artificial chemotaxis of self-phoretic active colloids: collective … · phoretic colloids near...

Artificial Chemotaxis of Self-Phoretic Active Colloids: CollectiveBehaviorPublished as part of the Accounts of Chemical Research special issue “Fundamental Aspects of Self-PoweredNano- and Micromotors”.

Holger Stark*

Technische Universitaẗ Berlin, Institute of Theoretical Physics, Hardenbergstrasse 36, D-10623 Berlin, Germany

CONSPECTUS: Microorganisms use chemotaxis, regulated by internal complexchemical pathways, to swim along chemical gradients to find better living conditions.Artificial microswimmers can mimic such a strategy by a pure physical process calleddiffusiophoresis, where they drift and orient along the gradient in a chemical density field.Similarly, for other forms of taxis in nature such as photo- or thermotaxis the phoreticcounterpart exists.In this Account, we concentrate on the chemotaxis of self-phoretic active colloids. Theyare driven by self-electro- and diffusiophoresis at the particle surface and thereby acquire aswimming speed. During this process, they also produce nonuniform chemical fields intheir surroundings through which they interact with other colloids by translational androtational diffusiophoresis. In combination with active motion, this gives rise to effectivephoretic attraction and repulsion and thereby to diverse emergent collective behavior. Aparticular appealing example is dynamic clustering in dilute suspensions first reported by agroup from Lyon. A subtle balance of attraction and repulsion causes very dynamicclusters, which form and resolve again. This is in stark contrast to the relatively static clusters of motility-induced phaseseparation at larger densities.To treat chemotaxis in active colloids confined to a plane, we formulate two Langevin equations for position and orientation,which include translational and rotational diffusiophoretic drift velocities. The colloids are chemical sinks and develop theirlong-range chemical profiles instantaneously. For dense packings, we include screening of the chemical fields.We present a state diagram in the two diffusiophoretic parameters governing translational, as well as rotational, drift and,thereby, explore the full range of phoretic attraction and repulsion. The identified states range from a gaslike phase overdynamic clustering states 1 and 2, which we distinguish through their cluster size distributions, to different types of collapsedstates. The latter include a full chemotactic collapse for translational phoretic attraction. Turning it into an effective repulsion,with increasing strength first the collapsed cluster starts to fluctuate at the rim, then oscillates, and ultimately becomes a staticcollapsed cloud. We also present a state diagram without screening. Finally, we summarize how the famous Keller−Segel modelderives from our Langevin equations through a multipole expansion of the full one-particle distribution function in position andorientation. The Keller−Segel model gives a continuum equation for treating chemotaxis of microorganisms on the level of theirspatial density.Our theory is extensible to mixtures of active and passive particles and allows to include a dipolar correction to the chemicalfield resulting from the dipolar symmetry of Janus colloids.

1. INTRODUCTION

In nature microorganisms have developed the ability to sensetheir environment to direct their motion. An importantbehavioral response is called taxis. Microorganisms can senseand move along the gradient of an external stimulus or fieldand thereby are able to find better living conditions.1 The mostprominent example is chemotaxis, where the gradient is formedby the density of a chemical species,2 and many micro-organisms employ this strategy.2−6 Other forms of taxis foundfor living organisms are gravitaxis,7 rheotaxis,8 magnetotaxis9

phototaxis,10 or thermotaxis.11 To implement a taxis strategy,microorganisms change their swimming mode in response tofield gradients. For example, bacteria modify their tumble rate

using chemical sensors and an internal biochemical signalingpathway12−15 or light sensing algae alter the breast stroke oftheir flagella, which is triggered by light receptors.10

The rapidly evolving field of artificially designed micro-swimmers16,17 is also driven by the idea of mimicking differenttaxis strategies, now based on pure physical principles, tocontrol their swimming paths and explore a wide field ofintriguing applications. Indeed, in colloidal systems phoreticmotion induced by field gradients (phoresis) is wellestablished.18 Combined with activity, colloids exhibit novel

Received: June 3, 2018Published: October 16, 2018

Article

pubs.acs.org/accountsCite This: Acc. Chem. Res. 2018, 51, 2681−2688

© 2018 American Chemical Society 2681 DOI: 10.1021/acs.accounts.8b00259Acc. Chem. Res. 2018, 51, 2681−2688

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complex behavior while they sense temperature gradients,19−22

gravitational fields,23 perform phototaxis,24 or show diffusio-phoresis,25−29 where they drift and orient along chemicalgradients.In this Account, we concentrate on how chemical fields

influence the collective motion of self-phoretic activecolloids,28,30−33 which produce nonuniform chemical fieldsby consuming chemical reactants and generating products.Prominently, they show dynamic clustering25−27,29,34 orstimulate the assembly of passive colloids.35 In suspensionsof active AgCl particles a colloidal collapse and clusteroscillations were observed.36,37 Continuum theories reveal awealth of pattern formation.28,38,39 A detailed diffusiophoreticand hydrodynamic modeling reproduces the motion of self-phoretic colloids near boundaries with step-like topography.40

On the molecular level enzymes are established as nanomotors,which undergo chemotaxis.41−44 Hydrodynamic modeling ofchemical nanomotors either of Janus type or sphere dimersshow different kinds of clustering45−47 or orientational orderwhen pinned to a substrate.48 Finally, also emulsion dropletsexhibit chemotaxis.49

This Account summarizes our work on the chemotaxis ofself-phoretic active colloids.27,29 It was very much inspired byexperiments of ref 25. Gold particles half covered withplatinum, which catalyzes the reaction of H2O2 into waterand oxygen, self-propel through a combination of self-diffusio-and electrophoresis. They create nonuniform chemical fields intheir surroundings with chemical gradients, along whichneighboring particles drift (translational diffusiophoresis) orreorient (rotational diffusiophoresis). In combination with self-propulsion this gives rise to an effective phoretic attraction andrepulsion. A subtle balance of these effective interactions thencauses dynamic clustering in dilute suspensions, where clustersform and resolve again.25,26 This is in stark contrast to therelatively static clusters of motility-induced phase separation atlarger densities.50,51

In the following we summarize our particle-based Langevinmodel for addressing collective motion of self-phoretic activecolloids. In formulating the model, we wanted to concentrateon the essential features. Exploring the full range of phoreticattraction and repulsion, we illustrate the collective dynamicsof self-phoretic colloids ranging from a gas-like state, overdynamic clustering 1 and 2, to different types of collapsedstates including a full chemotactic collapse. The latter can berationalized by the Keller−Segel equation,52,53 which derivesfrom our Langevin model.

2. MODELTo explore the collective dynamics of self-phoretic colloidsinduced by diffusiophoretic or chemotactic interactions, we setup a simplified model system. We consider a colloidalmonolayer close to a bounding plate with fluid in the infinitehalf-space above it. The active colloids are Janus particlespartially covered by a catalyst, which catalyzes a chemicalreaction in the surrounding fluid. In general, through acombined process of self-electro- and diffusiophoresis, thecolloids move with a velocity v0 along the unit vector e, whichgives a direction fixed in the particle. In the following, weassume that v0 and e are not changed by the presence of othernearby particles. Instead of taking into account all reactantsand products of the chemical reaction, we simply consider theself-phoretic colloid as a chemical sink. It consumes a chemicaland thereby produces a nonuniform chemical field with

concentration c, in which nearby colloids perform translationaland rotational diffusiophoretic motion. Hydrodynamic inter-actions, which have a reduced range close to a no-slip surface,are neglected against these effective interactions. The influenceof hydrodynamic flow has been studied in different systemsand is reviewed in ref 17.2.1. Translational and Rotational Diffusiophoresis

The molecules or solutes of the chemical field interact with thesurface of a colloid. This results in a body force on the fluid,which influences fluid pressure. Ultimately a gradient in theconcentration c along the particle surface causes a pressuregradient, which then drives a slip velocity vs = ζ∇c along thecolloidal surface, where the slip velocity coefficient ζ dependson the surface interaction potential between the chemical andthe colloidal surface. Averaging vs over the particle surfacegives the translational diffusiophoretic velocity18

ζ ζ ∇= [⟨ ⟩ − ⟨ ⊗ − ⟩] cv 1 n n 1(3 )/2D (1)

where ∇c is evaluated at the particle center, n is the localsurface normal, and ⟨...⟩ means average over the particlesurface. For particles with uniform surface properties but alsofor half-coated Janus colloids the quadrupolar term in eq 1vanishes and we obtain the diffusiophoretic drift velocity,which we use in the following

ζ ∇= − cvD tr (2)

Here, ζtr ≔ −⟨ζ⟩ is the translational diffusiophoreticparameter.The slip velocity field also causes a diffusiophoretic

rotational velocity18

ω ζ ζ∇ ∇= ⟨ ⟩ × = − ×a

c cn e9

4 iD rot (3)

where the rotational diffusiophoretic parameter ζrot is definedby (9/4a)⟨ζn⟩ = −ζrote. Spherical particles with a uniformsurface (constant ζ) do not rotate. However, for half-coatedJanus particles ωD is nonzero, as long as the solutes interactdifferently with the two sides, and symmetry dictates thatswimming direction e is parallel to ⟨ζn⟩.Since the slip velocity coefficient ζ can be controlled by

choosing appropriate materials for the Janus colloids and theircaps, but also by the geometry and number of the capscovering the colloidal surface,28 we expect the phoreticparameters ζrot and ζtr to be tunable to either positive ornegative values. This opens the possibility to induce andexplore a variety of collective colloidal dynamics. For example,since each active particle acts as a chemical sink in our setting,neighboring particles will drift toward the sink for positivetranslational diffusiophoretic parameter ζtr in eq 2, whichcorresponds to an effective attraction, while ζtr < 0 gives rise toan effective repulsion. Similarly, a positive rotational parameterζrot in eq 3 rotates the swimming direction of an active colloidtoward a neighboring chemical sink and the colloid movestoward the sink. Hence, rotational phoresis also acts like anattractive colloidal interaction while it becomes repulsive forζrot < 0.2.2. Langevin Dynamics and Chemical Field

At the micron scale inertia is negligible and position ri andorientation ei of the ith Janus colloid obey overdampedLangevin equations. Adding up the deterministic translationalvelocities from self-propulsion and diffusiophoretic drift (v0ei +

Accounts of Chemical Research Article

DOI: 10.1021/acs.accounts.8b00259Acc. Chem. Res. 2018, 51, 2681−2688

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http://dx.doi.org/10.1021/acs.accounts.8b00259

vD) and using the kinematic relation for the orientation vector,ei̇ = ωD × ei, we obtain

ξζ ∇̇ = − +v cr e r( )i i i i0 tr (4)

μζ ∇̇ = − − ⊗ + ×ce 1 e e r e( ) ( )i i i i i irot (5)

The additional vectors represent translational (ξi) androtational (μi) white noise of thermal origin with zero meanand respective time correlation functions ⟨ξi(t) ⊗ ξi(t′)⟩ =2Dtr1δ(t − t′) and ⟨μi(t)⊗μi(t′)⟩ = 2Drot1δ(t − t′), where Dtrand Drot are the translational and rotational diffusioncoefficients, respectively. All the results presented in thisreview are generated by a two-dimensional version of eqs 4 and5, since the Janus colloids move in a monolayer. An effectivehard-core repulsion between the colloids is implemented.Whenever they overlap during the simulations, we separatethem along the line connecting their centers to the point ofcontact. Finally, hydrodynamic flow fields are not includedhere. Close to a bounding plane they are less long-ranged andwe also assume that chemical interactions dominate.According to our model assumption the colloids consume a

chemical with rate k, which typically diffuses much faster onthe micron scale than the Janus colloids swim. Therefore, whenthe colloids move, they carry around with them a staticconcentration field c, which obeys the Poisson equation [for adiscussion of this approximation and its impact, see refs 38 and39]

∑ δ∇= − −=

D c k r r0 ( )i

N

ic2

1 (6)

where we approximate the Janus colloids as point-like chemicalsinks. We neglect here higher-order contributions that wouldbreak the radial symmetry of the concentration field because ofthe dipolar character of the Janus colloid. The solution is givenby

∑π

= −| − |=

c hckh

Dr

r r( )

41

i

N

i2D 0

c 1 (7)

Here, c0 is the concentration field far away from the colloids,and we have multiplied with the thickness of the colloidalmonolayer h = 2a to obtain a two-dimensional density in theplane of the monolayer. Note that according to our model thechemical field still diffuses in an infinite three-dimensional half-space. A no-flux boundary condition at the bounding surfacedoes not change the principal 1/r dependence in eq 7.If the chemotactic attraction between the active colloids is

sufficiently strong, they form compact clusters, where thechemical substance cannot diffuse freely between the colloids.We roughly take this into account by implementing a screenedchemical field, whenever a colloid is surrounded by six closelypacked neighbors with distances below rs = 2a(1 + ϵ). Then,we replace the algebraic decay 1/r = 1/|r − ri| in eq 7 byexp[−(r − ξ)/ξ]/r, where we introduce the screening length ξ≔ rs. We set ϵ = 0.3 but have checked that varying ϵ by 50%does not change the results. [In the Supporting Information ofref 26, the authors also mention a cutoff of the particleattraction at distances larger than three particle diameters. So,in particular, dynamic clustering should also be visible forlarger screening lengths.]

2.3. Essential Parameters

Rescaling the dimensions of all quantities in the equations ofmotion can be used to reduce the number of systemparameters and thereby reveals the essential parameters. Onepossibility is to set the rescaled diffusion constants, whichdetermine the noise strengths in the Langevin equations 4 and5, to one. This is achieved when rescaling time by tr = 1/(2Drot) and length by = =l D D a/ 2.33r tr rot . The thermaldiffusion coefficients in a bulk fluid would give

=D D a/ 1.15tr rot , while the experiments of ref 25 measurelr = 1.79a for colloids moving close to a bottom wall. Thehigher value used for all reported results in this Account doesnot change the qualitative behavior of our system.Ultimately, one obtains four essential system parameters: the

Peclet number = v D DPe /(2 )0 tr rot , the rescaled translational

diffusiophoretic parameter ζ π ζ→kh D D D/(8 ) /tr c rot tr3

tr, therescaled rotational diffusiophoretic parameter ζrotkh/(8πDcDtr)→ ζrot, and the area fraction σ defined as the ratio of projectedarea of all Janus colloids to the area of the simulation box. Notethat the factor kh/(4πDc) from eq 7 is already subsumed intothe rescaled diffusiophoretic parameters, when using ∇c2D(ri)in the Langevin eqs 4 and 5.2.4. Numerical Implementation

The Langevin eqs 4 and 5 are solved by a typical Euler schemein a two-dimensional square simulation box. Always 800particles are used and different area fractions σ are realized byadjusting the size of the simulation box. To implement thehard-core interactions mentioned earlier, a sufficiently smalltime step has to be chosen, which makes them the numericallymost expensive part of the simulations. Therefore, a neighborlist is implemented such that the search for overlappingparticles could be restricted to eight immediate neighbors, atmost. Furthermore, due to the small translational steps of thecolloids it is sufficient to update the chemical concentrationfield every 50th time step, which further saves computationaltime.Since only bulk properties are of interest here, boundary

conditions are implemented, which keep the colloids awayfrom the boundary of the simulation box. Thus, wheneverhitting the boundary, the particles are reflected into a randomlychosen direction.

3. RESULTS AND DISCUSSION

3.1. Overview: State Diagram

Figure1 shows a typical state diagram at low area fractions ofthe colloids for the two phoretic parameters ζtr and ζrot. It isroughly divided by a diagonal line, which separates collapsedstates, where all colloids form one cluster with differentproperties, from states with varying cluster size. In particular,for ζtr and ζrot both positive meaning that translational androtational phoretic motion act like an effective attractionbetween the colloids, a sharp transition from a gaslike to acollapsed state occurs, where all particles are packed into onesingle static cluster (see snapshot in Figure 2, bottom right).This behavior is reminiscent of the chemotactic collapse inbacterial systems.52,53 Making both phoretic interactionsrepulsive (ζtr < 0 and ζrot < 0), the gaslike state is realized,where small transient clusters are possible.From here two directions are possible. On the one hand,

inreasing ζtr to sufficiently large ζtr > 0, so that translational



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diffusiophoresis acts attractive, dynamic clustering occurs intwo different states 1 and 2. Motile clusters form that stronglyfluctuate in shape and size and may ultimately dissolve again(see snapshots in Figure 2, top right and bottom left). Thestrongly dynamic clusters form due to a delicate balance of theattractive translational phoresis (holding the clusters together)and the fact that active colloids turn and swim away from thecluster (effective repulsion). This behavior is decisivelydifferent from the motility-induced phase separation of hard-core active particles.50,51 On the other hand, keeping ζtr < 0(effective repulsion) but making ζrot sufficiently large so thatactive colloids orient and therefore swim toward each other(effective attraction), different types of collapsed states occur.With decreasing ζtr first the collapsed cluster fluctuates at the

rim due to particles leaving and rejoining it. Then the clusterdevelops nearly regular oscillations and, finally, for stronglynegative ζtr a static collapsed cloud appears, where colloids donot touch and their mutual distance increases toward the rimof the cloud.It is possible to derive the famous Keller−Segel equation

from the Langevin eqs 4 and 5 (see section 3.5 and ref 27). Asin bacterial systems it here predicts a chemotactic collapse ofthe active colloids at 8πσ(ζtr + ζrotPe)/(1 + 2Pe

2) = b, where bis a positive constant. In the ζrot−ζtr state diagram, the linearseparation line between the gaslike and collapsed state has anegative slope and it is shifted along the vertical to positivevalues. Thus, it gives the right trend for the state diagram ofFigure 1. The Keller−Segel equation is a mean-field equation,it only contains the density of the colloids. Therefore, it cannotdescribe subtle states such as dynamic clustering 1 and 2. Forexample, if ζtr is sufficiently large so that the phoretic driftvelocity vD exceeds the swim speed v0, the active colloids willalways stay attached to a cluster, regardless how negative ζrotbecomes. Hence, the separation line in Figure 1 deviates fromthe simple straight line.Finally, the state diagram of Figure 1 does not change

qualitatively with Pe and σ, as long as Pe is well above one andσ sufficiently low so that phase separation does not occur.In the following two sections we characterize the dynamic

clustering states and collapsed states in detail, report on thestate diagram, when screening of the chemical field in densecolloid clusters is switched off, and comment on how to derivethe Keller−Segel equation from our Langevin model.3.2. Gas Phase and Dynamic Clustering

3.2.1. Signature of Dynamic Clustering. Dynamicclustering states are distinguished from the gaslike state bythe occurrence of larger cluster and a pronounced increase incluster size between states 1 and 2. This is already pictured inFigure 1, where the dynamic clustering state 1 shows anincrease of the mean cluster size Nc in a small region. To bettercharacterize these states and justify, why there are twoclustering states, Figure 3 shows the cluster size distributionfor increasing ζtr for fixed rotational phoretic parameter ζrot =−0.38. The curves until ζtr = 15.4 are well fitted by

= −β−P n c n n n( ) exp( / )0 0 (8)

where the cutoff size n0 is a measure how large the clusters canbecome. While for pure steric interaction (ζtr = 0, blue curve)

Figure 1. Full state diagram ζtr versus ζrot at Pe = 19 and surfacefraction σ = 0.05. The mean cluster size Nc for the gaslike anddynamic-clustering state 1 are color-coded. A full discussion isprovided in the main text. Reprinted with permission from ref 29.Copyright 2015 European Physical Journal.

Figure 2. Snapshots of colloid configurations for increasing ζtrans atζrot = −0.38 and Pe = 19. Top left: Gas-like state. Top right: Dynamicclustering 1. Bottom left: Dynamic clustering 2. Bottom right:Collapsed state. Nc is the mean cluster size. Reprinted with permissionfrom ref 27. Copyright 2014 American Physical Society.

Figure 3. Cluster size distributions P(n) for increasing translationalphoretic parameter ζtr at Pe = 19 and ζrot = −0.38. The transitionbetween dynamic clustering states 1 and 2 occurs between the red andgreen curves. Inset: mean cluster size Nc versus ζtr. The transition isindicated. Reprinted with permission from ref 27. Copyright 2014American Physical Society.



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and small ζtr (gray curve) an exponential decay is predominant,the distributions for larger ζtr (orange and red curves) followfirst the power law before they are cut off by the exponential.The exponent β = 2.1 ± 0.1 gradually decreases for morenegative ζrot and the fit is robust against increasing particlenumber.27 So there is a smooth transition from the gaslike tothe dynamic clustering state 1.Further increasing ζtr, the distribution P(n) develops an

inflection point (green curves), which marks the onset of thedynamic clustering state 2. At the transition the mean clustersize strongly increases as the inset indicates. Now, the sum oftwo power-law-exponential curves have to be used for fittingthe distributions,

= − + −β β− −P n c n n n c n n n( ) exp( / ) exp( / )1 1 2 21 2 (9)with β1 = 2.1 ± 0.2 and β2 ≈ 1.5. Both dynamic clusteringstates are observed for all negative ζrot. However, turning offscreening of the chemical field within clusters, dynamicclustering is less pronounced and the clustering state 2 doesnot occur at all.27 Finally, all the exponents βi decrease forlarger area fraction σ and the sharp increase of Nc at thetransition from state 1 to 2 vanishes (see Figure 4). In

literature similar cluster-size distributions including thetransition indicated by the occurrence of an inflection pointwere observed in experiments on gliding bacteria, when thebacterial density was varied.54 While this system shows purehard-core interactions causing nematic alignment, we insteadvary the strength of the diffusiophoretic coupling.3.2.2. Dynamic Clustering for Varying Pećlet Num-

ber. In Figure 5, we plot the state diagram ζtr versus Pećletnumber Pe. Dynamic clustering becomes more pronounced forlarger ζtr and then also occurs at larger Pe, which is necessaryto establish the delicate balance between translational phoreticattraction (ζtr > 0) and effective repulsion (ζrot < 0), where theactive colloids swim away from the clusters. However, forconstant ζtr large clusters disappear with increasing Pe. This isin contrast to the experiments with diffusiophoretic coupling,which showed a linear scaling of the mean cluster size with Pe:Nc ∼ Pe, when increasing fuel concentration.25To rationalize the experimental scaling prediction, we note

that increasing the fuel concentration c0 means higher reactionrate k on the colloidal surface and therefore more self-activitybut also larger phoretic forces, which is encoded in the reduced

phoretic parameters ζtr, ζrot ∝ k introduced in section 2.3.Assuming Michaelis−Menten kinetics for the reaction rate inthe linear regime well before saturation, k ∼ c0, we not onlyfind Pe ∝ c055 but also ζtr ∝ ζrot ∝ c0. Therefore, varying c0defines a straight line in the ζtr−ζrot−Pe parameter space. Inthis three-dimensional space the dynamic clustering states 1and 2 are separated by a plane. We choose different lines,which always hit the transition plane and plot in Figure 6 the

mean cluster size Nc versus Pe along the lines. The blue andpurple curves show the strong increase of Nc when theclustering state 2 is entered, since the respective lines arenearly normal to the transition plane. Tilting the lines more,the increase of Nc becomes smoother. In particular, the greengraph shows an almost linear increase of Nc in the range Pe =10−20. Thus, Figure 6 demonstrates that the relation of clustersize and swimming speed might take different forms dependingon the relation between fuel concentration c0, activity Pe, andphoretic strengths ζtr and ζrot.

3.3. Clustering States

To classify the different collapsed states in Figure 1 and todescribe the hexagonal order in the N-particle cluster, weintroduce the global 6-fold bond orientational parameter

Figure 4. Mean cluster size Nc plotted against ζtr for different arealfractions σ. The dashed line indicates the transition between dynamicclustering states 1 and 2. The rotational diffusiophoretic parameter isζrot = −0.38. Reprinted with permission from ref 29. Copyright 2015European Physical Journal.

Figure 5. State diagram ζtr versus Pe at ζrot = −0.38 and surfacefraction σ = 0.05. The mean cluster size Nc for the gaslike anddynamic-clustering state 1 are color-coded. Reprinted with permissionfrom ref 27. Copyright 2014 American Physical Society.

Figure 6. Mean cluster size Nc versus Pe for different lines in theζtr−ζrot−Pe parameter space. The lines are defined via a para-metrization with x ∈ [0, 1], where Pe varies as in the experiments of,25Pe = 9.5 + 11.5x, ζtr = 4.8 + 16.6x, and ζrot = −0.16 − ζ0x. Theparameter ζ0 defines the different graphs. The transition betweenclustering states 1 and 2 roughly occurs at the intersection of the fittedtwo straight lines. Reprinted with permission from ref 27. Copyright2014 American Physical Society.



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∑

∑

≔ ∈ [ ]

≔ α

=

∈

qN

q

q e

10, 1

with16

k

Nk

k

j

i

61

6( )

6( ) 6

k

kj

6( ) (10)

The second sum goes over the set k6( ) of six nearest neighbors

of particle k and αkj is the angle between the line connectingparticle k to j and some prescribed axis.17 The local bondparameter q6

(k) becomes one if all six nearest neighbors form aregular hexagon around the central colloid and the global orderparameter is one in a hexagonal lattice.The temporal evolution of q6 is plotted in Figure 7 for

several ζtr at constant ζrot = 4.5. At positive ζtr the order

parameter is nearly constant in time indicating a staticcrystalline cluster, while q6 < 1 results from the colloids atthe rim of the cluster, which do not have six neighbors on ahexagon. At ζtr = 0 and especially for negative ζtr, whereparticles effectively repel each other due to translationalphoretic motion, the order parameter shows increasinglystrong fluctuations. The cluster fluctuates close to the rim,where particles leave and rejoin it frequently (ζtr = −6.4 inFigure 7). A video of the fluctuating collapsed state is attachedto ref 29. Further decreasing ζtr the order parameter developsnearly regular oscillations, where the cluster oscillates betweendensely packed and a cloud of confined colloids (ζtr = −12.8 inFigure 7). When the dense packing develops, the diffusiopho-retic interaction becomes strongly screened and the particleslose their orientations toward the cluster center. Repulsion dueto translational phoresis takes over and the cluster expandsuntil the particles are oriented back to the center. This initiatesthe collapse of the cloud and the cycle starts again. Thepulsating cluster is visualized in a video attached to ref 29. Thepower spectrum of q6 shows a broad peak at a nonzerofrequency and confirms the regular oscillations.29 Finally,further decreasing ζtr, the oscillations abruptly stop and a staticcollapsed cloud forms, where the strong effective repulsionprevents direct contact between the particles (ζtr = −16.0 inFigure 7). In this state the hexagonal bond order is small sinceq6 ≈ 0.35 is close to the value q6 = 1/3 of uniformly distributedparticles.

3.4. State Diagram without Screening

Some of the states in the full state diagram of Figure 1discussed so far depend on the screening of the chemical field,which is implemented when the colloids are densely packed.To demonstrate its influence, we show in Figure 8 the state

diagram in the absence of any screening for the sameparameters. A comparison reveals that without screeningdynamic clustering 1 is less pronounced producing on averagesmaller clusters and the dynamic clustering state 2 disappearscompletely. The rich phenomenology of the collapsed states atnegative ζtr is replaced by a single core−corona state, where adensely packed core is surrounded by a cloud of colloids (seeFigure 9). The extension of the core decreases, when ζtrbecomes more negative.

3.5. Relation to Keller−Segel ModelIn the end, we illustrate here how to derive the Keller-Segelequation starting from our model for the chemically interactingactive particles. More details are presented in ref 27. TheLangevin eqs 4 and 5 are equivalent to the Smoluchowskiequation for the full one-particle distribution function P(e, r,t), where we neglect direct interactions between the colloids:

ζ

ζ

∇ ∇ ∇ ∇

∇

∂ = − · + · +

+ ∂ [ ∂ · ] + ∂φ φ φ

P t v Pe P c D P

c P D P

e r

e

( , , ) ( ) ( )

( ) ( )t 0 tr tr

2

rot rot2

(11)

To formulate the Smoluchowski equation, we used e = (sin φ,cos φ) and rewrote eq 5 to ∂tφi = ζrot∂φ ei·∇c + μi. One thenderives dynamic equations for the colloidal density P0(r,t) =∫ P(e, r, t)dφ, as well as the polar [P1(r,t) = ∫ e P dφ ] and

Figure 7. Time evolution of the bond orientational parameter q6 fordifferent ζtr. Further parameters are Pe = 19, ζrot = 4.5, and σ = 0.05.Reprinted with permission from ref 29. Copyright 2015 EuropeanPhysical Journal.

Figure 8. Full state diagram ζtr versus ζrot at Pe = 19 and σ = 0.05 inthe absence of screening. Reprinted with permission from ref 29.Copyright 2015 European Physical Journal.

Figure 9. Snapshots of the colloidal configuration in the core−coronastate: (a) ζtr = −6.4 and (b) ζtr = −16. Other parameters are Pe = 19and ζrot = 4.1. Reprinted with permission from ref 29. Copyright 2015European Physical Journal.



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the nematic ∫ φ= ⊗ −( )t PP r e e( , ) d12 2ÄÇÅÅÅÅÅÅ

ÉÖÑÑÑÑÑÑ order parame-

ter. The resulting hierarchy of coupled equations is closed bysetting P3 to zero. Neglecting time derivatives of P1 and P2 ontime scales much larger than the rotational diffusion time 1/Drot and also higher-order spatial derivatives, one arrivesultimately at the Keller−Segel equation for the colloidaldensity

ζ ∇ ∇ ∇∂ = +P P c D P( )t 0 eff 0 eff2

0 (12)

with renormalized phoretic coefficient and effective transla-tional diffusion constant:

ζ ζζ

= + = +v

DD D

vD2

and2eff tr

rot 0

roteff tr

02

rot (13)

From the Keller−Segel equation the condition for thechemotactic collapse mentioned in section 3.1 can be derived.

4. CONCLUSIONSThis Account reviews our work on self-phoretic active colloidsinteracting by self-generated chemical gradients and therebymimicking chemotaxis well-known from the biological world.The combination of translational and rotational diffusiopho-retic drift velocities with self-propulsion gives rise to effectivephoretic attraction and repulsion. Exploration of the full rangeof the drift velocities reveals that a variety of dynamic statesoccur that range from a gas-like state, over dynamic clustering1 and 2, to different types of collapsed states including a fullchemotactic collapse. The latter can be rationalized by theKeller−Segel equation, which derives from our Langevinmodel.We shortly discuss one point. In our state diagram in Figure

1, the dynamic clustering states only appear in a narrow region,while in the experiments dynamic clustering seems to be ageneric feature.25,26 One reason could be that furtherattractive/repulsive forces between the active colloids arepresent in the experiments, which are not included in ourmodel. Indeed, in simulations with a Lennard-Jones potentialdynamic clustering with cluster size distributions similar to theones discussed in this article are reported.56,57

Possible extensions of the Langevin model presented in thisarticle are mixtures of active and passive particles. Anexperimental realization was published recently in ref 35.Furthermore, Janus particles do not just produce a monopoledisturbance in the chemical environment but also a dipolarcontribution due to their polar character. We are currentlyexploring the consequences of such a contribution. Ultimately,a full solution of the chemical field with appropriate boundaryconditions at the colloid surface is needed for being able totreat particles, which are really close to each other. However,this will also directly influence the swimming speed of thecolloids. In addition, a full hydrodynamic treatment needs tobe included, which has been addressed in recent publica-tions.45−47 Finally, the slip velocity coefficient ζ in eq 1determines the two diffusiophoretic parameters ζrot and ζtr. Itstrongly depends on material properties. So, as a challenge tothe fabrication of self-phoretic colloids, one can ask if it ispossible to purposely tune the phoretic parameters by choosingappropriate materials, catalysts, cap sizes, and physicalmechanisms for the phoretic process?All this will help to further develop the idea of how artificial

microswimmers can mimic biological taxis strategies to control

their swimming paths and explore a wide field of intriguingapplications.

■ AUTHOR INFORMATIONCorresponding Author

*E-mail: [email protected]

Holger Stark: 0000-0002-6388-5390Notes

The author declares no competing financial interest.

Biography

Holger Stark received his Ph.D. from University of Stuttgart andpursued postdoctoral studies at the University of Pennsylvania inPhiladelphia. After staying as a Heisenberg fellow at the University ofKonstanz and a group leader at the Max-Planck-Institute forDynamics and Self-Organization in Göttingen, he became a Professorof Theoretical Physics at the Technical University Berlin. His researchinterests lie in the areas of nonequilibrium statistical physics of softmatter and biological systems.

■ ACKNOWLEDGMENTSThe author thanks Oliver Pohl and Julian Stürmer forcollaborating on the topic and acknowledges funding fromthe DFG within the research training group GRK 1558 and thepriority program SPP 1726, project number STA 352/11. Theauthor also thanks Julian Stürmer for preparing the picture forthe Conspectus.

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