Dynamic Clustering and Chemotactic Collapse of Self-Phoretic Active Particles

Oliver Pohl1 and Holger Stark1,21Institut für Theoretische Physik, Technische Universität Berlin, Hardenbergstrasse 36, 10623 Berlin, Germany

2Kavli Institute for Theoretical Physics, Kohn Hall, University of California, Santa Barbara, California 93106, USA(Received 13 March 2014; published 10 June 2014)

Recent experiments with self-phoretic particles at low concentrations show a pronounced dynamicclustering [I. Theurkauff et al., Phys. Rev. Lett. 108, 268303 (2012)]. We model this situation by taking intoaccount the translational and rotational diffusiophoretic motion, which the active particles perform in theirself-generated chemical field. Our Brownian dynamics simulations show pronounced dynamic clusteringonly when these two phoretic contributions give rise to competing attractive and repulsive interactions,respectively. We identify two dynamic clustering states and characterize them by power-law-exponentialdistributions. In case of mere attraction a chemotactic collapse occurs directly from the gaslike into thecollapsed state, which we also predict by mapping our Langevin dynamics on the Keller-Segel model forbacterial chemotaxis.

DOI: 10.1103/PhysRevLett.112.238303 PACS numbers: 82.70.Dd, 05.10.Gg, 05.20.−y, 47.57.J−

The collective motion of self-propelling objects is amost fascinating subject that has been studied in a varietyof systems [1,2]. At the macro scale, collective patternsoccur, for example, in flocks of birds or fish schools [3–5],while at the microscopic scale bacterial cells in an aqueousenvironment generate intricate motional patterns [6–8]. Tounderstand basic features of structure formation in non-equilibrium, systems with spherical or circular micro-swimmers are investigated. A number of theoretical andexperimental studies have demonstrated that activity ofmicroswimmers alone can result in clustering and phaseseparation [9–17] due to reducedmotility in dense aggregates[9,15]. However, the colloidal density has to be large enoughthat the characteristic time for a particle to join a clusterbecomes comparable to its rotational diffusion time neededto dissolve from it [13]. Other investigations explore theinfluence of hydrodynamics on collective motion [18–24].In experiments with dilute suspensions of self-phoretic

active Janus colloids, dynamic clustering has been observed[25,26]. In this novel nonequilibrium phenomenon, particlesconstantly join and leave clusters that exhibit a very dynamicshape. Since the colloids consume a chemical, they create anonuniform chemical field around themselves. The chemicalgradients initiate diffusiophoresis [27], whereby colloids canattract each other as demonstrated in Ref. [26]. Thediffusiophoretic mechanism not only provides a novelcolloidal interaction, it also serves as a biomimetic versionof bacterial chemotaxis [28], where cells identify and swimalong chemical gradients. Autochemotactic cells typicallyconglomerate in large aggregates or even exhibit a chemo-tactic collapse [7,29–32].Recent theoretical and experimental studies included

short-range attraction between active colloids andobserved clustering at low colloidal densities [26,33–35].Reference [36] implements diffusiophoresis for concrete

surface properties of self-phoretic colloids and identifiesvarious states such as clumping and asters.The work presented here has very much been inspired

by the experiments of the Lyon group [25]. The diffusio-phoretic interaction has a translational and orientationalcontribution. Using Brownian dynamics simulations, wedemonstrate that pronounced dynamic clustering occursonly when these two contributions give rise to competingattractive and repulsive interactions, respectively. We iden-tify two dynamic clustering states and characterize them.Otherwise, in case of mere attraction a chemotactic collapseoccurs directly from the gaslike state before pronouncedclusters are able to form. We support this result by mappingour Langevin dynamics on the Keller-Segel model forbacterial chemotaxis.In our model we consider a dilute suspension of N self-

phoretic colloids confined in a quadratic box. For example,by adding the chemical H2O2, they become active due toself-electrophoresis [37] and the colloids move with aswimming velocity v0. It depends on the concentration cof the chemical that is a control parameter in the experi-ments [38,39]. We assume that v0 does not change notice-ably due to local inhomogeneities of c and consider it as asystem parameter [25,40]. In addition, the active colloidsexperience diffusiophoretic forces along chemical gra-dients, which are generated by the other colloids whenthey consume the chemical. Since the colloids are bipolar, atorque also aligns their swimming directions e along achemical gradient. The associated translational and rota-tional diffusiophoretic velocities are given by [27]

vD ¼ ½hζi1 − hζð3n ⊗ n − 1Þ=2i�∇c; ð1Þ

ωD ¼9

4ahζni ×∇c: ð2Þ

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http://dx.doi.org/10.1103/PhysRevLett.108.268303http://dx.doi.org/10.1103/PhysRevLett.112.238303http://dx.doi.org/10.1103/PhysRevLett.112.238303http://dx.doi.org/10.1103/PhysRevLett.112.238303http://dx.doi.org/10.1103/PhysRevLett.112.238303

Here the slip-velocity coefficient ζ, also called mobilityconstant in Ref. [41], depends on the interaction of thechemical with the colloid’s surface, h� � �i is an average overthe surface with local normal vector n, and a is the colloidradius. The quadrupolar term in Eq. (1) vanishes for half-coated colloids, whereas the angular velocity ωD withhζni ∝ ei is nonzero since the strength of the chemical-surface interaction and thereby the coefficient ζ is differentfor the two halfs of the Janus colloids [42]. Using _ei ¼ωD × ei and ðei ×∇cÞ × ei ¼ ð1 − ei ⊗ eiÞ∇c, we obtaina system of two Langevin equations describing the two-dimensional position ri and direction ei of the ith colloid inthe overdamped limit,

_ri ¼ v0ei − ζtr∇cðriÞ þ ξi; ð3Þ_ei ¼ −ζrotð1 − ei ⊗ eiÞ∇cðriÞ þ μi × ei; ð4Þ

where we introduce ei≔hζni=jhζnij and the respectivetranslational and rotational diffusiophoretic parametersζtr≔hζi, ζrot≔9jhζnij=ð4aÞ. Thermal and intrinsic fluctua-tions enter the equations by translational (ξi) and rotational(μi) noise with zero mean and correlations hξi ⊗ ξii ¼2Dtr1δðt − t0Þ and hμi ⊗ μii ¼ 2Drot1δðt − t0Þ. We modelthe active particles as hard spheres; so, whenever they startto overlap, we separate them along the line connecting theircenters. The chemical field diffuses and has sinks at thepositions of the particles since they consume the chemical[43]: _cðrÞ ¼ Dc∇2c − kPNi¼1 δðr − riÞ. Since the chemi-cal diffuses much faster than the colloids move, a stationarydensity field develops at each instance,

cðrÞ ¼ c0 −k

4πDc

XN

i¼1

1

jr − rij: ð5Þ

In experiments, active colloids settle on a surface and swimin two dimensions while the chemical field diffuses freelyin half-space. Implementing zero flux at the surface byimage colloids does not change the form of Eq. (5). Thesame holds when we integrate the concentration over alayer with thickness twice the colloid radius a to make therelevant concentration field two-dimensional.Within clusters of active colloids, the concentration field

cannot freely diffuse. So, whenever a colloid is surroundedby six closely packed neighbors, we introduce for it ascreened chemical field exp½−ðr − ξÞ=ξ�=r, with thescreening length ξ ¼ 2að1þ ϵÞ. In our simulations wetypically take ϵ ¼ 0.3 and have checked that our results donot change when ε is changed by 50%. We rescale time andlength by tr ¼ 1=ð2DrotÞ and lr ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiDtr=Drot

p ¼ 2.33a.respectively. For historical reasons this is above theexperimental value of 1.79a [25], while the thermalvalue is 1.15a. Again, the following results do not changedrastically, if we adjust the value. The noise intensitiesfor rescaled ξi and μi become one and we are leftwith three essential parameters: the Péclet number

Pe ¼ v0=ð2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiDtrDrot

p Þ and reduced translational aswell as rotational diffusiophoretic parametersζtrk=ð8πDtrDcÞ → ζtr and ζrotk=ð8πDc

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiDtrDrot

p Þ → ζrot,respectively, for which we keep the same symbols.Note that in our model each active colloid creates a

chemical sink. So, for ζtr > 0 and ζrot > 0 it pulls androtates other colloids towards itself, which means aneffective attraction. At boundaries of the simulation box,we let colloids move with a randomly chosen directionback into the box. Following experiments [25], we inves-tigate dilute systems with an area fraction of σ ¼ 5% andsimulate a total of 800 particles. We have checked that alarger number does not change our results qualitativelyas long as the area fraction is maintained. Finally, wechoose the swimming velocity v0 in the range 2–4.5 μm=sas in experiments [25] corresponding to Péclet num-bers Pe ¼ 10–22.In Fig. 1 we illustrate typical particle configurations for

increasing translational diffusiophoretic parameter ζtr. Atζtr ¼ 0 (upper left) the active colloids hardly cluster sincethere is no phoretic attraction between them. They assume agaslike state with a mean cluster size Nc close to 3. Todetermine Nc, we define a cluster as an assembly of morethan two colloids and average over many snapshots atdifferent times. Thus, by definition Nc ≥ 3. In contrast, atlarge ζtr (bottom right) the system collapses into a singlelarge cluster similar to the chemotactic collapse that occursin bacterial systems [29,44]. In between, we observedynamic clustering (top right, bottom left). Motile clustersform that strongly fluctuate in shape and size and ultimatelydissolve again (see movies 1 and 2 in the Supplemental

FIG. 1 (color online). Snapshots of colloid configurations forincreasing ζtrans at ζrot ¼ −0.38 and Pe ¼ 19. Top left: gaslikestate. Top right: dynamic clustering 1. Bottom left: dynamicclustering 2. Bottom right: collapsed state. Nc is the meancluster size.

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Material [45]). In Fig. 2 we plot a state diagram in thephoretic parameters ζtr, ζrot, where we also color code themean cluster size Nc. At ζrot ¼ 0 the maximal dynamiccluster size just before the collapse isNc ≈ 5. When we turnon rotational diffusiophoresis with ζrot > 0, the swimmingdirection e of a free active colloid points towards a cluster(chemical sink), which further supports the formation ofone cluster. This explains the fact that in Fig. 2 the collapseoccurs for smaller ζtr when ζrot increases. The mean clustersize just before the collapse decreases and dynamicclustering is hardly visible. In contrast, at ζrot < 0 activeparticles rotate away from chemical sinks and thus aneffective repulsion is introduced. Once active colloids join acluster, their swimming direction rotates outwards and thecolloids can leave the cluster again if the translationalphoretic attraction is not too large. This balance of effectivephoretic attraction and repulsion is the cause for pro-nounced dynamical clustering with large cluster sizes.Interestingly, the state diagram in Fig. 2 indicates twoclustering states: one where cluster sizes up to 6.5 areobserved (see snapshot at top right in Fig. 1) and a secondclustering state where much larger dynamic clusters occur(see snapshot at bottom left in Fig. 1).To further quantify the two dynamic clustering states 1

and 2, we determine the cluster-size distribution PðnÞ. InFig. 3 we plot it for fixed rotational phoretic parameter ζrot ¼−0.38 and increasing translational coupling ζtr. For puresteric interaction (ζtr ¼ 0, blue curve), an exponential decayis predominant. Closer to the transition line betweendynamic clustering states 1 and 2 in Fig. 2, the distributionfollows a power law at small n, before it falls off exponen-tially (orange and red curves in Fig. 3). Indeed, we can fit ourresults by PðnÞ ¼ c0n−β expð−n=n0Þ with exponentβ ¼ 2.1� 0.1, which gradually decreases for more negativeζrot. This fit is robust against increasing particle number asdemonstrated in the Supplemental Material [45]. Withfurther increase of ζtr we observe that the distributionPðnÞ develops an inflection point (green curves) andwe have to use a sum of two power-law-exponential curvesto fit our distributions, PðnÞ ¼ c1n−β1 expð−n=n1Þ þc2n−β2 expð−n=n2Þ with β1 ¼ 2.1� 0.2 and β2 ≈ 1.5.

This defines the dynamic clustering state 2, where verylarge clusters coexist with smaller ones and individualparticles. The transition in the cluster-size distribution isobserved for all negative ζrot. Typically, the mean cluster sizeNc increases strongly in state 2 as indicated in the inset ofFig. 3. In contrast, for ζrot > 0 the system exhibits thecollapse to a single cluster before large dynamic clusters canappear. When we turn off screening of the chemical fieldwithin clusters, the clustering state 2 disappears completelyas demonstrated in the Supplemental Material [45].Similar cluster-size distributions including the transition

indicated by the occurrence of an inflection point have beenobserved in experiments on gliding bacteria [7]. However,in this work bacterial density was varied in a system withpure hard-core interactions causing nematic alignment. Incontrast, we vary the strength of the diffusiophoreticcoupling ultimately leading to the collapsed state, whichhas not been observed in Ref. [7].The experiments with diffusiophoretic coupling showed a

linear scaling of the mean cluster size with Pe: Nc ∼ Pe [25].It appears counterintuitive that at low area fractions ofσ ¼ 5% faster colloids generate larger clusters and indeedthe simulations of our model show the contrary behavior. InFig. 4 we plot a state diagram in ζtr versus Pe while ζrot iskept constant. Clearly, for constant ζtr large clusters dis-appear with increasing Pe. However, large activity Pe isnecessary for observing dynamic clustering. At small Peonly small cluster sizes occur with increasing ζtr, while atsufficiently large Pe we observe both dynamic clusteringstates.In the experiments of Ref. [25] colloidal activity

increases with the concentration c0 of the activatingchemical H2O2. According to Ref. [39], the swimmingvelocity scales as c0 ∼ v0 ∝ Pe. The concentration c alsocouples to our reparametrized diffusiophoretic parametersζtr and ζrot through the reaction rate k introduced in Eq. (5).In the following, we assume Michaelis-Menten kinetics forthe reaction rate in the linear regime well before saturation,k ∼ c0, and thus find ζtr ∼ c0 ∼ ζrot. So, the linear

FIG. 2 (color online). State diagram: ζtr versus ζrot at Pe ¼ 19.The mean cluster size Nc is color coded.

10 5

10 4

10 3

10 2

10 1

1 5 10 20 50 100 200

100

tr

tr

08.813.415.416.318.2

P (n)

n

0 5 10 1505

101520

tr

Nc

Transition

FIG. 3 (color online). Cluster-size distributions PðnÞ for in-creasing translational phoretic parameter ζtr at Pe ¼ 19 andζrot ¼ −0.38. ζtr assumes the values 0, 8.8, 14.5, 15.4, 16.3,and 18.2. The transition between dynamic clustering states 1 and2 occurs between the red and green curves. Inset: mean clustersize Nc versus ζtr . The transition is indicated.

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dependence of the parameters Pe, ζtr, and ζrot on c0 definesa line in the parameter space. In this space the dynamicclustering states 1 and 2 are separated by a plane. Wechoose different lines that always hit the transition planeand plot in Fig. 5 the mean cluster size Nc versus Pe alongthe lines. The blue and purple curves show the strongincrease of Nc when the clustering state 2 is entered, sincethe respective lines hit the transition plane under anglescloser to 90°. Making this angle smaller, the increase ismore modest. In particular, the green graph shows analmost linear increase of Nc in the Pe range from 10 to 20.So, Fig. 5 demonstrates that the relation between Nc andswimming velocity does not have a simple dependence.To gain some more insight into the transition to the

collapsed state, we formulate a Smoluchowski equation forthe full spatial and orientational probability distribution.We determine its orientational moments and derive anequation for the spatial colloidal density Pðr; tÞ coupled tothe chemical field cðr; tÞ, similar to the approach inRef. [46,47]: _P ¼ ζeff∇ · ðP∇cÞ þDeff∇2PþOðð∇cÞ2Þ.

Details are given in the Supplemental Material [45]. Thisequation is reminiscent of one relation of the Keller-Segelmodel [29] used to describe the chemotaxis of bacteria buthere with effective chemotactic and diffusion constants:ζeff ¼ ζtr þ ðζrotv0=2DrotÞ and Deff ¼ Dtr þ ðv20=2DrotÞ. Intwo dimensions it exhibits an instability of the uniformstate towards a chemotactic collapse when its parameterssatisfy ðζeffkσ=DcDeffÞ > b, where σ is the area fraction ofthe colloids and b is a constant that depends on thegeometry of the system [44,48]). In our unitless parametersthe condition becomes 8πσðζtr þ ζrotPeÞ=ð1þ 2Pe2Þ > b.For constant Pe, this agrees nicely with the nearly straighttransition line in Fig. 2 between the gaslike and thecollapsed state at ζrot > 0. However, the condition doesnot reproduce the straight transition line in Fig. 4. Thereason is that at ζrot < 0 pronounced clustering occursbefore the collapse takes place. In the SupplementalMaterial [45] we test the collapse condition in regionswhere dynamic clustering is absent (ζrot > 0). Dynamicclustering, however, cannot be described by the Keller-Segel equation. Clustering occurs when a colloid movestowards the rim of a cluster and rotates away from it(ζrot < 0), and then a delicate balance between swimmingaway from the cluster and diffusiophoretic attraction by thecluster sets in: v0 ∼ ζtrj∇cj. This mechanism cruciallydepends on ballistic swimming with velocity v0. It cannotbe described by our Keller-Segel equation since v0 onlyappears in effective chemotactic and diffusion parameters.Assuming that the colloid attaches directly to two particlesat the rim of a cluster, we estimate in our reduced unitsj∇cj ≈ 1=ð2a=2.33aÞ2 ¼ 1=0.73 and obtain a condition fordynamic clustering: ζtr ≈ 0.73Pe. This simple estimatereproduces the straight transition line between clusteringstates 1 and 2 in Fig. 4 and the region where clusteringoccurs. A more thorough explanation of dynamic clusteringmight use the kinetic approach of Refs. [7,49] but this isbeyond the scope of this Letter.

To conclude, self-phoretic active colloids mediate dif-fusiophoretic interactions between each other. They consistof a translational and an orientational part, which togetherwith the active swimming can act either attractively orrepulsively. When they are both attractive, the colloidsshow a transition from the gaslike to a collapsed statereminiscent of a chemotactic collapse in bacterial systemsas our mapping on the Keller-Segel model demonstrates.When translational and rotational diffusiophoresis generatecounteracting attraction and repulsion, two dynamic clus-tering states with characteristic cluster-size distributions arestabilized similar to the dynamic clustering observed in theexperiments of Ref. [25].The present system mimics chemotaxis in bacterial

colonies without relying on a complex signalling pathwaynecessary in cells. Thereby, it may help to explorechemotactic structure formation and design novel dynamicpatterns in bacterial colonies [6].

FIG. 4 (color online). State diagram plotting mean cluster sizeNc against Péclet number Pe and chemotactic control parameterζtr . ζrot ¼ −0.38. The color code indicates the mean number ofparticles Nc in a dynamic cluster.

10 12 14 16 18 20

5

10

15

20

25

30

35

Pe

Nc

0

0.090.250.471.13

FIG. 5 (color online). Mean cluster size Nc versus Pe fordifferent lines in the full parameter space defined via a para-metrization with x ∈ ½0; 1�. We vary Pe as in experiments ofRef. [25], Pe ¼ 9.5þ 11.5x, and choose ζtr ¼ 4.8þ 16.6x andζrot ¼ −0.16 − ζ0x, where the parameter ζ0 defines the differentgraphs. The transition between clustering states 1 and 2 roughlyoccurs at the intersection of the two straight lines.

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We thank M. Bär, L. Bocquet, F. Peruani, C. Prohm,C. Valeriani, and K. Wolff for helpful discussions and onereferee for bringing Ref. [41] to our attention. We alsoacknowledge funding from the DFG within the researchtraining group GRK 1558. This research was supported inpart by the National Science Foundation under Grant No.PHY11-25915. H. S. thanks the Kavli Institute forTheoretical Physics for hospitality and financial support.

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