Living clusters and crystals from low density suspensions of active colloids
B. M. Mognetti, A. Šarić, S. Angioletti-Uberti, A. Cacciuto, C. Valeriani, D. Frenkel
LLiving clusters and crystals from low density suspensions of active colloids
B. M. Mognetti † , A. ˇSari´c † , S. Angioletti-Uberti,
1, 3
A. Cacciuto, C. Valeriani, ∗ and D. Frenkel Deparment of Chemistry, University of Cambridge,Lensfield Road, Cambridge, CB2 1EW, United Kingdom Department of Chemistry, Columbia University, 3000 Broadway, New York, NY 10027 † Institute of Physics, Humboldt Universit¨at zu Berlin, Newtonstr. 15, 12489 Berlin, Germany Departamento de Quimica Fisica I, Facultad de Ciencias Qumicas,Universidad Complutense de Madrid, 28040 Madrid, Spain
Recent studies aimed at investigating artificial analogues of bacterial colonies have shown that low–density suspensions of self–propelled particles confined in two dimensions can assemble into finiteaggregates that merge and split, but have a typical size that remains constant (living clusters). Inthis Letter we address the problem of the formation of living clusters and crystals of active particles inthree dimensions. We study two systems: self-propelled particles interacting via a generic attractivepotential and colloids that can move towards each other as a result of active agents (e.g. by molecularmotors). In both cases fluid–like ‘living’ clusters form. We explain this general feature in terms ofthe balance between active forces and regression to thermodynamic equilibrium. This balance canbe quantified in terms of a dimensionless number that allows us to collapse the observed clusteringbehaviour onto a universal curve. We also discuss how active motion affects the kinetics of crystalformation.
PACS numbers:
Active systems consume energy that keeps them in anout–of–equilibrium state [1, 2]. This is typical for manybiologically relevant systems which, by exploiting chemi-cal energy, can self-organise into complex structures thatlack any equilibrium counterpart. Examples are abun-dant and exist at different length-scales: from cytoskele-ton remodulation during cell mitosis [3] to swarming phe-nomena in micro-swimmers or flocks of birds [4, 5]. Thesimilarity of the patterns displayed by these systems leadmany to address the general principles behind their for-mation using simple models [6]. The reproducibility androbustness of the phenomena under a variety of externalconditions motivated a large body of research on self-assembly of active particle as a possible strategy towardsthe fabrication of new functional nano- and mesoscopicstructures [7]. In this respect there have been severalefforts to study active self-organisation in a tightly con-trolled environment; the most studied systems being self–propelled particles [6, 8–11] and active gels (e.g. [12]).Recently, two experimental groups have shown howtwo–dimensional suspensions of self–propelled (SP) col-loids (moving through the consumption of an appropri-ate ‘fuel’) self–assemble into dynamic clusters that con-stantly join and split, recombining with each other toreach a steady-state [13, 14]. A general understandingof active cluster formation is lacking, but different expla-nations have been proposed. The authors of Ref. [13] ar-gued that a net attraction between colloids is responsiblefor clustering. This attraction is due to non-uniformityin the chemical fuel concentration between two colloids.This was confirmed by Ref. [14], that also found a 1 /d dependence of the particle-particle attraction ( d beingthe inter-particle distance). The formation of finite-sizeclusters has been also observed in low density bacte- ria/polymer suspensions, at intermediate polymer con-centrations just before phase separation [10]. However,in other recent studies (e.g. [9, 11, 15, 16]) clusters wereshown to form due to dynamic instabilities, hence attrac-tion cannot be singled out as the only cause promotingaggregation in active systems.Doubts remain over the nature of the driving forcesbehind clustering observed in different experiments. Fur-thermore, it is not clear whether this phenomenon isspecific to these systems or a generic feature of non-equilibrium.To address these questions, in this Letter we reportand explain the formation of living clusters in two verydifferent active systems and at arbitrarily low packingfraction. The first system consists of SP particles similarto those used in Refs. [13, 14] with an added isotropic at-traction. The second are hard spheres that, when closerthan a given range, are brought together by molecularmotors acting as dynamical cross links between ‘tracks’(e.g. microtubules) grafted to the particles. The modelwas inspired by recent work in which centrosomes [17]were self–assembled with the microtubule polarity con-strained to point inward/outward [24]. Note that in thismodel, motors’ action leads to a net attraction betweenparticles. We refer to these particles as self-displacing(SD). To highlight the difference between the models,consider that in the absence of activity and at the pack-ing fractions used in this study (two orders of magnitudebelow hard sphere freezing) SP particles would condense,while SD particles would remain in the gas phase. Fur-thermore, the role of activity in the two systems is in-verted; in the SP model it tends to break up clusters,whereas for SD particles it drives their formation, andsplitting is instead due to diffusion. a r X i v : . [ c ond - m a t . s o f t ] F e b By comparing these two different systems, we showthat the assembly of finite sized clusters at low pack-ing fraction is a generic feature of suspensions of activecolloids and it is not limited to the systems studied inRefs [13, 14]. Moreover, we demonstrate that the for-mation of finite aggregates can be interpreted as a com-petition between equilibrium and active forces.
Simulation details . Self-propelled colloids are modelledas spherical particles of radius σ p = 10 σ (where σ isthe MD unit of length) interacting via a Lennard-Jonespotential V ( r ij ) = 4 (cid:15) (cid:34)(cid:18) σ p r ij (cid:19) − (cid:18) σ p r ij (cid:19) (cid:35) , (1)truncated and shifted at 25 σ p . Self–propulsion is im-plemented by adding a constant force F acting along apredefined axis through the particle (see Fig. 1 a ). To en-able the rotation of the axis of the colloid, and thereforethe direction of its propulsion, two small ideal particleswith diameter σ are placed inside each colloid along itsaxis and positioned 4.5 σ symmetrically with respect tothe center of the particle, forming a rigid body with thecolloid. Though the small particles do not interact withany other particle in the system, they are however sub-jected to the thermal fluctuations induced by the bath,thus leading to a net Brownian rotation of their axis.The motion of each particle is governed by the Langevinequation: m ¨ r i = − (cid:88) j (cid:54) = i ∂V ( r ij ) ∂ r i − ζ ˙ r i + F i + F R,i (2)where m is the sphere’s mass, set to 1, and ζ is thefriction coefficient ( ζ = mγ with damping coefficient γ ). F R,i = √ k B T ζR i ( t ) is the the random force due tothe solvent, where R i ( t ) is a stationary Gaussian noisewith zero mean and variance (cid:104) R i ( t ) R j ( t (cid:48) ) (cid:105) = δ ( t − t (cid:48) ) δ ij .The damping parameter is γ = τ − , τ being the MDtime unit. Simulations were performed at two differ-ent packing fractions φ = 0 .
01 and φ = 0 .
1, using thecode LAMMPS [18] with a total number of particlesof N part = 1728 for at least 15 · time-steps (being∆ t = 0 . τ ). Each simulation was repeated between 4and 16 times with different initial random velocities.SD colloids are modelled as hard spheres with diame-ter σ following Brownian motion with diffusion constant D . Whenever two (randomly chosen) particles are closerthan a given distance ( R A ), they can be self–displacedtoward each other along the direction joining their cen-tres. The activity of SD colloids is then specified by therange R A within which particles can be cross–linked, therate ν at which pulling events occur and ∆ x A , the sizeof the active displacement (see Fig. 1 b ). In this workwe set R A = 2 σ . The dynamics is implemented using a a b FF D x A / < R A r ij FIG. 1: a ) Self–propelled (SP) particles (with propulsionforce F ) interacting via a Lennard–Jones potential. b ) Self-displacing (SD) particles moving towards each other over adistance ∆ x A with a rate ν if they are within a distance R A .Particles also undergo diffusion ( D ). Depending on the valuesof ∆ x A , ν and D , either clustering or collisions dominates. Monte Carlo algorithm generating diffusive or attractivedisplacements with the correct frequencies. For a given‘pulling’ rate ν we randomly select a colloid i and one ofits neighbours j . Particle i and j are then moved towardeach other over a distance min [∆ x A , r ij − σ ], unless thismotion results in an overlap with a third colloid. Theactive displacement does not alter the centre of mass ofthe two colloids. If ∆ x A ≥ R A − σ the colloids are dis-placed to their closest possible configuration. We call thisthe ”processive limit” because molecular motors woulddrag the two colloids until the end of the microtubule isreached. Simulations with SD colloids were performedwith N part = 1000 and N part = 2000. Aggregation of living clusters . One of the most inter-esting results of our simulations is that, despite beingfundamentally different, both models lead to the forma-tion of disordered ‘living’ aggregates. The degree ofclustering is measured by theΘ = 1 − (cid:104) S clust (cid:105) , (3)where (cid:104) S clust (cid:105) is the average number of particles in acluster. Θ ranges from 0 in the gas phase ( (cid:104) S clust (cid:105) =1)to 1 − /N part ≈ (cid:104) S clust (cid:105) = N part ). Two particles are definedas clustered whenever r ij < . · σ p (for SP particles)or if r ij < R A (for SD particles). The number andmorphology of the clusters depend on the relative ra-tio between equilibrium and out-of-equilibrium controlparameters, as measured by a dimensionless quantitywhich we call the propensity for aggregation P agg . Inthe case of SP colloids P agg is simply the ratio betweenthe strength of the attraction between particles and thepropelling force: P agg , SP ≡ ε/ ( F σ p ), while for SD col-loids, P agg is the ratio between the times to move col-loids actively and diffusively over a distance of R A − σ : P agg , SD = ν/ [2 D/ ( R A − σ ) ].Fig. 2 a shows the number of clusters per particles inthe SP colloidal suspension for different values of (cid:15) and FIG. 2: Degree of clustering Θ versus aggregation propen-sity (see text for the definitions) for the SP ( a ) and the SDsystems ( b ). Results for several propulsion forces F ( a ) anddiffusion coefficients D ( b ) collapse onto two universal curves.The insets represent typical snapshots of living clusters, thecluster size distribution function ( a ), and the pair distribu-tion function ( b ) at significant P agg where finite clusters form.Particles belonging to different clusters in (a) are colored dif-ferently and, for clarity, monomers and small clusters are notshown. In the SP case φ = 0 . F ref σ p = 10 k B T , whilein the SD case d cm /R A = 2 ( φ = 0 . N part = 1000 and D ref = 0 . σ/τ MC , τ MC being the simulation time unit. F . These data represent the steady-state value for thecluster distribution, to which different initial states con-verge after a short transient time. Strikingly, all datacollapse onto a single master curve when plotted as afunction of P agg , SP . Furthermore, the figure shows alinear increase in the degree of clustering with decreas-ing activity (i.e. increasing P agg,SP ), which supports thesimple picture of the mechanism of cluster formation interms of two competing terms: the LJ attraction drivesaggregation, whilst activity counteracts the formation oflarge clusters.Fig. 2 b shows the same analysis for SD colloids in theprocessive limit (∆ x A > R A − σ ) at a colloidal packingfraction of 8 · − . In this case clustering is observedwhen activity increases. The trend is opposite to thatobserved in Fig. 2 a for SP particles because, as discussed above, in the SD suspension activity is responsible foraggregation, whereas for SP particles activity is the lim-iting factor for cluster formation. Increasing the col-loidal diffusivity increases the probability of a colloid todetach from the cluster. Interestingly, as in the SP sys-tem, the number of clusters exhibits a linear dependenceon P agg , SD . However, as we approach the limit in whicha single big cluster forms (see Fig. 2), we observe thatthe relaxation time required to reach the steady state in-creases sensibly, suggesting the presence of a phase tran-sition. We defer this issue to a future study.In both systems we find that the cluster size distri-bution appears to follow a power law (see insets in Fig.2 a ). Clusters coexist with a monomers–rich gas phase.The clusters grow and shrink dynamically. An analysisof the pair correlation function (inset of Fig. 2 b ) doesnot reveal any sign of structural order at longer rangethan the typical cluster size. Notice that living clus-ters form in the region where P agg , SP ≈ . − . P agg , SD ≈ −
5, i.e. where active and thermal forceshave comparable magnitude. This highlights the synergicnature of these non–equilibrium aggregates. This power-law behavior for cluster-size distributions has also beenobserved as a function of energy-intake and particle den-sity in a recent study on active Brownian particles [19].Interestingly, when SP particles were confined to movein 2D, for moderate P agg , SP we obtained living clusterswith crystalline order analogous to those observed in theexperiments of Ref. [14]. However, we never observedliving clusters with crystalline order in 3D in the regimewhere active and thermal forces have comparable magni-tude. Driven phase diagram of the SD suspension
In the SDsuspension, activity is controlled not only by the pullingrate but also by the size of the active displacement ∆ x A .In Fig. 3 a we show how clustering is altered by the latterparameter. As expected, decreasing ∆ x A the degree ofclustering can be kept constant by increasing ν .In Fig. 3 b we investigate the change in the numberof clusters as a function of the number density ρ . Morespecifically we plot it as a function of the average distancebetween colloids in the bulk, d cm = ρ − / . Not surpris-ingly, a larger degree of activity is required to maintainclustering at large values of d cm . Both Figs. 2 and 3suggest the presence of a dynamical transition in whichthe number of clusters varies continuously from one to afinite fraction. Active crystallisation
Beyond a well-defined onsetvalue of P agg we observe cluster crystallisation. Crys-tallinity of the aggregates is detected by evaluating the q bond–order parameter and using the criteria describedin Ref [20, 21] Fig. 4 shows the fraction of solid-like par-ticles as a function of P agg , SP for suspensions of SP ( a )and SD ( b ) particles. For SP particles (Fig. 4 a ) the frac-tion of crystalline particles exhibits a non–monotonic be-haviour. For (cid:15) >> F σ p (in our simulations, (cid:15) ≥ k B T d c m / R A x A / ( R A ) a b
2 4 6 8 10 12 14 16 18 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.2 0.4 0.6 0.8 1
2 4 6 8 10 12 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 0 0.2 0.4 0.6 0.8 1 P agg , SD P agg , SD ⇥ FIG. 3: ( a ) Fraction of clusters in the SD suspension as afunction of the processivity parameter ∆ x A / ( R A − σ ) versusthe aggregation propensity P agg , SD and ( b ) as a function oftheir average distance d cm = ρ − / versus P agg , SD . In ( a ) d cm /R A = 2 ( φ = 0 . a ) and SD ( b ) case. The snapshots in a ) show thecrystallization process of a single cluster ( φ = 0 . (cid:15) = 60 k B T and F σ p = 80 k B T ) during the course of a simulation. Fluid-like particles are depicted in blue, solid-like particles in red.Crystals span the whole simulation box b ) SD particles atlow ∆ x A /σ form small faceted crystals that are replaced byarrested aggregated when ∆ x A increases (inset snapshots). Inthis case d cm /R A = 2 ( φ = 0 . and F σ p ∼ k B T ), we observe the formation of disor-dered structures that survive throughout the simulation.Formation of the equilibrium crystalline phase at highvalues of (cid:15) is kinetically hindered by the strong inter-particle attraction, which leads to low diffusivity andhence long equilibration times. In the opposite regime,when (cid:15) ≈ . F σ p (i.e. P agg,SP ≈ . (cid:15)/F σ p (see Fig. 4 a ) instead, thebalance between particle attraction and activity leads tothe formation of crystalline clusters. A possible interpre-tation of these results is that in this intermediate regime,the role of activity is that of speeding-up the kinetics ofcrystal formation by increasing the diffusivity, allowing afaster annealing of defects.It is instructive to relate the simulation parametersin the SP case to the experimental systems discussed inRef. [13–16]. If a colloid of σ p = 1 µm is propelling with v ∼ µm/s , its propulsion force in water is F ∼ . pN .Our analysis suggests that the interaction strength be-tween the colloids required to form a single macroscopicfluid cluster should be (cid:15) ∼ k B T ( P agg , SP ∼ . a , a well-ordered crystal of active particles should beexpected within a few seconds for (cid:15) ∼ k B T .Fig. 4 b shows the crystallisation route followed bySD particles. In this system, crystals only form forsmall displacements. When the processivity parameter∆ x A > .
1, aggregates instead undergo structural arrestbefore they can rearrange into an ordered structure. Webelieve the reason behind the formation of regular struc-tures with SD particles is similar to the mechanism lead-ing to crystallisation in an external field (e.g. [22, 23]).However, here the effective external forces push the par-ticles toward the center of a cluster. It should be stressedthat in both our systems crystallisation typically leads toirreversible clustering rather than to an equilibrium sizedistribution of living clusters.
Conclusion . In this Letter we have investigated the for-mation of living fluid-like clusters and crystals from lowdensity suspensions of active colloids. We have studiedtwo different systems: self–propelled particles interactingvia an attractive potential and hard spheres in which col-loids are pairwise self–displaced toward each others. Inboth cases we observed the formation of living fluid clus-ters (as in Ref.[13, 14]) and irreversible crystals of activeparticles. We showed how the formation of finite aggre-gates can be ascribed to a balanced competition betweenequilibrium forces and activity, regardless of the nature ofthe latter . Furthermore, we demonstrated that activityaids annealing defects in crystalline clusters.
Acknowledgments:
This work was supported bythe ERC Advanced Grant 227758, the National Sci-ence Foundation under Career Grant No. DMR-0846426,the Wolfson Merit Award 2007/R3 of the Royal So-ciety of London and the EPSRC Programme GrantEP/I001352/1. BMM acknowledge T. Curk and A. Bal-lard for useful discussions. C. V. acknowledges financialsupport from a Juan de la Cierva Fellowship, from theMarie Curie Integration Grant PCIG-GA-2011-303941ANISOKINEQ, and from the National Project FIS2010-16159. S. A-U acknowledges support from the Alexandervon Humboldt Foundation. ∗ Electronic address: [email protected]; Corre-sponding author † These authors contributed equally[1] B. Schmittmann and R. K. P. Zia “Statistical Mechanicsof Driven Diffusive Systems” in Phase Transitions andCritical Phenomena , , 1-220, eds. C. Domb and J. L.Lebowitz (Academic, N.Y., 1995)[2] M. C. Marchetti, J.-F. Joanny, S. Ramaswamy, T. B.Liverpool, J. Prost, M. Rao, R. S. Simha, Reviews ofModern Physics, , 18720 (2011);A. Cavagna, A. Cimarelli, I. Giardina, G. Parisi, R. San-tagati, F. Stefanini, M. Viale, Proc. Natl. Acad. Sci.USA, , 1226 (1995);[7] R. Galland, P. Leduc, C. Gurin, D. Peyrade, L. Blan-choin, M. Thry, Nat. Materials , 416 (2013)[8] A. Baskaran, M. C. Marchetti, Phys. Rev. Lett., ,268101 (2008)[9] J. Stenhammar, A. Tiribocchi, R. J. Allen, D. Maren- duzzo, and M. Cates, Phys. Rev. Lett. , 145702(2013).[10] J. Schwarz-Linek, C. Valeriani, A. Cacciuto, M. E. Cates,D. Marenduzzo, A. N. Morozov, and W. C. K. Poon,Proc. Natl. Acad. Sci. U.S.A. , 4052 (2012)[11] G. S. Redner, A. Baskaran, and M. F. Hagan, Phys. Rev.E , 012305 (2013).[12] S. Wang, P. G. Wolynes, Proc. Natl. Acad. Sci. USA, , 15184 (2011)[13] I. Theurkauff, C. Cottin-Bizonne, J. Palacci, C. Ybert,L. Bocquet, Phys. Rev. Lett. , 268303 (2012).[14] J. Palacci, S. Sacanna, A. Preska Steinberg, D.J. Pine, P. M. Chaikin, P. M. Sci. Expresshttp://dx.doi.org/10.1126/science.1230020 (2013).[15] I. Buttinoni, J. Bialke, F. K¨ummel, Hartmut L¨owen, C.Bechinger, T. Speck, Phys. Rev. Lett. , 238301 (2013)[16] Y. Fily and M. C. Marchetti, Phys. Rev. Lett. ,235702 (2012)[17] E. D. Spoerke, G. D. Bachand, J. Liu, D. Sasaki, B. C.Bunker, Langmuir, , 7039 (2008)[18] S. J. Plimpton, J. Comp. Phys. 117, 1 (1995).[19] V. Lobaskin and M. Romenskyy, Phys. Rev. E ,052135 (2013)[20] S. Auer and D. Frenkel, Annu. Rev. Phys. Chem. , 333(2004)[21] Two particles i and j are considered bonded if their cen-ter of the mass distance is r ij < . σ p (in the case of SPparticles) or if r ij < R A (for SD particles). Therefore, aparticle is identified as solidlike if it is bonded to morethan 7 neighbors and if q ( i ) q ( j ) > . , 4267 (1993). R. P. A. Dullens, D. G. A. L. Aarts, andW. K. Kegel, Phys. Rev. Lett. , 228301 (2006).[23] E. Allahyarov and H. L¨owen, Europhys. Lett.95