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Eur. Phys. J. E (2015) 38: 93 DOI 10.1140/epje/i2015-15093-4

Self-phoretic active particles interactingby diffusiophoresis: A numerical studyof the collapsed state and dynamic clustering

Oliver Pohl and Holger Stark

DOI 10.1140/epje/i2015-15093-4

Regular Article

Eur. Phys. J. E (2015) 38: 93 THE EUROPEAN

PHYSICAL JOURNAL E

Self-phoretic active particles interacting by diffusiophoresis:A numerical study of the collapsed state and dynamic clustering⋆

Oliver Pohla and Holger Stark

Institut fur Theoretische Physik, Technische Universitat Berlin, Hardenbergstrasse 36, 10623 Berlin, Germany

Received 7 May 2015 and Received in final form 6 July 2015Published online: 31 August 2015 – c© EDP Sciences / Societa Italiana di Fisica / Springer-Verlag 2015

Abstract. Self-phoretic active colloids move and orient along self-generated chemical gradients by diffusio-phoresis, a mechanism reminiscent of bacterial chemotaxis. In combination with the activity of the colloids,this creates effective repulsive and attractive interactions between particles depending on the sign of thetranslational and rotational diffusiophoretic parameters. A delicate balance of these interactions causesdynamic clustering and for overall strong effective attraction the particles collapse to one single cluster.Using Langevin dynamics simulations, we extend the state diagram of our earlier work (Phys. Rev. Lett.112, 238303 (2014)) to regions with translational phoretic repulsion. With increasing repulsive strength,the collapsed cluster first starts to fluctuate strongly, then oscillates between a compact form and a col-loidal cloud, and ultimately the colloidal cloud becomes static. The oscillations disappear if the phoreticinteractions within compact clusters are not screened. We also study dynamic clustering at larger areafractions by exploiting cluster size distributions and mean cluster sizes. In particular, we identify the dy-namic clustering 2 state as a signature of phoretic interactions. We analyze fusion and fission rate functionsto quantify the kinetics of cluster formation and identify them as local signatures of phoretic interactions,since they can be measured on single clusters.

1 Introduction

The collective motion of synthetic microswimmers has re-cently gained increasing attention [1–4]. Active particlesconstantly convert chemical or other types of energy intodirected motion and thereby form systems far from equi-librium [2, 5–8]. Already in relatively simple setups thesesystems give rise to surprising phenomena. Phase separa-tion [9–15], large density fluctuations [16, 17], pump for-mation in a harmonic trap [18, 19], periodic motion inPoiseuille flow [20–23], active sedimentation under grav-ity [24–26], and dynamic clustering [27, 28] are amongthe plethora of non-equilibrium effects. In biology, col-lective motion of motile organisms forms intriguing dy-namic patterns such as flocking of birds [29], phase sep-aration in mussel beds [30], traveling waves in penguinhuddles [31], swarming of midges [32], or turbulence inbacterial colonies [33].

A common property of these biological and syntheticsystems is the active motion of their individual agents.More importantly, biological as well as synthetic mi-croswimmers are able to sense and move along field gra-

⋆ Supplementary material in the form of two .avi files avail-able from the Journal web page athttp://dx.doi.org/10.1140/epje/i2015-15093-4

a e-mail: oliver.pohl@tu-berlin.de

dients, which allows to design biomimetic systems andthereby learn more about their biological counterparts.For example, in chemotaxis cellular organisms detectchemical gradients via multiple transmembrane recep-tors [34]. Other types of taxis like gravitaxis [35, 36] orthermotaxis [37] refer to directed motion in a gravita-tional field or along temperature gradients. In colloidalsystems phoretic motion induced by field gradients is wellestablished [38]. Recently, active particle systems havebeen explored in connection with thermophoresis [6,39–41]or diffusiophoresis [17, 27, 28, 42, 43]. In the latter, activeparticles consume chemicals from the ambient fluid andthereby generate chemical field gradients, which influencethe swimming paths of neighboring particles and give riseto collective motion.

Experiments with active Janus colloids or light-activated particles show pronounced dynamic cluster-ing [17, 27]. In dilute suspensions clusters of colloids arevery dynamic, they form and resolve again. This is in starkcontrast to the relatively static clusters observed whenactive colloids, interacting only by hard-core potentials,phase-separate at large densities [3]. Dynamic clusteringhas been attributed to diffusiophoretic motion [17,27,28].Particles not only move towards or away from each otherdue to chemical gradients, but they also orient along thesegradients [28,42]. Indeed, with our theoretical modeling inref. [28] we demonstrated that dynamic clustering is the

Page 2 of 11 Eur. Phys. J. E (2015) 38: 93

result of a delicate balance between translational diffusio-phoretic attraction and rotational diffusiophoresis actingrepulsively. The latter occurs when neighboring particlesorient antiparallel and thereby try to swim away from eachother. Moreover, when both contributions were attractive,we also predicted a collapsed state, in which all particlesgathered in one single cluster, reminiscent of a chemotac-tic collapse [28,44].

In this work we continue our previous study on self-phoretic active colloids, where diffusiophoresis determinestheir emergent collectice dynamics. The active colloidsmove in a single layer as in the experiments of ref. [27]. Weextend the state diagram presented in ref. [28] to regionswith translational phoretic repulsion. If it competes withrotational attraction, the collapsed cluster is no longerstatic. With increasing repulsive strength, the cluster firststarts to fluctuate strongly, then oscillates between a com-pact form and a colloidal cloud, and ultimately the col-loidal cloud becomes static. In a continuum theory theonset of spontaneous oscillations in the same parameterregion has been predicted in ref. [42].

In ref. [28] we identified dynamic clustering states 1and 2. The first is characterized by a power-law-dominatedcluster size distribution function whereas in the secondstate only a sum of two power-law-exponential functionsmatch the measured distributions. In this article we inves-tigate them in more detail extending our studies to largerdensities. Without phoretic interactions active colloidsphase-separate into a gas-like phase and compact clus-ters at high densities [3, 45]. This has also been predictedin theory [12, 14]. By analyzing cluster size distributionfunctions and mean cluster sizes, we show that dynamicclustering still exists at larger densities and that dynamicclustering 2 only occurs in the presence of phoretic inter-actions. In experiments, it is difficult to identify phoreticinteractions. The absence of dynamic clustering 2 wouldbe an indication that they do not exist. In addition, weanalyze fusion and fission rate functions to quantify thekinetics of cluster formation. We demonstrate that theyalso are an efficient tool to identify phoretic interactions.

The article is structured as follows. In sect. 2 we in-troduce our model for phoretically interacting colloids. Insect. 3 we present the full state diagram in the transla-tional and rotational phoretic parameters and analyze thecollapsed state. In sect. 4 we examine dynamic clusteringat different densities investigating cluster size distributionfunctions and mean cluster sizes as well as the kinetics ofcluster formation with the help of fusion and fission ratefunctions. We conclude in sect. 5.

2 Model

Following our earlier work in ref. [28], we model a sys-tem of N active colloids in a square simulation box thateffectively interact via diffusiophoretic forces induced bya chemical field. Typically, the colloids are heavier thanthe suspending solvent. They settle to the bottom of theexperimental cell, where they form a colloidal monolayer.To induce activity, Janus colloids together with different

mechanisms are used (see, for example, [5,46,47]). For in-stance, ref. [27] employs gold particles half covered withplatinum, which catalyzes the separation of H2O2 towardswater and oxygen. Through a combination of self-diffusio-and electrophoresis the particles self-propel with a swim-ming speed v0 along direction e, which we assume to befixed in the particle. During the catalytic reactions atthe surfaces of the colloids fuel is converted into severalchemical products. This has two consequences. First, inthe surrounding of the particles fuel is diminished. If theswimming speed is very sensible to the fuel concentration,particles will move slower in regions with large colloidaldensity. Literature discusses a motility-induced phase sep-aration due to the slow down of colloids [12]. Here, we referto experiments [27], where fuel is abundant and, therefore,one can set the swimming speed v0 constant.

Second, spatial gradients of reactants and productsform around the particles and induce diffusiophoretic mo-tion of neighboring particles. Anderson in ref. [38] explainsthe main mechanism. A gradient in the concentration c ofa chemical causes a pressure difference along the particlesurface and thereby generates fluid flow close to the sur-face, which one can treat as an effective slip velocity. Usingit as boundary condition in the Stokes equations and inte-grating over the surface, one obtains the diffusiophoretictranslational velocity of a colloidal particle [38]:

vD = [〈ζ〉1 − 〈ζ(3n ⊗ n − 1)/2〉]∇c. (1)

The slip velocity coefficient ζ depends on the surface in-teraction potential between the chemical and the colloidalsurface. The brackets 〈. . .〉 refer to a surface average. Notethat particles with uniform surface properties also exhibittranslational phoretic motion in a chemical gradient. Inthe following, we have half-coated Janus colloids in mind,where the quadrupolar term in eq. (1) vanishes. Then, wearrive at

vD = −ζtr∇c, (2)

where we have introduced the translational diffusio-phoretic parameter ζtr := −〈ζ〉.

Similarly, one also calculates the diffusiophoretic rota-tional velocity [38]

ωD =9

4a〈ζn〉 × ∇c = −ζrotei × ∇c, (3)

where n is the surface normal vector. For uniformly cov-ered spherical particles, ζ is constant and the angular ve-locity vanishes. For half-coated Janus particles, however,ωD is non-zero as long as the solutes interact differentlywith the two sides. Since this asymmetry also generatesthe active colloidal motion, we have (9/4a)〈ζn〉 = −ζrotei.Here, ei is the swimming direction and we have definedthe rotational diffusiophoretic parameter ζrot. The slip ve-locity coefficient ζ depends on the surface potential withwhich the solute interacts with the colloid surface. So, bychoosing appropriate materials for the Janus colloids andtheir caps, the phoretic parameters ζrot and ζtr = −〈ζ〉might each be tuned to have a positive or negative value.The phoretic parameters also depend on the geometry andnumber of caps covering the colloidal surface [42].

Eur. Phys. J. E (2015) 38: 93 Page 3 of 11

The Reyolds number of the colloids in our system issmall, Re ≈ 10−5, so we formulate the Langevin equa-tions for position ri and orientation ei of the i-th colloidin the overdamped limit. Adding up active and phoreticcontributions, the deterministic translational velocity be-comes ri = v0ei + vD, and the time variation of the ori-entation vector is described by the kinematic equationei = ωD × ei. Then, we use eqs. (2) and (3) togetherwith (ei × ∇c) × ei = (1 − ei ⊗ ei)∇c to write down theLangevin equations for the i-th colloid:

ri = v0ei − ζtr∇c(ri) + ξi, (4)

ei = −ζrot(1 − ei ⊗ ei)∇c(ri) + µi × ei. (5)

Here, ξi is translational and µi rotational white noisewith zero mean and respective time correlation functions〈ξi(t) ⊗ ξi(t

′)〉 = 2Dtr1δ(t − t′) and 〈µi(t) ⊗ µi(t′)〉 =

2Drot1δ(t−t′), where we have introduced the translational(Dtr) and rotational (Drot) diffusion coefficients. We con-sider here their thermal values but they will differ fromthe usual bulk coefficients since the colloids move closeto the bottom boundary. For the same reason, we modeltheir dynamics described in eqs. (4) and (5) in two dimen-sions. Finally, we let the particles with radius a interactvia a hard-core repulsion. Whenever they overlap duringthe simulations, we separate them along the line connect-ing their centers.

The chemical concentration field c evolves accordingto the diffusion equation

c(r) = Dc∇2c − k

N∑

i=1

δ(r − ri). (6)

We make a simplification here and only consider one chem-ical, which the colloids consume with rate k. Therefore,they act as a sink. Typically, the chemical diffuses muchfaster along a particle radius than the swimming colloidsneed for the same distance. So, we can neglect the timederivative of c in eq. (6) meaning each colloid instantly es-tablishes a static chemical sink around itself, which moveswith the colloid. The chemical diffuses in three dimen-sions and we use the static solution of eq. (6), c3d(r) =

c0− (k/4πDc)∑N

i=1 1/|r−ri|, where c0 is the backgroundchemical concentration. Strictly speaking, we would needto implement a non-flux boundary condition at the bound-ary of the infinite half-space, but this will not change thebasic 1/r dependence of the chemical field. To introducea two-dimensional concentration field, in which the col-loidal monolayer moves, we integrate over a thin layer of

thickness h = 2a and obtain c2d(r) =∫ h

0c3d(r)dz ≈ hc3d.

Finally, the chemical concentration field reads

c2d(r) = hc0 −kh

4πDc

N∑

i=1

1

|r − ri|. (7)

So, self-phoretic colloids induce long-range chemical- con-centration gradients and via eqs. (4) and (5) long-rangeinteractions between the colloids, which have indeed beenmeasured [17].

When colloids form compact clusters, the chemicalsubstance cannot diffuse freely between the particles.Therefore, we implement a screened chemical field as fol-lows. Whenever a colloid is surrounded by six closelypacked neighbors with distances below rs = 2a(1 + ǫ), wereplace the term 1/|r− ri| in eq. (7) by exp[−(r− ξ)/ξ]/rwith r = |r− ri|, where we introduce the screening lengthξ := rs. We set ǫ = 0.3 and checked that our results donot change, if ǫ varies by 50%. In ref. [42] screening oc-curs if the reaction rate k depends on concentration c.However, this does not apply to the experimental situa-tion of ref. [27], we aim to model, since the chemical isabundant.

In our approach according to eq. (7), each particle actsas a chemical sink. So, if the translational diffusiophoreticparameter ζtr is positive, the diffusiophoretic velocity vD

from eq. (2) is directed towards a neighboring particle andit acts as an effective attraction, while ζtr < 0 gives riseto an effective repulsion. Similarly, a positive rotationalparameter ζrot in eq. (3) rotates the swimming directionof an active colloid towards a neighboring chemical sinkand the colloid moves towards the sink. Hence, rotationalphoresis also acts like an attractive colloidal interactionwhile it becomes repulsive for ζrot < 0. Tuning the signsof the phoretic parameters, one can realize different typesand combinations of effective interactions anf thereby in-duce a variety of collective dynamics.

To reduce the parameters of our system, we rescaletime by tr = 1/(2Drot) and length by lr =

√

Dtr/Drot =2.33a such that the rescaled diffusion constants are setto one in the Langevin equations (4) and (5). Usingthe thermal values from the Stokes-Einstein relations forthe diffusion coeffcients, one has

√

Dtr/Drot = 1.15a.Since the colloids move close to the bottom wall, thisvalue is changed. Indeed, in the experiments of ref. [27]lr = 1.79a is measured. For historical reasons, we usehere lr = 2.33a, which does not change the qualitativebehavior of our system. The actual value of lr is neededfor implementing screening of the phoretic interaction in-side a colloidal cluster, as explained above. After rescal-ing we are left with four relevant system parameters: thePeclet number Pe = v0/(2

√DtrDrot), the rescaled trans-

lational diffusiophoretic parameter ζtrkh/(8πDtrDc) →ζtr, the rescaled rotational diffusiophoretic parameterζrotkh/(8πDc

√DtrDrot) → ζrot, and the area fraction σ

defined as the projected area of all particles divided by thearea of the simulation box. Note that the factor kh/(4πDc)from eq. (7) is subsumed into the rescaled diffusiophoreticparameters, when using ∇c2d(ri) in the Langevin equa-tions (4) and (5).

We implemented a numerical solution of the Langevinequations in a two-dimensional square simulation box. Tostudy different area fractions σ, we always use 800 par-ticles. The interaction of active particles with confiningboundaries is highly diverse and depends, for example, onthe swimmer type [48]. In our simulations we are inter-ested in the bulk properties, so when hitting the bound-ary of the simulation box, the particles are reflected intoa randomly chosen direction. The exact realization of the

Page 4 of 11 Eur. Phys. J. E (2015) 38: 93

Fig. 1. Full state diagram ζtr versus ζrot at Pe = 19 anddensity σ = 0.05. More details are given in the main text.

reflection does not alter the results, which we are goingto present in this article. Typically, swimmers stay atboundaries for long times and for our system may act asconglomeration sites, where larger clusters emerge. Yet,the experiments we address do not have such confiningboundaries in lateral directions and show pure bulk be-havior, which we are interested in.

3 State diagram in ζrot and ζtr

In ref. [28] we investigated the state behavior of our systemat low area fraction σ. We found that for ζrot and ζtr, bothpositive and increasing, a sudden transition occurs froma gas-like to a collapsed state, in which all particles arepacked into one single cluster. This behavior is reminiscentof the chemotactic collapse in bacterial systems [49, 50].Only for ζrot < 0 and ζtr > 0, when rotational dif-fusiophoresis acts repulsively, we found pronounced dy-namic clustering as observed in experiments [27]. Usingthe shapes of the cluster size distribution functions, we dis-tinguished three cluster states, namely gas-like, dynamicclustering 1, and dynamic clustering 2. We will discuss dy-namic clustering in sect. 4 also for higher densities. In fig. 1we present the full state diagram in the diffusiophoreticparameters ζrot and ζtr including the region ζtr < 0 atarea fraction σ = 0.05. In particular, we will show thatfor ζtr < 0 the collapsed state has to be divided into threeregimes including an oscillating state.

If not stated otherwise throughout this article, we setPe = 19, which is a typical value in experiments [27].However, we have checked that the state diagram of fig. 1does not change qualitatively for sufficiently large Pecletnumbers. Only the parameter regions in ζrot and ζtr, wherethe states are found, are different. In addition, at areafractions larger than σ = 0.05 still dynamic clusteringstates 1 and 2 as well as all collapsed states are observed,as long as phase separation does not occur.

3.1 Line of collapse

In ref. [28] we derived the Keller-Segel equation from theLangevin equations (4) and (5). It predicts a chemotacticcollapse of active colloids at 8πσ(ζtr+ζrot Pe)/(1+2Pe2) =b, where b is a positive constant. According to this con-dition, a linear separation line between the gas-like andthe collapsed state occurs. As we mentioned already inref. [28], the condition is no longer valid for sufficientlylarge ζtr where dynamic clustering occurs. Indeed, the sep-aration line eventually has to saturate to a constant valueof ζtr. This happens when the translational velocity vD ofa colloid due to the diffusiophoretic coupling with neigh-boring particles becomes larger than its own swimmingspeed v0 ∝ Pe. Then, for any value of ζrot the systemwill always collapse since, however an active colloid is ori-ented at the rim of a cluster, it cannot swim away fromthe cluster.

3.2 Collapsed state

For sufficiently large ζtr or ζrot all N particles in thesystem collapse and accumulate in one single cluster. Toquantify the crystalline hexagonal order in such a cluster,we introduce the global 6-fold bond orientational param-eter

q6 :=

∣

∣

∣

∣

∣

1

N

N∑

k=1

q(k)6

∣

∣

∣

∣

∣

∈ [0, 1], with q(k)6 :=

1

6

∑

j∈N(k)6

ei6αkj .

(8)

Here N (k)6 is the set of six nearest neighbors of particle k

and αkj is the angle between the vector connecting parti-cle k to j and some prescribed axis [15]. The local bond

parameter q(k)6 becomes one if all six nearest neighbors

form a regular hexagon around the central colloid and theglobal order parameter becomes one in a hexagonal lattice.

We now use the temporal behavior of q6 to identifydifferent regimes in the collapsed state as indicated in thestate diagram of fig. 1. In fig. 2(a) we plot q6 versus timefor several ζtr. We set ζrot = 4.5 to a sufficiently largevalue to guarantee the collapse of the system for all cho-sen ζtr. For positive ζtr we find the order-parameter valueq6 ≈ 0.89 nearly constant in time. In this situation allparticles are packed in one crystalline cluster and q6 isonly smaller than 1 since colloids at the rim of the clusterare not surrounded by six particles on a hexagon. Alreadyfor ζtr = 0 small fluctuations in the order parameter arevisible. The cluster is no longer static. It even can be-come more circular in shape and q6 assumes values above0.89. The fluctuations increase for ζtr < 0, i.e., when theparticles effectively repel each other due to translationalphoretic motion. Single particles or even clusters occasion-ally leave the main large cluster and rejoin it after a while.For decreasing ζtr the fluctuations in q6 strongly increase.For example, for ζtr = −6.4 q6 significantly varies in timemeaning that the integrity of the main cluster stronglyfluctuates (see video M1 in the Supplemental Material).

Eur. Phys. J. E (2015) 38: 93 Page 5 of 11

Fig. 2. (a) Time evolution of the bond orientational parameterq6 for different ζtr. Further parameters are Pe = 19, ζrot = 4.5,and σ = 0.05. (b) Standard deviation ∆q6 of the fluctuatingq6 plotted versus ζtr for different ζrot. The curve at ζrot = 4.5quantifies the fluctuations in the graphs of (a).

Further decreasing ζtr enters a regime, where thesefluctuations transform into nearly regular oscillations,which marks the oscillation regime in the state diagram offig. 1. In video M2 in the Supplemental Material the ori-gin of the oscillating bond orientational parameter can beseen: the cluster evolves dynamically (large q6), it resolvesand particles disperse (small q6), they rejoin to form thecluster again, and so on. So, the cluster oscillates betweena crystalline structure and a cloud of confined colloids.In the crystalline cluster the diffusiophoretic interactionis strongly screened and the particles are not perfectlyoriented towards the cluster center. Thermal fluctuationslocally disturb the hexagonal packing. The repulsion dueto translational diffusiophoresis becomes more long-range.This destabilizes the cluster, which appears more like acloud. Now, the long-range diffusiophoretic interaction ori-ents all particles back to the cloud center and the compactcluster forms again. This exlains the pulsation. For verynegative ζtr the oscillations stop. The effective repulsionforces are strong and constantly separate particles fromeach other preventing direct contact similar to the snap-shot in fig. 5(b). In this collapsed cloud the hexagonal

bond order nearly vanishes leading to q6 ≈ 0.35, which isclose to the value q6 = 1/3 when particles are uniformlydistributed.

To quantify the fluctuations of the bond orientationalorder parameter q6, we plot its standard deviation Δq6 :=[〈(q6 − 〈q6〉)2〉]1/2 in the full range of ζtr in fig. 2(b). Forlarge values of ζrot fluctuations continuously increase withdecreasing ζtr and then when entering the collapsed-cloudregime a sharp decrease occurs. However, the fluctuationsdo not indicate a transition to the oscillatory regime. Thedependence on ζtr is smooth at the transition between thetwo regimes of fluctuating and oscillating clusters. Notethat for ζrot = 2.2 and 3.0 the sharp drop with decreasingζtr indicates the transition into the gas-like state.

We identified the oscillating regime by determining thepower spectrum of the bond orientational parameter. Tothis purpose we first define the time-autocorrelation func-tion

C(jτ) =1

n

n∑

i=1

[q6(ti + jτ) − 〈q6〉][q6(ti) − 〈q6〉](Δq6)2

. (9)

Here {t1, . . . , tn} is a set of equally spaced time pointsfrom the stationary state with n typically around 10000,τ = ti − ti−1, and j ranges from 1 to 1000. We perform adiscrete Fourier transform,

Q6(ω) =

k∑

j=1

C(jτ) exp(−iωjτ), (10)

which according to Wiener-Khinchin’s theorem is equalto the power spectrum of q6. The results for different ζtr

are plotted in fig. 3(a). We fit the spectrum with a non-normalized Gaussian function and detect its maximumat the position ωmax. In the fluctuating-cluster regime apeak in the power spectrum does not occur. The curvefor ζtr = −3.2 decreases monotonically. In the oscillating-cluster regime a clear maximum at non-zero frequencyωmax exists. We identify the oscillation state in the statediagram if ωmax > 0.01. This value is slightly larger thanzero, in order to being able to clearly identify a maximum.In ref. [42] the authors formulated continuum equationsfor the diffusiophoretically coupled active colloids. In thecase where the diffusiophoretic translational velocity actsrepulsively, i.e., for ζtr < 0, they predict an instabilitywith the onset of spontaneous oscillations. We show thatoscillations persist in steady state in a defined region inthe state diagram.

The activity or colloidal swimming velocity stronglydetermines the frequency of the pulsating cluster. Infig. 3(b) we plot ωmax versus ζtr. From above (fluctuat-ing cluster) and from below (collapsed cloud) a relativelysharp increase of ωmax indicates the onset of the oscil-lation regime. The curve for ωmax displays a plateau-likemaximum with a value practically independent of ζrot. Forexample, the curves in fig. 3(b) belong to Pe = 19 and wefind ωmax ≈ 0.012 ± 0.002 for the maxium value. Indeed,the oscillation frequencies are strongly determined by theactivity of the particles. In the inset of fig. 3(b) we plot

Page 6 of 11 Eur. Phys. J. E (2015) 38: 93max

max

Fig. 3. (a) The power spectrum Q6(ω) of the bond orien-tational parameter plotted against frequency ω (symbols) fordifferent ζtr at Pe = 19, ζrot = 4.5 and σ = 0.05. The lines arefits with a Gaussian function in order to determine the charac-teristic frequency ωmax. (b) Characteristic frequency ωmax ofthe oscillating cluster plotted against ζtr. Inset: ωmax versus

Peclet number Pe in a double logarithmic plot. The error barsshow the standard deviation of ωmax of the maximum valuesfor each Pe.

the maxium value of ωmax versus Pe. Beyond the regimewhere thermal fluctuations dominate, which is set by a de-fined threshold value Pe ≈ 20, the maxium value exhibitsa nearly linear increase in Pe. So, the oscillations becomefaster if the active colloids are faster.

3.3 No screening

Within a cluster or more specfic when one particle isclosely surrounded by six neighboring particles, the dif-fusiophoretic interaction between particles is screened. Toillustrate the effect of such a screening, we switch it offcompletely in this section. Therefore, we now study thestate behavior in the presence of purely long-range diffu-siophoretic interactions. Figure 4 shows the correspondingstate diagram. As reported in ref. [28], without screen-ing dynamic clustering for ζrot < 0 and ζtr > 0 is lesspronounced and the dynamic clustering 2 state does not

Fig. 4. Full state diagram ζtr versus ζrot at Pe = 19 andσ = 0.05 in the absence of screening.

(b)(a)

(c)

Fig. 5. Snapshots of the system in the core-corona state at (a)ζtr = −6.4 and (b) ζtr = −16. Other parameters are Pe = 19and ζrot = 4.1. (c) Radial particle distribution g(r) plotted forvarious ζtr. The dashed line shows an analytic approximationfor ζtr = −16.

occur. At negative ζtr the collapsed state also behavesdifferently. Typically, the collapsed cluster has an innercore with radius rc, where particles are ordered in a two-dimensional hexagonal lattice, surrounded by a corona ofparticles. The core-corona state, as we call it, is illustratedin fig. 5(a). The core radius rc becomes smaller with de-creasing ζtr and eventually vanishes. The resulting stateresembles very much the collapsed cloud discussed before.In fig. 5(b) we show an example, where a compact core isno longer present. So, as long as compact cores in the col-lapsed state form, screening of the diffusiophoretic interac-

Eur. Phys. J. E (2015) 38: 93 Page 7 of 11

tion leads to fluctuating or oscillating clusters, as we havedemonstrated in the previous section, whereas for suffi-ciently negative ζtr we observe the static collapsed cloudsince screening does not play a role in this configuration.

To quantify the density profile of the core-corona clus-ter, we determine its center-of-mass location rcm andcalculate the radial distribution function of the particles

g(r) :=1

r

N∑

j=1

K(rjc − r). (11)

Here

K =1

v√

2πexp

[

− (rjc − r)2

2v2

]

(12)

is a 1D Gaussian kernel function with zero mean and widthv = 2a, where a is the particle radius. Furthermore, rjc =|rj − rcm| is the particle distance to the center of mass.With this definition one obtains

∫

g(r)rdr = N , where Nis the total particle number. In fig. 5(c) we plot the radialparticle density g(r) for various ζtr. In the crystalline coreat r < rc, g(r) is basically constant. For larger distancesr, g(r) drops off, which indicates the increasing distancesbetween particles when approaching the rim of the cluster.

In the following we seek to quantify g(r) for sufficientlynegative ζtr, where a crystalline core does not exist so thatdirect contacts between the particle are negligible. We con-sider the stationary state, where the total velocity of eachparticle vanishes. Furthermore, we assume that all parti-cles are exactly directed towards the center of the cluster.This is a good approximation in the collapsed cloud stateas a closer inspection of the snapshot in fig. 5(b) reveals.Then, the balance of active and diffusiophoretic velocitiesfor each particle i gives

Pe ei = ζtr∇i

N∑

j=1

1

rij, (13)

where ei is a unit vector pointing from particle i to the cen-ter of the collapsed cloud and rij is the distance betweenparticle i and j. Without loss of generality we assume par-ticle i to sit on the y-axis of a cartesian coordinate systemat some distance r to the center. Assuming a sphericalcluster cloud with radius R, we only consider the y com-ponent of eq. (13) and obtain

Pe = ζtr∂r

N∑

j=1

1√

(r − yj)2 + x2j

≈ ζtr

πa2∂r

∫ R

0

∫ 2π

0

g(r′)r′√

r2 + r′2 − 2 cos(φ)rr′dφ dr′

:= F (r). (14)

So, we search for a radial density distribution g(r) suchthat the phoretic interactions cancel the active velocity v0

of every particle.The corresponding problem in three dimensions can

readily be solved. Here, F3d(r) is equivalent to the negative

force, which a test particle of unit mass experiences at adistance r from the center of a radially symmetric massdistribution g3d(r). It is well known that

F3d(r) = γM(r)

r2, (15)

where M(r) = 4π∫ r

0g3d(r′)r′2dr′ is the mass inside a

sphere of radius r and γ is the gravitational constant.Now, according to eq. (15) a constant gravitational forceF3d(r) = F within the whole mass distribution meansM(r) ∝ r2 which implies g3d(r) ∝ 1/r. For a uniformmass distribution ghom(r) ∝ 1, one obtains F hom

3d ∝ r,and we can write

g3d(r) ∝ F

F hom3d (r)

. (16)

Based on this finding, we suggest an approximate so-lution for the two-dimensional problem. For a uni-form particle density ghom(r) ∝ 1, the integral ineq. (14) is readily solved and one obtains F hom

2d (r) =[2ζtrr/(πa2R)]E[(r/R)2], where E[m] is the complete el-liptic integral of the second kind [51]. With F = Pe wepropose in analogy to eq. (16) the approximate solution

g(r) =p Pe

1 + F hom2d (r)

, (17)

where p is a fitting parameter and the number one inthe denominator prevents an unphysical singularity. Fig-ure 5(c) demonstrates that eq. (17) is a good approxima-tion for g(r) determined in our simulations for ζtr = −16.

4 Signatures of dynamic clustering

In the previous section we thoroughly reviewed the col-lapsed state occuring at ζrot > 0. Here, we discuss theregime of dynamic clustering, which we observe at nega-tive ζrot. Since the results are qualitatively the same fordifferent negative values of ζrot, we always set ζrot = −0.38in the following. In ref. [28] we characterized gas-like anddynamic clustering states by their cluster size distributionfunctions. For the low area fraction σ = 0.05 and increas-ing ζtr, we illustrate the distributions in fig. 7(a). In thegas-like and dynamic clustering 1 states, the size distribu-tions follow a power-law-exponential curve,

P (n) = c0n−β exp(−n/n0), (18)

with exponent β = 2.1±0.1 at low densities. In the gas-likestate n0 is small and the exponential dominates (blue andgray curves in fig. 7(a)). In the dynamic clustering state 1n0 is large and for n < n0 the power-law is observed by thelinear decrease in the double-logarithmic plot (orange andred curves in fig. 7(a)). In the dynamic clustering state 2the cluster size distributions develop an inflection point inthe log-log plot (green curves in fig. 7(a)), which we fit bya sum of two power-law-exponentials,

P (n) = c1n−β1 exp(−n/n1) + c2n

−β2 exp(−n/n2), (19)

Page 8 of 11 Eur. Phys. J. E (2015) 38: 93

Fig. 6. (a) Mean cluster size Nc plotted against ζtr for differentareal fractions σ. (b) Nc plotted against σ for different ζtr. Inboth plots the dashed line indicates the transition between dy-namic clustering states 1 and 2. The rotational diffusiophoreticparameter is ζrot = −0.38.

with β1 = 2.1 ± 0.2 and β2 ≈ 1.5. In the following weexplore various aspects of dynamic clustering also for dif-ferent area fractions σ. We analyze mean cluster sizes andcluster size distributions as well as the kinetics of dynamicclustering. Thereby, we reveal the specific behavior of sys-tems with phoretic interactions.

4.1 Mean cluster sizes and cluster size distributions

In fig. 6(a) we plot the mean cluster size Nc against ζtr atfixed area fraction σ, which we call density for short in thefollowing. The dashed line roughly indicates the transitionbetween dynamic clustering states 1 and 2. In the case oflowest density σ = 0.05, Nc grows strongly when enter-ing the regime of dynamic clustering 2. The mean clustersizes increase with density and the transition point be-tween the two dynamic clustering states is strongly shiftedto smaller ζtr. Moreover, at larger densities the growth ofNc at the transition point becomes smoother. In fig. 7(b)we plot the cluster size distributions for σ = 0.19. Com-pared to σ = 0.05 both dynamic clustering states occur atsmaller phoretic strengths ζtr and the main signature of

Fig. 7. Cluster size distribution functions for different ζtr

at densities (a) σ = 0.05 and (b) σ = 0.19. Furthermoreζrot = −0.38. Color code: blue/gray: gas-like state; orange/red:dynamic clustering 1; green: dynamic clustering 2.

dynamic clustering state 2, the inflection point, is less pro-nounced. The exponents in the power-law-exponential fitsof eqs. (18) and (19) are smaller compared to σ = 0.05. Wefind β = 1.7 ± 0.15 in the dynamic clustering state 1 andβ1 = 1.7±0.2 and β2 = 1.1±0.2 for dynamic clustering 2.

In fig. 6(b) we plot Nc versus density σ at differentphoretic strengths ζtr. Again, mean cluster sizes grow withincreasing σ and ζtr. Note, when the strengths of phoreticforces increases, the curvature of the curves in the semi-logarithmic plot changes. For small ζtr we observe a convexshape, while at larger ζtr it becomes concave.

In fig. 8 we show cluster size distribution functions fordifferent densities. We compare the case without phoreticinteractions (a) to the case with a moderate phoreticstrength (b). Whereas in fig. 8(b) for increasing densityσ a transition between the two dynamic clustering statesoccurs, colloids interacting only by hard-core potential donot show dynamic clustering 2. In the latter case, we canfit all curves by the power-law-exponential curve P (n) =c0n

−β exp(−n/n0) with exponent β ≈ 1.77 ± 0.05. Thedistributions approach a clear power-law at large densities.At even larger densities phase separation into large clus-ters with a gas-like phase in between occurs [52]. To con-clude, the dynamic clustering 2 state is unique to phoreticsystems and its occurrence indicates that phoretic inter-actions are present.

Eur. Phys. J. E (2015) 38: 93 Page 9 of 11

Fig. 8. Cluster size distribution functions for different densi-ties. (a) Without phoretic interactions (ζtr = ζrot = 0); (b) atmoderate phoretic interactions with ζtr = 8.0 and ζrot = −0.38.

4.2 Kinetics of dynamic clustering

Cluster size distribution functions have been studied in abiological context, for example, in refs. [53–55]. In experi-ments with a strain of non-chemotactic rod-shaped bacte-ria, the authors of ref. [54] measured such distributions forincreasing density1. For low densities they also fit their ex-perimental distributions by a power-law-exponential func-tion, whereas for large densities a clear maximum at alarge cluster size n develops. The corresponding distribu-tions, as before, are reproduced by a sum of two power-law-exponential functions. In ref. [56] the authors developa kinetic model for clustering and predict the distributionfunctions. They define “reactions” between the clusters byintroducing conglomeration and dissociation rates. Thesequantities can be written as linear functions of a fissionfunction ρ(n), which is the rate with which a cluster orsingle particle detaches from a cluster of size n, and thefusion rate function a(n), which quantifies the event wherea cluster or single particle joins a cluster of size n [53]. As-suming power-law functions for the rate functions [57],

ρ(n) = c1nγρ + cρ

a(n) = c2nγa + ca, (20)

1 The cluster size distribution p(n) in ref. [54] is related toour distribution according to nP (n) = p(n).

Fig. 9. (a) Fusion rate function a(n) for different ζtr at ζrot =−0.38. The solid lines are fits to ca+c2n

γa . Inset: γa versus ζtr.(b) Fission rate function ρ(n) for different ζtr at ζrot = −0.38.The solid lines are fits to cρ + c1n

γρ . Inset: γρ versus ζtr. Thecolor code refers to the gas-like state (blue, gray), dynamicclustering state 1 (orange, red) and state 2 (green).

Peruani and Bar are indeed able to reproduce the tran-sition from a power-law-exponential to a peaked clustersize distribution, as observed in the experiments. To doso, the coeffcients c1 and c2 have to increase with den-sity while the exponents γρ and γa are kept constant. Inparticular, for circular particles in two dimensions one hasγρ = γa = 0.5. Since particles join or leave a cluster onlyat its border, the rate functions have to grow with thecircumference of the cluster ∝ n0.5. However, for an eventwhere a cluster breaks in two nearly even pieces, the ex-ponent γρ deviates from 0.5.

Motivated by the work of ref. [57], we study fissionand fusion rate functions in our phoretic system at lowdensity σ = 0.05 and for varying ζtr. To this end we takesnapshots of our system at different times and measure thetime intervals between two events, where either a particleor cluster leaves a cluster with size n or merges with it.By averaging over all inverse time intervalls, we obtainthe respective rate statistics for a(n) and ρ(n). They areplotted in fig. 9 for different ζtr. For systems with smallclusters (gas-like state), we of course obtain little data for

Page 10 of 11 Eur. Phys. J. E (2015) 38: 93

the rates at large n. We only plot a(n) and ρ(n), when wehave at least 100 data points to average over. Therefore,the curves in fig. 9 do not span the whole size range butcease depending on the mean cluster size. The rates shownin fig. 9 clearly depend on ζtr. In particular, the exponentsγa and γρ plotted in the insets are no longer constantas in the non-phoretic case. The exponent γa (inset offig. 9(a)) decreases with increasing ζtr, in particular, in thedynamic clustering 2 state, which is marked in green. Theexponent γa is always positive meaning that the fusionrates increase with cluster size n. The dependence of thefission rate functions on ζtr is much more pronounced. Therelevant exponent γρ decreases rapidly with increasing ζtr

and becomes negative in the dynamic clustering 2 state asthe inset of fig. 9(b) demonstrates. Hence, large clustershave a smaller probability to loose particles than smallclusters.

Both exponents, γa and γρ, strongly deviate from 0.5predicted for systems with hard-core interactions and,therefore, indicate the presence of phoretic interactions. Inaddition, the exponent γρ of the fission rate ρ(n) clearlysignals the transition between the two dynamic clusteringstates when becoming negative. In contrast to cluster sizedistributions, the fission rate is easier to determine since itcan be measured locally by monitoring breaking clusters.

5 Summary

In this article we modeled the collective motion of self-propelled Janus particles in a two-dimensional confinedregion. Each particle performs active motion, is subjectto thermal noise, and acts as a chemical sink which in-duces translational and rotational diffusiophoretic motionof neighboring particles. This gives rise to long-range ef-fective phoretic interactions, which are either attractive orrepulsive. In densely packed colloidal clusters, the chemi-cal field and the effective interactions are screened.

In such a system dynamic clustering occurs and forstrong effective attractions the system collapses to onesingle cluster [28]. By exploring the full state diagramin the translational and rotational phoretic parameters,ζtr and ζrot, we have shown that the collapsed state canbe further subclassified by the dynamics of the collapsedcluster. Indeed, its dynamics depends on the translationalphoretic strength ζtr. When particles attract each other(ζtr > 0), one large cluster with crystalline order buildsup. Gradually decreasing ζtr to negative values, the col-loidal cluster starts to fluctuate strongly and then enterspulsing dynamics, where it oscillates between a compactcluster and a colloidal cloud. Calculating the power spec-trum of the bond orientational order parameter, we wereable to determine a characteristic oscillation frequency,which increases with the Peclet number. When the effec-tive repulsion is sufficiently strong, a sharp transition toa static colloidal cloud occurs, where the particles gatherin one cluster without crystalline order.

In suspensions of AgCl particles, a colloidal collapseand cluster oscillations were observed [58, 59]. They were

traced back to oscillating reactions at the particle sur-faces, which in turn lead to oscillating chemical gradi-ents in the solvent. In contrast, in our case screening ofthe chemical field in dense colloidal clusters plays a cru-cial role for the oscillations to occur. To elucidate theeffect of screening, we also studied the case, where thecolloids always interact by long-range fields. For suffi-ciently large rotational phoretic parameter ζrot the col-lapsed state occurs but oscillations are not observed. In-stead, at ζtr < 0 a core-corona state appears, in whicha compact crystalline core is surrounded by a corona ofparticles.

We also studied the dynamic clustering states for in-creasing densities by determining cluster size distributionsand mean cluster sizes. One expects that increasing den-sity may have similar effects as strengthening the phoreticinteractions since they depend on the inverse colloidal dis-tances. Indeed, we found that dynamic clustering evolvesat smaller phoretic strengths. In particular, we showedthat dynamic clustering 2 also appears at large densitiesand only when phoretic interactions are present. So, bydetermining cluster size distributions in dense suspensionsof active particles, one is able to identify phoretic interac-tions.

Another method is to measure fission and fusion ratesfor the dynamic clusters. We wrote them as power-lawfunctions in the cluster size. In the presence of phoreticinteractions, the exponents depend on the phoretic param-eters and deviate from 0.5, which is expected for activecolloids with pure hard-core interaction. In particular, theexponent of the fission rate function becomes negative inthe dynamic clustering 2 state. This means the probabilityfor any particle to dissolve from a cluster decreases withthe size of the cluster, which is a clear signature of attrac-tive interactions between particles. Identifying phoretic in-teractions in dense suspensions of active colloids is notstraightforward. In this article we provided methods to doso. It would be interesting to apply them to the exper-imental system of ref. [45], where phase separation wasobserved.

In our model we introduce screening of the chemicalfield in dense colloidal clusters in an approximate fash-ion. Clearly, solving the diffusion equation with a fluxcondition at all colloidal surfaces would be a more ex-act approach towards screening. In ref. [60] reactions ofreactants catalyzed by dumbbell particles were explic-itly simulated and dynamic clustering for small clusterswas indeed observed. Finally, we propose experiments tocheck the predictions being made in this article. In partic-ular, being able to explicitely control phoretic interactionstrengths in experiments would help to explore the richstate space of active colloidal systems. This would gen-erate new insights in the emergent collective behavior ofactive colloids, which can sense chemical fields by diffusio-phoresis. Last but not least, the analogy of diffusiophoresisand chemotaxis, without the need for a complex signallingpathway, provides a biomimetic system to explore not onlythe collective patterns of cells but also how they react andorganize in external chemical fields.

Eur. Phys. J. E (2015) 38: 93 Page 11 of 11

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