Reentrant Phase Behavior in Active Colloids with Attraction
Motivated by recent experiments, we study a system of self-propelled colloids that experience short-range attractive interactions and are confined to a surface. Using simulations we find that the phase behavior for such a system is reentrant as a function of activity: phase-separated states exist in both the low- and high-activity regimes, with a homogeneous active fluid in between. To understand 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 describe the varied kinetics of phase separation, which range from the familiar nucleation and growth of clusters to the complex coarsening of active particle gels.
The collective behaviors of swarming organisms such as birds, fish, insects, and bacteria have long been subjects of wonder and fascination, as well as scientific study Vicsek and Zafeiris (2012). From a physicist’s perspective, such systems can be understood as fluids driven far from equilibrium by the injection of kinetic energy at the scale of individual particles, leading to a zoo of unusual phenomena such as dynamical self-regulation Gopinath et al. (2012), clustering Peruani et al. (2006); Tailleur and Cates (2008); Cates and Tailleur (2013); Redner et al. (2013); Fily and Marchetti (2012); Buttinoni et al. (2013), segregation McCandlish et al. (2012), anomalous density fluctuations Ramaswamy et al. (2003), and strange rheological and phase behavior Giomi et al. (2010); Saintillan (2010); Cates et al. (2008); Shen and Wolynes (2004); Bialké et al. (2012). Recently, nonliving systems that also exhibit collective behaviors have been constructed from chemically propelled particles undergoing self-diffusophoresis Palacci et al. (2013, 2010); Paxton et al. (2004); Hong et al. (2007), squirming droplets Thutupalli et al. (2011), Janus particles undergoing thermophoresis Jiang et al. (2010); Volpe et al. (2011), and vibrated monolayers of granular particles Narayan et al. (2007); Kudrolli et al. (2008); Deseigne et al. (2010), suggesting the possibility of creating a new class of active materials with properties not achievable with traditional materials. However, designing such systems is presently hindered by an incomplete understanding of how emergent patterns and dynamics depend on the interplay between activity and microscopic interparticle interactions.
In this work we investigate an apparent conflict between two recently-reported effects of activity on the phase behavior of active fluids. We recently studied Redner et al. (2013) a system of self-propelled hard spheres (with no attractive interactions) in which activity induces a continuous phase transition to a state in which a high density solid coexists with a low density fluid, complete with a binodal coexistence curve and critical point. This athermal phase separation is driven by self-trapping Tailleur and Cates (2008); Cates and Tailleur (2013). Separately, a study of swimming bacteria with depletion-induced attractive interactions Schwarz-Linek et al. (2012) demonstrated that activity suppresses phase separation, an effect which those authors postulate is generic.
We resolve this apparent paradox by demonstrating that the phase diagram for a system of particles endowed with both attractive interactions and nonequilibrium self-propulsion is reentrant as a function of activity. Depending on parameter values, activity can either compete with interparticle attractions to suppress phase separation or act cooperatively to enhance it. At low activity the system is phase-separated due to attraction, while moderate activity levels suppress this clustering to produce a homogeneous fluid. Increasing activity even further induces self-trapping which returns the system to a phase-separated state. We construct a simple kinetic model whose analytic solution captures the form of this unusual phase diagram and explains the mechanism by which activity can both suppress and promote phase separation in different regimes. We also describe the kinetics of phase separation, which differ significantly between the near-equilibrium and high-activity phase-separated states. The behaviors we observe are robust to variations in parameter values, and thus could likely be observed in experimental systems of self-propelled, attractive colloids such as those studied in Refs. Theurkauff et al. (2012); Schwarz-Linek et al. (2012); Palacci et al. (2013).
Our model is motivated by recently developed experimental systems of self-propelled colloids sedimented at an interface Theurkauff et al. (2012); Palacci et al. (2013), and consists of smooth spheres immersed in a solvent and confined to a two-dimensional plane Redner et al. (2013). Each particle is active, propelling itself forward at a constant speed. Since the particles are smooth spheres and we neglect all hydrodynamic coupling 111Neglecting hydrodynamic coupling can be justified either by restricting the domain of applicability to motile cells on a surface Fily and Marchetti (2012), by noting that hydrodynamic coupling is screened and decays rapidly in space for particles close to a hard wall Diamant et al. (2005); Dufresne et al. (2000), or by observing as in two recent studies of active colloids Palacci et al. (2013); Buttinoni et al. (2013) that the collective phenomena observed in experiments can be well reproduced in simulations without including hydrodynamic interactions., they do not interchange angular momentum and thus there are no systematic torques which might lead to alignment. However, the particles’ self-propulsion directions undergo rotational diffusion; based on experimental observations Theurkauff et al. (2012), we confine the propulsion directions to be always parallel to the surface. For simplicity, interparticle interactions are modeled by the standard Lennard-Jones potential which provides hard-core repulsion as well as short-range attraction, with the nominal particle diameter, and the depth of the attractive well.
The state of the system is represented by the positions and self-propulsion directions of the particles, and their evolution is governed by the coupled overdamped Langevin equations:
Here , is the magnitude of the self-propulsion velocity, and . The Stokes drag coefficient is related to the diffusion constant by the Einstein relation . is the rotational diffusion constant, which for a sphere in the low-Reynolds-number regime is . The are Gaussian white noise variables with and .
We non-dimensionalized the equations of motion using and as basic units of length and energy, and as the unit of time. Our Brownian dynamics simulations employed the stochastic Runge-Kutta method Brańka and Heyes (1999) with an adaptive time step, with maximum value . The potential was cut off and shifted at .
Iii Phase Behavior
We parametrize the system by three dimensionless variables: the area fraction , the Péclet number , and the strength of attraction . In order to limit our investigation to regions with nontrivial phase behavior, we fix the area fraction at . At this density, a passive system () is supercritical for , and phase-separated for stronger interactions Smit and Frenkel (1991). For purely repulsive self-propelled particles, the system undergoes athermal phase separation as a result of self-trapping for , and remains a homogeneous fluid for smaller Pe Redner et al. (2013).
To understand the phase behavior away from these limits, we performed simulations in a periodic box with side length (with resulting particle count ) for a range of attraction strengths and propulsion strengths . Except where noted, each simulation was run until time . Systems were initialized with random particle positions and orientations except that (1) particles were not allowed to overlap and (2) each system initially contained a close-packed hexagonal cluster comprised of particles to overcome any nucleation barriers. To quantify clustering, we consider two particles bonded if their centers are closer than a threshold, and identify clusters as bonded sets of more than 200 particles. The cluster fraction is then calculated as the total number of particles in clusters divided by .
The behavior of the system is illustrated in Fig. 1 by representative snapshots (see also LABEL:Supplement-fig:ljas-0.40-004-004, LABEL:Supplement-fig:ljas-0.40-020-004, and LABEL:Supplement-fig:ljas-0.40-100-004 in the SI Sup ()), and in Fig. 2 with a contour plot of . The most striking result is that the phase diagram is reentrant as a function of Pe. As shown in Fig. 1, low-Pe systems form kinetically arrested gels Trappe and Sandkühler (2004) which gradually coarsen toward bulk phase separation. Increasing Pe to a moderate level destabilizes these aggregates and produces a homogeneous fluid, while increasing activity beyond a second threshold accesses a high-Pe regime in which self-trapping Redner et al. (2013); Tailleur and Cates (2008); Cates and Tailleur (2013) restores the system to a phase-separated state.
As evident in Fig. 1, the width of the intermediate single-phase region shrinks as the attraction strength increases, eliminating reentrance for . This trend can be schematically understood as follows. In the low-activity gel states, particles are reversibly bonded by energetic attraction. Particles thus arrested have random orientations, and so the mean effect of self-propulsion is to break bonds and pull aggregates apart. This opposes the influence of attraction, and so the width of the low-Pe gel region increases with . By contrast, at high Pe we find that self-trapping is the primary driver of aggregation. As shown in the next section, energetic attractions act cooperatively with self-trapping in this regime to enable phase separation at lower Pe than would be possible with activity alone.
Iv Kinetic Model
To better understand the physical mechanisms underlying the reentrant phase behavior, we develop a minimal kinetic model to describe the phase separated state. By analytically solving the model we obtain a form for which captures the major features of the phase behavior observed in our simulations. In the model we consider a single large close-packed cluster coexisting with a dilute gas which is assumed to be homogeneous and isotropic (Fig. 3). Particles in the cluster interior are assumed to be held stationary in cages formed by their neighbors, but their propulsion directions continue to evolve diffusively.
To calculate the rate at which gas-phase particles condense onto the cluster, we observe that the flux of gas particles traveling in a direction through a flat surface is , with the number density of the gas and normal to the surface. Integrating over angles for particles traveling toward the surface of our cluster yields the condensation flux .
Next we estimate the rate of evaporation. We note that a particle on the cluster surface remains bound so long as the component of its effective propulsion force along the outward normal is less than , the maximum restoring force exerted on a particle being pulled away from the surface. This force may involve multiple bonds and is not simply related to the interparticle attraction force. As shown in Fig. 3, this implies an “escape cone” in which the particle’s director must point in order for it to escape. The critical angle is , with the non-dimensionalized maximum restoring force scaled by the depth of the attractive well, which subsumes all relevant details of the binding force and is treated as a fitting parameter: .
We now consider the steady-state angular probability distribution of particles on the cluster surface . In the absence of condensation, this distribution evolves according to the diffusion equation with absorbing boundaries at the edges of the escape cone: and , with general solution . The flux of particles leaving the cluster is then . To simplify the analysis we note that higher-order terms decay rapidly in time, so the steady-state behavior is dominated by the term. We therefore discard higher-order terms and solve to find .
From visual observations it is clear that this minimal model does not capture all microscopic details of the interfacial region. In reality the cluster surface is neither flat nor close packed, but has a complex form which is constantly reshaped by fluctuations in both phases (see LABEL:Supplement-fig:interface-high and LABEL:Supplement-fig:interface-low in the SI Sup ()). We therefore expect quantitative deviations from the model predictions, which we capture in a general fitting parameter which modifies the evaporative flux: .
Equating and yields a steady-state condition which can be solved for the gas density . Since the densities of the two phases are known (the cluster is assumed to be close-packed with density ) and the number of particles is fixed, we can calculate the cluster fraction :
As shown in Fig. 2, this model reproduces the essential features of our system, including active suppression of phase separation at low Pe, activity-induced phase separation at high Pe, and a reentrant phase diagram. The model thus extends the analysis in Ref. Redner et al. (2013) to describe the 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 of Tailleur and Cates Tailleur and Cates (2008); Cates and Tailleur (2013) in which a self-propulsion velocity that decreases with density leads to an instability of the homogeneous initial state.
V Phase Separation Kinetics
The kinetics of phase separation differ significantly between the low-Pe gel and high-Pe self-trapping regions. In low-Pe systems, thermal influences dominate and the kinetics are those of a colloidal particle gel Trappe and Sandkühler (2004). Since the area fraction in our simulations is high, the gels we observe appear nonfractal. Thermal agitation gradually reorganizes the gel into increasingly dense structures Poon (1998); d’Arjuzon et al. (2003), leading toward a single compact cluster in the infinite-time limit. The presence of activity greatly accelerates the rate at which the gel evolves, as shown in Figs. 4 and 5. This effect is also visible in Fig. 1, as the apparent correlation length in the gel states (each observed after a fixed amount of simulation time) increases with Pe.
As Pe is increased beyond the threshold value , activity begins to overwhelm energetic attraction and the gel is ripped apart. This arrests the compaction, resulting in a plateau in the system’s total potential energy (Fig. 5). Just above this transition, the system resembles a fluid of large mobile clusters which rapidly split, translate, and merge (Fig. 4). As Pe is further increased, the characteristic mobile cluster size decreases until the appearance of an ordinary active fluid of free particles is recovered. The fluid phase is clearly identified by superdiffusive mean-square displacement measurements (Fig. 5), distinct from the subdiffusive behavior found in gels.
Additional increase of Pe will eventually cross a second threshold into a phase-separated regime whose behavior is dominated by self-trapping. As reported previously Redner et al. (2013), these systems undergo nucleation, growth, and coarsening stages in a manner familiar from the kinetics of quenched fluid systems, albeit with the unfamiliar control parameter Pe instead of temperature.
These three regimes are characterized by dramatically different particle dynamics. To illustrate these behaviors and their effect on particle reorganization timescales, in Fig. 4 we present snapshots from simulations in which initial particle positions are identified by color. We see that when attraction is dominant (), each particle’s set of neighbors remains nearly static over the timescales simulated, indicative of long relaxation timescales due to kinetic arrest. In contrast, when activity dominates () particles rapidly exchange neighbors and the system becomes well-mixed. Importantly, note that particle dynamics cannot be directly inferred from the instantaneous spatial structures in the system. For example, while systems with low activity (, Fig. 4 second row) and moderate activity (, Fig. 4 third row) appear structurally similar, the rate of particle mixing differs by orders of magnitude.
Activity can both suppress and induce phase separation, and we have shown that these opposing effects can coexist in the same simple system. The resulting counterpoint produces a reentrant phase diagram in which two distinct types of phase separation exist, separated by a homogeneous fluid regime. This surprising result makes it possible to use two experimentally accessible control parameters (Pe and ) in concert to tune the phase behavior of active suspensions. This control is especially valuable because attractive interparticle interactions are common in experimental active systems, being either intrinsic Theurkauff et al. (2012); Palacci et al. (2013) or easily imposed, such as by the addition of depletants Schwarz-Linek et al. (2012). An understanding of the complex phase behavior accessible to these systems is a critical stepping stone toward designing smart active materials whose phases and structural properties can dynamically respond to conditions around them.
This work was supported by NSF-MRSEC-0820492 (GSR, AB, MFH), as well as NSF-DMR-1149266 (AB). Computational support was provided by the National Science Foundation through XSEDE computing resources (Trestles) and the Brandeis HPC.
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