current Opinion in Colloid & Interface Science, 21:14– 23, February 2016.

[47] Huan J. Keh and Shing B. Chen. Diﬀusiophoresis and electrophoresis of colloidal cylinders. Langmuir, 9(4):1142–1149, April 1993.

[48] Dennis C. Prieve and Ronald Roman. Diﬀusiophoresis of a rigid sphere through a viscous electrolyte solution. Journal of the Chemical Society, Faraday Trans- actions 2: Molecular and Chemical Physics, 83(8):1287–1306, January 1987.

[49] J. P. Ebel, John L. Anderson, and D. C. Prieve. Diﬀusiophoresis of latex particles in electrolyte gradients. Langmuir, 4(2):396–406, March 1988.

92 [50] Juliane Simmchen, Jaideep Katuri, William E. Uspal, Mihail N. Popescu, Mykola Tasinkevych, and Samuel Sànchez. Topographical pathways guide chemical microswimmers. Nature Communications, 7:ncomms10598, February 2016. [51] S. Ebbens, D. A. Gregory, G. Dunderdale, J. R. Howse, Y. Ibrahim, T. B. Liverpool, and R. Golestanian. Electrokinetic effects in catalytic platinum- insulator Janus swimmers. EPL (Europhysics Letters), 106(5):58003, 2014. [52] J. Palacci, Stefano Sacanna, Adrian Vatchinsky, Paul M. Chaikin, and David J. Pine. Photoactivated Colloidal Dockers for Cargo Transportation. J. Am. Chem. Soc., 135(43):15978–15981, 2013. [53] Marco Leoni, Jurij Kotar, Bruno Bassetti, Pietro Cicuta, and Marco Cosentino Lagomarsino. A basic swimmer at low Reynolds number. Soft Matter, 5(2):472–476, 2009. [54] A. Snezhko, M. Belkin, I. S. Aranson, and W.-K. Kwok. Self-Assembled Mag- netic Surface Swimmers. Physical Review Letters, 102(11):118103, March 2009. [55] J R Melcher and G I Taylor. Electrohydrodynamics: A Review of the Role of Interfacial Shear Stresses. Annual Review of Fluid Mechanics, 1(1):111–146, January 1969. [56] Antoine Bricard, Jean-Baptiste Caussin, Nicolas Desreumaux, Olivier Dauchot, and Denis Bartolo. Emergence of macroscopic directed motion in populations of motile colloids. Nature, 503(7474):95–98, November 2013. [57] James R. Baylis, Ju Hun Yeon, Max H. Thomson, Amir Kazerooni, Xu Wang, Alex E. St John, Esther B. Lim, Diana Chien, Anna Lee, Jesse Q. Zhang, James M. Piret, Lindsay S. Machan, Thomas F. Burke, Nathan J. White, and Christian J. Kastrup. Self-propelled particles that transport cargo through flowing blood and halt hemorrhage. Science Advances, 1(9):e1500379, October 2015. [58] Nitin Kumar, Harsh Soni, Sriram Ramaswamy, and A. K. Sood. Flocking at a distance in active granular matter. Nature Communications, 5:ncomms5688, September 2014. [59] Vijay Narayan, Sriram Ramaswamy, and Narayanan Menon. Long-Lived Giant Number Fluctuations in a Swarming Granular Nematic. Science, 317(5834):105–108, July 2007. [60] Xiaolu Wang, Martin In, Christophe Blanc, Maurizio Nobili, and Antonio Stocco. Enhanced active motion of Janus colloids at the water surface. Soft Matter, 11(37):7376–7384, 2015. [61] Sriram Ramaswamy. The Mechanics and Statistics of Active Matter. Annual Review of Condensed Matter Physics, 1(1):323–345, 2010.

93 [62] JérémiePalacci, CécileCottin-Bizonne, Christophe Ybert, and LydéricBoc- quet. Sedimentation and Effective Temperature of Active Colloidal Suspensions. Physical Review Letters, 105(8):088304, August 2010.

[63] Krishna Kanti Dey and Ayusman Sen. Chemically Propelled Molecules and Machines. Journal of the American Chemical Society, May 2017.

[64] Walter F. Paxton, Shakuntala Sundararajan, Thomas E. Mallouk, and Ayus- man Sen. Chemical Locomotion. Angewandte Chemie International Edition, 45(33):5420–5429, August 2006.

[65] Leonardo F. Valadares, Yu-Guo Tao, Nicole S. Zacharia, Vladimir Kitaev, Fer- nando Galembeck, Raymond Kapral, and Geoﬀrey A. Ozin. Catalytic Nanomo- tors: Self-Propelled Sphere Dimers. Small, 6(4):565–572, February 2010.

[66] Ivo Buttinoni, Julian Bialké, Felix Kümmel, Hartmut Löwen, Clemens Bechinger, and Thomas Speck. Dynamical Clustering and Phase Separation in Suspensions of Self-Propelled Colloidal Particles. Physical Review Letters, 110(23):238301, June 2013.

[67] Wei Gao, Mattia D’Agostino, Victor Garcia-Gradilla, Jahir Orozco, and Joseph Wang. Multi-Fuel Driven Janus Micromotors. Small, 9(3):467–471, 2013.

[68] Jing Yan, Ming Han, Jie Zhang, Cong Xu, Erik Luijten, and Steve Granick. Reconﬁguring active particles by electrostatic imbalance. Nature Materials, 15(10):1095–1099, October 2016.

[69] Stephen Ebbens, Richard A. L. Jones, Anthony J. Ryan, Ramin Golestanian, and Jonathan R. Howse. Self-assembled autonomous runners and tumblers. Physical Review E, 82(1):015304, July 2010.

[70] Howard C. Berg and Douglas A. Brown. Chemotaxis in Escherichia coli analysed by Three-dimensional Tracking. Nature, 239(5374):500–504, October 1972.

[71] W. R. Schneider and R. N. Doetsch. Velocity Measurements of Motile Bacteria by Use of a Videotape Recording Technique. Applied Microbiology, 27(1):283– 284, January 1974.

[72] H. G. Othmer, S. R. Dunbar, and W. Alt. Models of dispersal in biological systems. Journal of Mathematical Biology, 26(3):263–298, 1988.

[73] K. Martens, L. Angelani, R. Di Leonardo, and L. Bocquet. Probability distributions for the run-and-tumble bacterial dynamics: An analogy to the Lorentz model. The European Physical Journal E, 35(9):84, September 2012.

[74] L. G. Wilson, V. A. Martinez, J. Schwarz-Linek, J. Tailleur, G. Bryant, P. N. Pusey, and W. C. K. Poon. Diﬀerential Dynamic Microscopy of Bacterial Motil- ity. Physical Review Letters, 106(1):018101, January 2011.

94 [75] Borge ten Hagen, Sven van Teeﬀelen, and Hartmut L¨owen. Brownian motion of a self-propelled particle. Journal of Physics: Condensed Matter, 23(19):194119, May 2011. arXiv: 1005.1343.

[76] Xu Zheng, Borge ten Hagen, Andreas Kaiser, Meiling Wu, Haihang Cui, Zhan- hua Silber-Li, and Hartmut L¨owen. Non-Gaussian statistics for the motion of self-propelled Janus particles: Experiment versus theory. Physical Review E, 88(3):032304, September 2013.

[77] M. E. Cates and J. Tailleur. When are active Brownian particles and run-and- tumble particles equivalent? Consequences for motility-induced phase separation. EPL (Europhysics Letters), 101(2):20010, 2013.

[78] A. P. Solon, M. E. Cates, and J. Tailleur. Active brownian particles and run- and-tumble particles: A comparative study. The European Physical Journal Special Topics, 224(7):1231–1262, July 2015.

[79] W. C. K. Poon. From Clarkia to Escherichia and Janus: the physics of natural and synthetic active colloids. arXiv:1306.4799 [cond-mat, physics:physics, q- bio], June 2013. arXiv: 1306.4799.

[80] Michael E. Cates and Julien Tailleur. Motility-Induced Phase Separation. An- nual Review of Condensed Matter Physics, 6(1):219–244, 2015.

[81] Julien Barr´e,Raphal Ch´etrite,Massimiliano Muratori, and Fernando Peruani. Motility-Induced Phase Separation of Active Particles in the Presence of Veloc- ity Alignment. Journal of Statistical Physics, 158(3):589–600, February 2015.

[82] Giuseppe Gonnella, Davide Marenduzzo, Antonio Suma, and Adriano Tiri- bocchi. Motility-induced phase separation and coarsening in active matter. Comptes Rendus Physique, 16(3):316–331, April 2015.

[83] A. Suma, G. Gonnella, D. Marenduzzo, and E. Orlandini. Motility-induced phase separation in an active dumbbell ﬂuid. EPL (Europhysics Letters), 108(5):56004, 2014.

[84] Joakim Stenhammar, Raphael Wittkowski, Davide Marenduzzo, and Michael E. Cates. Activity-Induced Phase Separation and Self-Assembly in Mixtures of Active and Passive Particles. Physical Review Letters, 114(1):018301, January 2015.

[85] S. C. Takatori and J. F. Brady. Towards a thermodynamics of active matter. Physical Review E, 91(3):032117, March 2015.

[86] S. C. Takatori, W. Yan, and J. F. Brady. Swim Pressure: Stress Generation in Active Matter. Physical Review Letters, 113(2):028103, July 2014.

[87] Sho C. Takatori and John F. Brady. Swim stress, motion, and deformation of active matter: eﬀect of an external ﬁeld. Soft Matter, 10(47):9433–9445, 2014.

95 [88] Yaouen Fily and M. Cristina Marchetti. Athermal Phase Separation of Self-Propelled Particles with No Alignment. Physical Review Letters, 108(23):235702, June 2012.

[89] M. Cristina Marchetti, Yaouen Fily, Silke Henkes, Adam Patch, and David Yllanes. Minimal model of active colloids highlights the role of mechanical interactions in controlling the emergent behavior of active matter. Current Opinion in Colloid & Interface Science, 21:34–43, February 2016.

[90] J. Tailleur and M. E. Cates. Statistical Mechanics of Interacting Run-and- Tumble Bacteria. Physical Review Letters, 100(21):218103, May 2008.

[91] J. Harder, S. A. Mallory, C. Tung, C. Valeriani, and A. Cacciuto. The role of particle shape in active depletion. The Journal of Chemical Physics, 141(19):194901, November 2014.

[92] J. Harder, C. Valeriani, and A. Cacciuto. Activity-induced collapse and reex- pansion of rigid polymers. Physical Review E, 90(6):062312, December 2014.

[93] Clarion Tung, Joseph Harder, C. Valeriani, and A. Cacciuto. Micro-phase separation in two dimensional suspensions of self-propelled spheres and dumbbells. Soft Matter, 12(2):555–561, 2016.

[94] S. A. Mallory, A. Sari´c,C.ˇ Valeriani, and A. Cacciuto. Anomalous Thermo- mechanical Properties of a Self-propelled Colloidal Fluid. Physical Review E, 89(5), May 2014.

[95] Yaouen Fily, Aparna Baskaran, and Michael F. Hagan. Dynamics and density distribution of strongly conﬁned noninteracting nonaligning self-propelled particles in a nonconvex boundary. Phys. Rev. E, 91(1):012125, January 2015.

[96] Yaouen Fily, Aparna Baskaran, and Michael F. Hagan. Dynamics of Self- Propelled Particles Under Strong Conﬁnement. Soft Matter, 10:5609, 2014.

[97] Jens Elgeti and Gerhard Gompper. Wall accumulation of self-propelled spheres. EPL (Europhysics Letters), 101(4):48003, 2013.

[98] Jens Elgeti and Gerhard Gompper. Run-and-tumble dynamics of self-propelled particles in conﬁnement. EPL (Europhysics Letters), 109(5):58003, 2015.

[99] Chiu Fan Lee. Active particles under conﬁnement: aggregation at the wall and gradient formation inside a channel. New Journal of Physics, 15(5):055007, 2013.

[100] Nikolai Nikola, Alexandre P. Solon, Yariv Kafri, Mehran Kardar, Julien Tailleur, and Raphal Voituriez. Active Particles with Soft and Curved Walls: Equation of State, Ratchets, and Instabilities. Physical Review Letters, 117(9):098001, August 2016.

96 [101] H. H. Wensink and H. L¨owen. Aggregation of self-propelled colloidal rods near conﬁning walls. Phys. Rev. E, 78(3):031409, 2008.

[102] Peter Galajda, Juan Keymer, Paul Chaikin, and Robert Austin. A Wall of Funnels Concentrates Swimming Bacteria. J. Bacteriol., 189(23):8704–8707, 2007.

[103] Frank Smallenburg and Hartmut L¨owen. Swim pressure on walls with curves and corners. Physical Review E, 92(3):032304, September 2015.

[104] Fabricio Q. Potiguar, G. A. Farias, and W. P. Ferreira. Self-propelled particle transport in regular arrays of rigid asymmetric obstacles. Physical Review E, 90(1):012307, July 2014.

[105] Luca Angelani, Roberto Di Leonardo, and Giancarlo Ruocco. Self-Starting Micromotors in a Bacterial Bath. Phys. Rev. Lett., 102(4):048104, January 2009.

[106] R. Di Leonardo, L. Angelani, D. DellArciprete, G. Ruocco, V. Iebba, S. Schippa, M. P. Conte, F. Mecarini, F. De Angelis, and E. Di Fabrizio. Bacterial ratchet motors. Proc. Natl. Acad. Sci. USA, 107(21):9541–9545, 2010.

[107] M. E. Cates. Diﬀusive transport without detailed balance in motile bacteria: does microbiology need statistical physics? Reports on Progress in Physics, 75(4):042601, 2012.

[108] SA Mallory, C Valeriani, and A Cacciuto. Curvature-induced activation of a passive tracer in an active bath. Physical Review E, 90(3):032309, 2014.

[109] Luca Angelani and Roberto Di Leonardo. Geometrically biased random walks in bacteria-driven micro-shuttles. New J. Phys., 12(11):113017, 2010.

[110] Bao-quan Ai, Qiu-yan Chen, Ya-feng He, Feng-guo Li, and Wei-rong Zhong. Rectiﬁcation and diﬀusion of self-propelled particles in a two-dimensional cor- rugated channel. Phys. Rev. E, 88(6):062129, 2013.

[111] A. Pototsky, A. M. Hahn, and H. Stark. Rectiﬁcation of self-propelled particles by symmetric barriers. Phys. Rev. E, 87(4):042124, 2013.

[112] M. B. Wan, C. J. Olson Reichhardt, Z. Nussinov, and C. Reichhardt. Recti- ﬁcation of Swimming Bacteria and Self-Driven Particle Systems by Arrays of Asymmetric Barriers. Phys. Rev. Lett., 101(1):018102, 2008.

[113] J. Tailleur and M. E. Cates. Sedimentation, trapping, and rectiﬁcation of dilute bacteria. EPL (Europhysics Letters), 86(6):60002, 2009.

[114] Andrey Sokolov, Mario M. Apodaca, Bartosz A. Grzybowski, and Igor S. Aran- son. Swimming bacteria power microscopic gears. Proc. Natl. Acad. Sci. USA, 107(3):969–974, 2010.

97 [115] Yuichi Hiratsuka, Makoto Miyata, Tetsuya Tada, and Taro Q. P. Uyeda. A microrotary motor powered by bacteria. Proceedings of the National Academy of Sciences, 103(37):13618–13623, September 2006.

[116] N Koumakis, A Lepore, C Maggi, and R Di Leonardo. Targeted delivery of colloids by swimming bacteria. Nat. Commun., 4, 2013.

[117] Sho Asakura and Fumio Oosawa. Interaction between particles suspended in solutions of macromolecules. J. Polym. Sci., 33(126):183–192, 1958.

[118] Davide Marenduzzo, Kieran Finan, and Peter R. Cook. The depletion attraction: an underappreciated force driving cellular organization. The Journal of Cell Biology, 175(5):681–686, December 2006.

[119] JérémiePalacci, Benjamin Abécassis,CécileCottin-Bizonne, Christophe Ybert, and LydéricBocquet. Colloidal Motility and Pattern Formation under Rectified Diffusiophoresis. Phys. Rev. Lett., 104(13):138302, April 2010.

[120] Walter F. Paxton, Paul T. Baker, Timothy R. Kline, Yang Wang, Thomas E. Mallouk, and Ayusman Sen. Catalytically Induced Electrokinetics for Motors and Micropumps. J. Am. Chem. Soc., 128(46):14881–14888, 2006.

[121] Joakim Stenhammar, Davide Marenduzzo, Rosalind J. Allen, and Michael E. Cates. Phase behaviour of active Brownian particles: the role of dimensionality. Soft Matter, 10(10):1489–1499, 2014.

[122] Marjolein Dijkstra, Joseph M. Brader, and Robert Evans. Phase behaviour and structure of model colloid-polymer mixtures. J. Phys.: Condens. Matter, 11(50):10079, 1999.

[123] Marjolein Dijkstra, Ren´evan Roij, and Robert Evans. Phase diagram of highly asymmetric binary hard-sphere mixtures. Phys. Rev. E, 59(5):5744–5771, May 1999.

[124] Marjolein Dijkstra, Ren´evan Roij, Roland Roth, and Andrea Fortini. Eﬀect of many-body interactions on the bulk and interfacial phase behavior of a model colloid-polymer mixture. Phys. Rev. E, 73(4):041404, April 2006.

[125] J. Dzubiella, H. L¨owen, and C. N. Likos. Depletion Forces in Nonequilibrium. Phys. Rev. Lett., 91(24):248301, December 2003.

[126] J. Schwarz-Linek, C. Valeriani, A. Cacciuto, M. E. Cates, D. Marenduzzo, A. N. Morozov, and W. C. K. Poon. Phase separation and rotor self-assembly in active particle suspensions. Proc. Natl. Acad. Sci. USA, 109(11):4052–4057, 2012.

[127] Subir K. Das, Sergei A. Egorov, Benjamin Trefz, Peter Virnau, and Kurt Binder. Phase Behavior of Active Swimmers in Depletants: Molecular Dy- namics and Integral Equation Theory. Phys. Rev. Lett., 112(19):198301, May 2014.

98 [128] D. Ray, C. Reichhardt, and C. J. Olson Reichhardt. Casimir effect in active matter systems. Physical Review E, 90(1):013019, July 2014. [129] C. Parra-Rojas and R. Soto. Casimir Effect in Swimmer Suspensions. Phys. Rev. E, 90:013024, 2014. [130] Ran Ni, Martien A. Cohen Stuart, and Peter G. Bolhuis. Tunable Long Range Forces Mediated by Self-Propelled Colloidal Hard Spheres. Physical Review Letters, 114(1):018302, January 2015. [131] L. Angelani, C. Maggi, M. L. Bernardini, A. Rizzo, and R. Di Leonardo. Effec- tive Interactions between Colloidal Particles Suspended in a Bath of Swimming Cells. Phys. Rev. Lett., 107(13):138302, September 2011. [132] Steve Plimpton. Fast parallel algorithms for short-range molecular dynamics. J. Comput. Phys., 117(1):1–19, 1995. [133] D. Chandler. Introduction to Modern Statistical Mechanics. Oxford University Press, 1987. [134] William Humphrey, Andrew Dalke, and Klaus Schulten. VMD Visual Molec- ular Dynamics. Journal of Molecular Graphics, 14:33–38, 1996. [135] E. J. Marsden, C. Valeriani, I. Sullivan, M. E. Cates, and D. Marenduzzo. Chemotactic clusters in confined run-and-tumble bacteria: a numerical investi- gation. Soft Matter, 10(1):157–165, November 2013. [136] Alexander Morozov and Davide Marenduzzo. Enhanced diffusion of tracer particles in dilute bacterial suspensions. Soft Matter, 10(16):2748–2758, 2014. [137] Andreas Kaiser, Anton Peshkov, Andrey Sokolov, Borge ten Hagen, Hartmut Löwen, and Igor S. Aranson. Transport Powered by Bacterial Turbulence. Phys. Rev. Lett., 112(15):158101, April 2014. [138] A. Kaiser, H. H. Wensink, and H. Löwen. How to Capture Active Particles. Phys. Rev. Lett., 108(26):268307, 2012. [139] A. Kaiser, K. Popowa, H. H. Wensink, and H. Löwen. Capturing self-propelled particles in a moving microwedge. Phys. Rev. E, 88(2):022311, 2013. [140] H. H. Wensink, V. Kantsler, R. E. Goldstein, and J. Dunkel. Controlling active self-assembly through broken particle-shape symmetry. Phys. Rev. E, 89(1):010302, 2014. [141] Xiao-Lun Wu and Albert Libchaber. Particle Diffusion in a Quasi-Two- Dimensional Bacterial Bath. Phys. Rev. Lett., 84(13):3017–3020, March 2000. [142] Kyriacos C. Leptos, Jeffrey S. Guasto, J. P. Gollub, Adriana I. Pesci, and Raymond E. Goldstein. Dynamics of Enhanced Tracer Diffusion in Suspensions of Swimming Eukaryotic Microorganisms. Phys. Rev. Lett., 103(19):198103, November 2009.

99 [143] A. Kaiser and H. L¨owen. Unusual swelling of a polymer in a bacterial bath. J. Chem. Phys., 141(4):–, 2014.

[144] Pai-Yi Hsiao and Erik Luijten. Salt-Induced Collapse and Reexpan- sion of Highly Charged Flexible Polyelectrolytes. Physical Review Letters, 97(14):148301, October 2006.

[145] J. Rudnick and G. Gaspari. The aspherity of random walks. J. Phys. A, 19(4):L191, 1986.

[146] S. A. Mallory, C. Valeriani, and A. Cacciuto. Anomalous dynamics of an elastic membrane in an active ﬂuid. Physical Review E, 92(1):012314, July 2015.

[147] M. Ballerini, N. Cabibbo, R. Candelier, A. Cavagna, E. Cisbani, I. Giardina, V. Lecomte, A. Orlandi, G. Parisi, A. Procaccini, M. Viale, and V. Zdravkovic. Interaction ruling animal collective behavior depends on topological rather than metric distance: Evidence from a ﬁeld study. Proceedings of the National Academy of Sciences of the United States of America, 105(4):1232–1237, Jan- uary 2008.

[148] Simon Hubbard, Petro Babak, Sven Th. Sigurdsson, and Kjartan G. Magnsson. A model of the formation of ﬁsh schools and migrations of ﬁsh. Ecological Modelling, 174(4):359–374, June 2004.

[149] Jesse L. Silverberg, Matthew Bierbaum, James P. Sethna, and Itai Cohen. Collective Motion of Humans in Mosh and Circle Pits at Heavy Metal Concerts. Physical Review Letters, 110(22):228701, May 2013.

[150] Henricus H. Wensink, Jrn Dunkel, Sebastian Heidenreich, Knut Drescher, Ray- mond E. Goldstein, Hartmut L¨owen, and Julia M. Yeomans. Meso-scale turbulence in living ﬂuids. Proceedings of the National Academy of Sciences, 109(36):14308–14313, September 2012.

[151] H. H. Wensink and H. L¨owen. Emergent states in dense systems of active rods: from swarming to turbulence. Journal of Physics: Condensed Matter, 24(46):464130, 2012.

[152] A. J. Koch and H. Meinhardt. Biological pattern formation: from basic mechanisms to complex structures. Reviews of Modern Physics, 66(4):1481–1507, October 1994.

[153] Elena Vedemenko. Competing Interactions and Pattern Formation in Nanoworld. John Wiley & Sons, 2007.

[154] Ludwik Leibler. Theory of Microphase Separation in Block Copolymers. Macro- molecules, 13(6):1602–1617, November 1980.

100 [155] Adam Blanazs, Steven P. Armes, and Anthony J. Ryan. Self-Assembled Block Copolymer Aggregates: From Micelles to Vesicles and their Biological Appli- cations. Macromolecular Rapid Communications, 30(4-5):267–277, February 2009.

[156] Thomas Smart, Hannah Lomas, Marzia Massignani, Miriam V. Flores-Merino, Lorena Ruiz Perez, and Giuseppe Battaglia. Block copolymer nanostructures. Nano Today, 3(3):38–46, June 2008.

[157] Ho-Cheol Kim, Sang-Min Park, and William D. Hinsberg. Block Copolymer Based Nanostructures: Materials, Processes, and Applications to Electronics. Chemical Reviews, 110(1):146–177, January 2010.

[158] Anna Stradner, Helen Sedgwick, Fr´ed´ericCardinaux, Wilson C. K. Poon, Ste- fan U. Egelhaaf, and Peter Schurtenberger. Equilibrium cluster formation in concentrated protein solutions and colloids. Nature, 432(7016):492–495, Novem- ber 2004.

[159] Francesco Sciortino, Stefano Mossa, Emanuela Zaccarelli, and Piero Tartaglia. Equilibrium Cluster Phases and Low-Density Arrested Disordered States: The Role of Short-Range Attraction and Long-Range Repulsion. Physical Review Letters, 93(5):055701, July 2004.

[160] A. Imperio and L. Reatto. A bidimensional ﬂuid system with competing interactions: spontaneous and induced pattern formation. Journal of Physics: Condensed Matter, 16(38):S3769, 2004.

[161] A. Imperio and L. Reatto. Microphase separation in two-dimensional systems with competing interactions. The Journal of Chemical Physics, 124(16):164712, April 2006.

[162] Alessandra Imperio and Luciano Reatto. Microphase morphology in two- dimensional ﬂuids under lateral conﬁnement. Physical Review E, 76(4):040402, October 2007.

[163] Alessandra Imperio, Luciano Reatto, and Stefano Zapperi. Rheology of colloidal microphases in a model with competing interactions. Physical Review E, 78(2):021402, August 2008.

[164] K. DeBell, A. B. MacIsaac, and J. P. Whitehead. Dipolar eﬀects in magnetic thin ﬁlms and quasi-two-dimensional systems. Reviews of Modern Physics, 72(1):225–257, January 2000.

[165] C. B. Muratov. Theory of domain patterns in systems with long-range interactions of Coulomb type. Physical Review E, 66(6):066108, December 2002.

[166] F. Ghezzi and J. C. Earnshaw. Formation of meso-structures in colloidal monolayers. Journal of Physics: Condensed Matter, 9(37):L517, 1997.

101 [167] F. Ghezzi, J. C. Earnshaw, M. Finnis, and M. McCluney. Pattern Formation in Colloidal Monolayers at the AirWater Interface. Journal of Colloid and Interface Science, 238(2):433–446, June 2001. [168] I. Theurkauff, C. Cottin-Bizonne, J. Palacci, C. Ybert, and L. Bocquet. Dy- namic Clustering in Active Colloidal Suspensions with Chemical Signaling. Physical Review Letters, 108(26):268303, June 2012. [169] B. M. Mognetti, A. Sarić,S.ˇ Angioletti-Uberti, A. Cacciuto, C. Valeriani, and D. Frenkel. Living Clusters and Crystals from Low-Density Suspensions of Active Colloids. Physical Review Letters, 111(24):245702, December 2013. [170] Vasileios Prymidis, Harmen Sielcken, and Laura Filion. Self-assembly of active attractive spheres. Soft Matter, 11(21):4158–4166, May 2015. [171] Philip Ball. Particle Clustering Phenomena Inspire Multiple Explanations. Physics Online Journal, 6:134, December 2013. [172] Nils Heptner and Joachim Dzubiella. Equilibrium structure and fluctuations of suspensions of colloidal dumbbells. Molecular Physics, 113(17-18):2523–2530, September 2015. [173] John G. Kirkwood and Jacob Riseman. The Intrinsic Viscosities and Diffusion Constants of Flexible Macromolecules in Solution. The Journal of Chemical Physics, 16(6):565–573, June 1948. [174] Angel J. Moreno and Christos N. Likos. Diffusion and Relaxation Dynamics in Cluster Crystals. Physical Review Letters, 99(10):107801, September 2007. [175] Bianca M. Mladek, Patrick Charbonneau, and Daan Frenkel. Phase Coexistence of Cluster Crystals: Beyond the Gibbs Phase Rule. Physical Review Letters, 99(23):235702, December 2007. [176] David R. Nelson and B. I. Halperin. Dislocation-mediated melting in two dimensions. Physical Review B, 19(5):2457–2484, March 1979. [177] Gabriel S. Redner, Michael F. Hagan, and Aparna Baskaran. Structure and Dy- namics of a Phase-Separating Active Colloidal Fluid. Physical Review Letters, 110(5):055701, January 2013. [178] Gabriel S. Redner, Aparna Baskaran, and Michael F. Hagan. Reentrant phase behavior in active colloids with attraction. Physical Review E, 88(1):012305, July 2013. [179] Ethayaraja Mani and Hartmut Löwen. Effect of self-propulsion on equilibrium clustering. Physical Review E, 92(3):032301, September 2015. [180] Ricard Matas Navarro and Suzanne M. Fielding. Clustering and phase behaviour of attractive active particles with hydrodynamics. Soft Matter, 11(38):7525–7546, September 2015.

102 [181] Clemens Bechinger, Roberto Di Leonardo, Hartmut Löwen, Charles Reichhardt, Giorgio Volpe, and Giovanni Volpe. Active particles in complex and crowded environments. Reviews of Modern Physics, 88(4):045006, November 2016. [182] Larysa Baraban, Denys Makarov, Robert Streubel, Ingolf Mnch, Daniel Grimm, Samuel Sanchez, and Oliver G. Schmidt. Catalytic Janus Motors on Microflu- idic Chip: Deterministic Motion for Targeted Cargo Delivery. ACS Nano, 6(4):3383–3389, April 2012. [183] Claudio Maggi, Juliane Simmchen, Filippo Saglimbeni, Jaideep Katuri, Michele Dipalo, Francesco De Angelis, Samuel Sanchez, and Roberto Di Leonardo. Self- Assembly of Micromachining Systems Powered by Janus Micromotors. Small, 12(4):446–451, January 2016. [184] Sabine H. L. Klapp. Collective dynamics of dipolar and multipolar colloids: From passive to active systems. Current Opinion in Colloid & Interface Science, 21:76–85, February 2016. [185] Andreas Walther and Axel H. E. Müller. Janus Particles: Synthesis, Self-Assembly, Physical Properties, and Applications. Chemical Reviews, 113(7):5194–5261, July 2013. [186] Shan Jiang, Qian Chen, Mukta Tripathy, Erik Luijten, Kenneth S. Schweizer, and Steve Granick. Janus Particle Synthesis and Assembly. Advanced Materials, 22(10):1060–1071, March 2010. [187] Jie Zhang, Bartosz A. Grzybowski, and Steve Granick. Janus Particle Synthesis, Assembly, and Application. Langmuir, 33(28):6964–6977, July 2017. [188] Wei Gao, Allen Pei, Xiaomiao Feng, Camille Hennessy, and Joseph Wang. Or- ganized Self-Assembly of Janus Micromotors with Hydrophobic Hemispheres. Journal of the American Chemical Society, 135(3):998–1001, January 2013. [189] Joel N. Johnson, Amir Nourhani, Robert Peralta, Christopher McDonald, Benjamin Thiesing, Christopher J. Mann, Paul E. Lammert, and John G. Gibbs. Dynamic stabilization of Janus sphere trans-dimers. Physical Review E, 95(4):042609, April 2017. [190] David M. Rutkowski, Orlin D. Velev, Sabine H. L. Klapp, and Carol K. Hall. Simulation study on the structural properties of colloidal particles with offset dipoles. Soft Matter, 13(17):3134–3146, 2017. [191] Bhuvnesh Bharti and Orlin D. Velev. Assembly of Reconfigurable Col- loidal Structures by Multidirectional Field-Induced Interactions. Langmuir, 31(29):7897–7908, July 2015. [192] Heiko Schmidle, Carol K. Hall, Orlin D. Velev, and Sabine H. L. Klapp. Phase diagram of two-dimensional systems of dipole-like colloids. Soft Mat- ter, 8(5):1521–1531, January 2012.

103 [193] Bin Ren, Aleksey Ruditskiy, Jung Hun (Kevin) Song, and Ilona Kretzschmar. Assembly Behavior of Iron Oxide-Capped Janus Particles in a Magnetic Field. Langmuir, 28(2):1149–1156, January 2012.

[194] Stoyan K. Smoukov, Sumit Gangwal, Manuel Marquez, and Orlin D. Velev. Reconﬁgurable responsive structures assembled from magnetic Janus particles. Soft Matter, 5(6):1285–1292, March 2009.

[195] Qian Chen, Jonathan K. Whitmer, Shan Jiang, Sung Chul Bae, Erik Luijten, and Steve Granick. Supracolloidal Reaction Kinetics of Janus Spheres. Science, 331(6014):199–202, January 2011.

[196] Liang Hong, Angelo Cacciuto, Erik Luijten, and Steve Granick. Clusters of Charged Janus Spheres. Nano Letters, 6(11):2510–2514, November 2006.

[197] Liang Hong, Angelo Cacciuto, Erik Luijten, and Steve Granick. Clusters of Amphiphilic Colloidal Spheres. Langmuir, 24(3):621–625, February 2008.

[198] Andreas Kaiser, Katarina Popowa, and Hartmut L¨owen. Active dipole clusters: From helical motion to ﬁssion. Physical Review E, 92(1):012301, July 2015.

[199] Francisca Guzm`an-Lastra,Andreas Kaiser, and Hartmut L¨owen. Fission and fusion scenarios for magnetic microswimmer clusters. Nature Communications, 7, November 2016.

[200] Juliane Simmchen, Alejandro Baeza, Daniel Ruiz, Maria Jos´e Esplandiu, and Maria Vallet-Reg. Asymmetric Hybrid Silica Nanomotors for Capture and Cargo Transport: Towards a Novel Motion-Based DNA Sensor. Small, 8(13):2053–2059, July 2012.

[201] Shakuntala Sundararajan, Paul E. Lammert, Andrew W. Zudans, Vincent H. Crespi, and Ayusman Sen. Catalytic Motors for Transport of Colloidal Cargo. Nano Letters, 8(5):1271–1276, May 2008.

[202] Eric W. Burkholder and John F. Brady. Tracer diﬀusion in active suspensions. Physical Review E, 95(5):052605, May 2017.

[203] Matthew C. Hagy and Rigoberto Hernandez. Dynamical simulation of dipolar Janus colloids: Equilibrium structure and thermodynamics. The Journal of Chemical Physics, 137(4):044505, July 2012.

[204] William Humphrey, Andrew Dalke, and Klaus Schulten. VMD: Visual molecular dynamics. Journal of Molecular Graphics, 14(1):33–38, February 1996.

[205] Masoud Abkenar, Kristian Marx, Thorsten Auth, and Gerhard Gompper. Col- lective behavior of penetrable self-propelled rods in two dimensions. Physical Review E, 88(6):062314, December 2013.

104 [206] Yingzi Yang, Vincent Marceau, and Gerhard Gompper. Swarm behavior of self-propelled rods and swimming ﬂagella. Physical Review E, 82(3):031904, September 2010.

[207] Stephen J. DeCamp, Gabriel S. Redner, Aparna Baskaran, Michael F. Hagan, and Zvonimir Dogic. Orientational order of motile defects in active nematics. Nature Materials, 14(11):1110–1115, November 2015.

[208] Clara Abaurrea Velasco, Sepehr Dehghani Ghahnaviyeh, Hossein Nejat Pishke- nari, Thorsten Auth, and Gerhard Gompper. Complex self-propelled rings: a minimal model for cell motility. Soft Matter, 2017.

[209] Chantal Valeriani, Martin Li, John Novosel, Jochen Arlt, and Davide Maren- duzzo. Colloids in a bacterial bath: simulations and experiments. Soft Matter, 7(11):5228–5238, 2011.

105