Reentrant Phase Behavior in Active Colloids with Attraction

Gabriel S. Redner, Aparna Baskaran, and Michael F. Hagan∗

Martin Fisher School of Physics, Brandeis University, Waltham, MA, USA.

Motivated by recent experiments, we study a system of self-propelled colloids that experienceshort-range attractive interactions and are confined to a surface. Using simulations we find thatthe phase behavior for such a system is reentrant as a function of activity: phase-separated statesexist in both the low- and high-activity regimes, with a homogeneous active fluid in between. Tounderstand the physical origins of reentrance, we develop a kinetic model for the system’s steady-state dynamics whose solution captures the main features of the phase behavior. We also describethe varied kinetics of phase separation, which range from the familiar nucleation and growth ofclusters to the complex coarsening of active particle gels.

I. INTRODUCTION

The collective behaviors of swarming organisms suchas birds, fish, insects, and bacteria have long been sub-jects of wonder and fascination, as well as scientific study[1]. From a physicist’s perspective, such systems can beunderstood as fluids driven far from equilibrium by theinjection of kinetic energy at the scale of individual parti-cles, leading to a zoo of unusual phenomena such as dy-namical self-regulation [2], clustering [3–8], segregation[9], anomalous density fluctuations [10], and strange rhe-ological and phase behavior [11–15]. Recently, nonlivingsystems that also exhibit collective behaviors have beenconstructed from chemically propelled particles undergo-ing self-diffusophoresis [16–19], squirming droplets [20],Janus particles undergoing thermophoresis [21, 22], andvibrated monolayers of granular particles [23–25], sug-gesting the possibility of creating a new class of activematerials with properties not achievable with traditionalmaterials. However, designing such systems is presentlyhindered by an incomplete understanding of how emer-gent patterns and dynamics depend on the interplay be-tween activity and microscopic interparticle interactions.

In this work we investigate an apparent conflict be-tween two recently-reported effects of activity on thephase behavior of active fluids. We recently studied [6]a system of self-propelled hard spheres (with no attrac-tive interactions) in which activity induces a continuousphase transition to a state in which a high density solidcoexists with a low density fluid, complete with a binodalcoexistence curve and critical point. This athermal phaseseparation is driven by self-trapping [4, 5]. Separately, astudy of swimming bacteria with depletion-induced at-tractive interactions [26] demonstrated that activity sup-presses phase separation, an effect which those authorspostulate is generic.

We resolve this apparent paradox by demonstratingthat the phase diagram for a system of particles endowedwith both attractive interactions and nonequilibrium self-propulsion is reentrant as a function of activity. De-pending on parameter values, activity can either compete

∗ hagan@brandeis.edu

U=1

34

68

15

25

40

Pe=0 4 8 10 20 30 50 70 100 140

U=4

Pe=4 Pe=20 Pe=100

FIG. 1. (Color online) Top: Phase diagram illustrated bysimulation snapshots at time t = 1000τ with area fractionφ = 0.4, as a function of interparticle attraction strength Uand propulsion strength Pe. The colors are a guide to the eyedistinguishing near-equilibrium gel states (upper-left) fromsingle-phase active fluids (center) and self-trapping-inducedphase-separated states (right). Bottom: Detail of reentrantphase behavior at U = 4 as Pe is increased. At Pe = 4 (left),the system is nearly thermal and forms a kinetically arrestedattractive gel. Increasing Pe to 20 (center) suppresses phaseseparation and produces a homogeneous fluid characterizedby transient density fluctuations. Increasing Pe further to100 (right) results in athermal phase separation induced byself-trapping.

with interparticle attractions to suppress phase separa-tion or act cooperatively to enhance it. At low activitythe system is phase-separated due to attraction, whilemoderate activity levels suppress this clustering to pro-

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duce a homogeneous fluid. Increasing activity even fur-ther induces self-trapping which returns the system toa phase-separated state. We construct a simple kineticmodel whose analytic solution captures the form of thisunusual phase diagram and explains the mechanism bywhich activity can both suppress and promote phase sep-aration in different regimes. We also describe the kineticsof phase separation, which differ significantly betweenthe near-equilibrium and high-activity phase-separatedstates. The behaviors we observe are robust to variationsin parameter values, and thus could likely be observed inexperimental systems of self-propelled, attractive colloidssuch as those studied in Refs. [16, 26, 27].

II. MODEL

Our model is motivated by recently developed exper-imental systems of self-propelled colloids sedimented atan interface [16, 27], and consists of smooth spheres im-mersed in a solvent and confined to a two-dimensionalplane [6]. Each particle is active, propelling itself for-ward at a constant speed. Since the particles are smoothspheres and we neglect all hydrodynamic coupling [28],they do not interchange angular momentum and thusthere are no systematic torques which might lead toalignment. However, the particles’ self-propulsion di-rections undergo rotational diffusion; based on experi-mental observations [27], we confine the propulsion direc-tions to be always parallel to the surface. For simplicity,interparticle interactions are modeled by the standard

Lennard-Jones potential VLJ = 4ε[(σr

)12 − (σr )6] which

provides hard-core repulsion as well as short-range at-traction, with σ the nominal particle diameter, and ε thedepth of the attractive well.

The state of the system is represented by the posi-tions and self-propulsion directions {ri, θi}Ni=1 of the par-ticles, and their evolution is governed by the coupledoverdamped Langevin equations:

ri =1

γF LJ({ri}) + vpνi +

√2D ηTi (1)

θi =√

2Dr ηRi (2)

Here F LJ = −∇VLJ, vp is the magnitude of the self-propulsion velocity, and νi = (cos θi, sin θi). The Stokesdrag coefficient γ is related to the diffusion constantby the Einstein relation D = kBT

γ . Dr is the ro-

tational diffusion constant, which for a sphere in thelow-Reynolds-number regime is Dr = 3D

σ2 . The η areGaussian white noise variables with 〈ηi(t)〉 = 0 and〈ηi(t)ηj(t′)〉 = δijδ(t− t′).

We non-dimensionalized the equations of motion using

σ and kBT as basic units of length and energy, and τ = σ2

Das the unit of time. Our Brownian dynamics simulationsemployed the stochastic Runge-Kutta method [31] withan adaptive time step, with maximum value 2 × 10−5τ .The potential VLJ was cut off and shifted at r = 2.5σ.

III. PHASE BEHAVIOR

We parametrize the system by three dimensionlessvariables: the area fraction φ, the Peclet number Pe =vp

τσ , and the strength of attraction U = ε

kBT. In or-

der to limit our investigation to regions with nontrivialphase behavior, we fix the area fraction at φ = 0.4. Atthis density, a passive system (Pe = 0) is supercriticalfor U <∼ 2.2, and phase-separated for stronger interac-tions [32]. For purely repulsive self-propelled particles,the system undergoes athermal phase separation as a re-sult of self-trapping for Pe >∼ 85, and remains a homoge-neous fluid for smaller Pe [6].

To understand the phase behavior away from these lim-its, we performed simulations in a periodic box with sidelength L = 200 (with resulting particle count N = 20371)for a range of attraction strengths U ∈ [1, 50] and propul-sion strengths Pe ∈ [0, 160]. Except where noted, eachsimulation was run until time 1000τ . Systems were ini-tialized with random particle positions and orientationsexcept that (1) particles were not allowed to overlap and(2) each system initially contained a close-packed hexag-onal cluster comprised of 1000 particles to overcome anynucleation barriers. To quantify clustering, we considertwo particles bonded if their centers are closer than athreshold, and identify clusters as bonded sets of morethan 200 particles. The cluster fraction fc is then calcu-lated as the total number of particles in clusters dividedby N .

The behavior of the system is illustrated in Fig. 1 byrepresentative snapshots (see also S1, S2, and S3 in theSI [33]), and in Fig. 2 with a contour plot of fc. Themost striking result is that the phase diagram is reen-trant as a function of Pe. As shown in Fig. 1, low-Pesystems form kinetically arrested gels [34] which grad-ually coarsen toward bulk phase separation. IncreasingPe to a moderate level destabilizes these aggregates andproduces a homogeneous fluid, while increasing activitybeyond a second threshold accesses a high-Pe regime inwhich self-trapping [4–6] restores the system to a phase-separated state.

As evident in Fig. 1, the width of the intermediatesingle-phase region shrinks as the attraction strength Uincreases, eliminating reentrance for U >∼ 40. This trendcan be schematically understood as follows. In the low-activity gel states, particles are reversibly bonded by en-ergetic attraction. Particles thus arrested have randomorientations, and so the mean effect of self-propulsion isto break bonds and pull aggregates apart. This opposesthe influence of attraction, and so the width of the low-Pegel region increases with U . By contrast, at high Pe wefind that self-trapping is the primary driver of aggrega-tion. As shown in the next section, energetic attractionsact cooperatively with self-trapping in this regime to en-able phase separation at lower Pe than would be possiblewith activity alone.

3

0 50 100 150

10

20

30

40

50

Pe

U

0

1

0 50 100 1500

10

20

30

40

50

Pe

U

0

1

FIG. 2. (Color online) Left: Fraction of particles in clus-ters fc measured from simulations as a function of Pe and U .Right: fc as predicted by our analytic theory (Eq. 3) whichreproduces the major features of the phase diagram, includingthe gel region for Pe < U , the self-trapping region for highPe, and the low-fc fluid regime in between. The values of theadjustable parameters are κ = 2 and fmax = 1.7, with the fitmade by eye.

IV. KINETIC MODEL

To better understand the physical mechanisms under-lying the reentrant phase behavior, we develop a minimalkinetic model to describe the phase separated state. Byanalytically solving the model we obtain a form for fcwhich captures the major features of the phase behaviorobserved in our simulations. In the model we consider asingle large close-packed cluster coexisting with a dilutegas which is assumed to be homogeneous and isotropic(Fig. 3). Particles in the cluster interior are assumedto be held stationary in cages formed by their neighbors,but their propulsion directions θi continue to evolve dif-fusively.

To calculate the rate at which gas-phase particles con-dense onto the cluster, we observe that the flux of gasparticles traveling in a direction ν through a flat surfaceis 1

2πρgvp(ν · n), with ρg the number density of the gasand n normal to the surface. Integrating over angles forparticles traveling toward the surface of our cluster yieldsthe condensation flux kin =

ρgvpπ .

Next we estimate the rate of evaporation. We notethat a particle on the cluster surface remains bound solong as the component of its effective propulsion forcealong the outward normal γvp(ν · n) is less than Fmax,the maximum restoring force exerted on a particle be-ing pulled away from the surface. This force may in-volve multiple bonds and is not simply related to theinterparticle attraction force. As shown in Fig. 3, thisimplies an “escape cone” in which the particle’s direc-tor must point in order for it to escape. The critical

angle is α = π − cos−1(Ufmax

Pe

), with fmax the non-

dimensionalized maximum restoring force scaled by thedepth of the attractive well, which subsumes all relevantdetails of the binding force and is treated as a fittingparameter: fmax = Fmax

UσkBT

.

������������

�� ���������� ���� ����������

FIG. 3. (Color online) Schematic representation of our kineticmodel. The high- and low-density phases fill the left andright half-spaces. A particle on the cluster surface (center)escapes only if its direction points within the “escape cone”defined by γvp(ν · n) > Fmax, while particles in the gas landon the surface at a rate proportional to the gas density andpropulsion speed.

We now consider the steady-state angular probabilitydistribution of particles on the cluster surface P (θ). Inthe absence of condensation, this distribution evolves ac-cording to the diffusion equation with absorbing bound-

aries at the edges of the escape cone: ∂P (θ,t)∂t = Dr

∂2P (θ,t)∂θ2

and P (±α, t) = 0, with general solution P (θ, t) =∑∞q=1Aq cos

(qπθ2α

)e−Dr

q2π2

4α2 t. The flux of particles leav-

ing the cluster is then kout = − 1σ

∂∂t

∫ α−α P (θ, t)dθ

∣∣∣t=0

.

To simplify the analysis we note that higher-order termsdecay rapidly in time, so the steady-state behavior isdominated by the q = 1 term. We therefore discard

higher-order terms and solve to find kout = Drπ2

4σα2 .From visual observations it is clear that this minimal

model does not capture all microscopic details of the in-terfacial region. In reality the cluster surface is neitherflat nor close packed, but has a complex form which isconstantly reshaped by fluctuations in both phases (seeS4 and S5 in the SI [33]). We therefore expect quanti-tative deviations from the model predictions, which wecapture in a general fitting parameter κ which modifies

the evaporative flux: kout = Drπ2κ

4σα2 .Equating kin and kout yields a steady-state condition

which can be solved for the gas density ρg. Since thedensities of the two phases are known (the cluster is as-sumed to be close-packed with density ρc = 2

σ2√3) and

the number of particles is fixed, we can calculate the clus-ter fraction fc:

fc =16φα2Pe− 3π4κ

16φα2Pe− 6√

3π3φκ(3)

As shown in Fig. 2, this model reproduces the essen-tial features of our system, including active suppressionof phase separation at low Pe, activity-induced phase sep-aration at high Pe, and a reentrant phase diagram. Themodel thus extends the analysis in Ref. [6] to describethe coupled effects of activity and energetic attraction.As noted in that reference, our model description of self-trapping can be considered a limiting case of the theory

4

Pe=

0Pe=

25

Pe=

40

Pe=

140

t = 1 10 100 1000

FIG. 4. (Color online) Visual guide to phase-separation ki-netics at fixed U = 30. To make mixing visible, particles arelabeled in two colors based on their positions at t = 1. Apassive system (top row) forms a space-spanning gel whichgradually coarsens. The addition of activity (second row)greatly increases the rate at which the gel evolves. In bothcases particles remain ‘local’ and largely retain the same set ofneighbors. When Pe exceeds U (third row), activity is strongenough to break the gel filaments and a fluid of large mo-bile clusters results. While the instantaneous configurationsappear structurally similar to the gels above, these systemsquickly become well-mixed due to splitting and merging ofclusters (see also Fig. 5). In the high-Pe limit (bottom row),self-trapping drives the emergence of a single well-mixed clus-ter surrounded by a dilute gas.

of Tailleur and Cates [4, 5] in which a self-propulsion ve-locity that decreases with density leads to an instabilityof the homogeneous initial state.

V. PHASE SEPARATION KINETICS

The kinetics of phase separation differ significantly be-tween the low-Pe gel and high-Pe self-trapping regions.In low-Pe systems, thermal influences dominate and thekinetics are those of a colloidal particle gel [34]. Sincethe area fraction in our simulations is high, the gelswe observe appear nonfractal. Thermal agitation gradu-ally reorganizes the gel into increasingly dense structures[35, 36], leading toward a single compact cluster in theinfinite-time limit. The presence of activity greatly accel-erates the rate at which the gel evolves, as shown in Figs.4 and 5. This effect is also visible in Fig. 1, as the ap-parent correlation length in the gel states (each observedafter a fixed amount of simulation time) increases with

Pe=0102030

4050

100 101 102 103 104

-6.0

-5.8

-5.6

-5.4

-5.2

-5.0

-4.8

time

2E

�UN

Pe=0102030

4050

100 101 102

10-1

100

101

102

103

104

time

XDr2\

FIG. 5. (Color online) Quantitative measurements of stateswith U = 30. Left: Mean interaction energy per parti-cle (E represents the total system potential energy). Low-Pegels are strongly arrested and do not approach bulk phase-separation on our simulation timescales; however, higher-Pesystems evolve much faster and nearly reach the bulk limit.When Pe > U (dashed lines), coarsening is arrested as thegel breaks into a fluid of mobile clusters with a Pe-dependentcharacteristic size, leading to a plateau in the system poten-tial energy. Right: Discrimination of gel from fluid statesby mean-square displacement. Dotted lines have slope 1 andhighlight the distinction at short times between gels (subdif-fusive), and active fluids (superdiffusive).

Pe.

As Pe is increased beyond the threshold value Pe ≈ U ,activity begins to overwhelm energetic attraction and thegel is ripped apart. This arrests the compaction, result-ing in a plateau in the system’s total potential energy(Fig. 5). Just above this transition, the system resemblesa fluid of large mobile clusters which rapidly split, trans-late, and merge (Fig. 4). As Pe is further increased, thecharacteristic mobile cluster size decreases until the ap-pearance of an ordinary active fluid of free particles is re-covered. The fluid phase is clearly identified by superdif-fusive mean-square displacement measurements (Fig. 5),distinct from the subdiffusive behavior found in gels.

Additional increase of Pe will eventually cross a sec-ond threshold into a phase-separated regime whose be-havior is dominated by self-trapping. As reported previ-ously [6], these systems undergo nucleation, growth, andcoarsening stages in a manner familiar from the kinet-ics of quenched fluid systems, albeit with the unfamiliarcontrol parameter Pe instead of temperature.

These three regimes are characterized by dramaticallydifferent particle dynamics. To illustrate these behaviorsand their effect on particle reorganization timescales, inFig. 4 we present snapshots from simulations in which ini-tial particle positions are identified by color. We see thatwhen attraction is dominant (U > Pe), each particle’sset of neighbors remains nearly static over the timescalessimulated, indicative of long relaxation timescales dueto kinetic arrest. In contrast, when activity dominates(Pe > U) particles rapidly exchange neighbors and thesystem becomes well-mixed. Importantly, note that par-ticle dynamics cannot be directly inferred from the in-

5

stantaneous spatial structures in the system. For exam-ple, while systems with low activity (Pe < U , Fig. 4second row) and moderate activity (Pe >∼ U , Fig. 4 thirdrow) appear structurally similar, the rate of particle mix-ing differs by orders of magnitude.

VI. CONCLUSIONS

Activity can both suppress and induce phase separa-tion, and we have shown that these opposing effects cancoexist in the same simple system. The resulting counter-point produces a reentrant phase diagram in which twodistinct types of phase separation exist, separated by ahomogeneous fluid regime. This surprising result makesit possible to use two experimentally accessible controlparameters (Pe and U) in concert to tune the phase be-

havior of active suspensions. This control is especiallyvaluable because attractive interparticle interactions arecommon in experimental active systems, being either in-trinsic [16, 27] or easily imposed, such as by the addi-tion of depletants [26]. An understanding of the complexphase behavior accessible to these systems is a criticalstepping stone toward designing smart active materialswhose phases and structural properties can dynamicallyrespond to conditions around them.

VII. ACKNOWLEDGMENTS

This work was supported by NSF-MRSEC-0820492(GSR, AB, MFH), as well as NSF-DMR-1149266 (AB).Computational support was provided by the NationalScience Foundation through XSEDE computing re-sources (Trestles) and the Brandeis HPC.

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