Flow-induced phase separation of active particles is controlled by boundary conditions

Significance

Active particle suspensions comprise energy-consuming and hydrodynamically interacting units, such as swimming microorganisms, autophoretic colloids, and active droplets. Though it is now recognized that emergent self-organization in such systems is driven by their spontaneous hydrodynamic flow, it is still not well-understood how the modification of this flow by confining boundaries impacts self-organization. Here we combine experiments, theory, and simulations to elucidate the effect of boundaries on the spontaneous flow in a suspension of active emulsion droplets. Our results establish a widely applicable paradigm for flow-induced phase separation in active fluids and offer routes to manipulating their microstructure.

Abstract

Active particles, including swimming microorganisms, autophoretic colloids, and droplets, are known to self-organize into ordered structures at fluid–solid boundaries. The entrainment of particles in the attractive parts of their spontaneous flows has been postulated as a possible mechanism underlying this phenomenon. Here, combining experiments, theory, and numerical simulations, we demonstrate the validity of this flow-induced ordering mechanism in a suspension of active emulsion droplets. We show that the mechanism can be controlled, with a variety of resultant ordered structures, by simply altering hydrodynamic boundary conditions. Thus, for flow in Hele–Shaw cells, metastable lines or stable traveling bands can be obtained by varying the cell height. Similarly, for flow bounded by a plane, dynamic crystallites are formed. At a no-slip wall, the crystallites are characterized by a continuous out-of-plane flux of particles that circulate and re-enter at the crystallite edges, thereby stabilizing them. At an interface where the tangential stress vanishes, the crystallites are strictly 2D, with no out-of-plane flux. We rationalize these experimental results by calculating, in each case, the slow viscous flow produced by the droplets and the long-ranged, many-body active forces and torques between them. The results of numerical simulations of motion under the action of the active forces and torques are in excellent agreement with experiments. Our work elucidates the mechanism of flow-induced phase separation in active fluids, particularly active colloidal suspensions, and demonstrates its control by boundaries, suggesting routes to geometric and topological phenomena in an active matter.

There are many instances, drawn from biological, physicochemical, and technological contexts, in which microscopic particles produce spontaneous flow in a viscous fluid. The energy necessary to maintain this flow is supplied by a variety of mechanisms, of which there are a wide variety, at the interface between the particles and the fluid. The ciliary layer in cells (1), the chemically reacting boundary layer in autophoretic colloids (2), and the dissolution layer in auto-osmophoretic drops (3) provide three distinct examples. In each case, the activity within the layer drives the exterior fluid into motion, which appears as if it were a spontaneous fluid flow around the particles. It is possible, though not necessary, for the particles to translate and/or rotate in response to the spontaneous flow. Irrespective of the property of self-propulsion and/or self-rotation, such active particles in a suspension will each produce a spontaneous flow in which other particles will be entrained. This mutual entrainment, if sufficiently strong, can produce states of organization with no analogue in an equilibrium suspension of passive particles.

The above-mentioned mechanism has been conjectured (4) to underlie the spontaneous crystallization of Janus particles (5) and fast-moving bacteria (6) at a plane wall. However, a conclusive experimental demonstration of the validity of this flow-induced phase separation (FIPS) mechanism is still lacking. If bulk hydrodynamic flow is the principal cause of self-organization, any alteration of the flow should manifest itself in altered states of self-organization. The simplest way of altering the bulk flow, keeping other experimental conditions constant, is to vary the hydrodynamic conditions at the boundaries of the flow. If this produces correspondingly distinct states of self-organization, both the FIPS mechanism and the role of boundaries in controlling it are, thereby, established.

Here, we use a suspension of active, self-propelled emulsion droplets to investigate the role of hydrodynamics on their collective behavior. The system has been used previously (3, 7–10) to provide insights into out-of-equilibrium phenomena relevant to self-organization in both natural (6, 11, 12) and synthetic active particle settings. The typical size $b\sim 50\hspace{0.17em}\mathrm{\mu }$m of the emulsion droplets and their typical self-propulsion speed $v_s\sim 5\hspace{0.17em}\mathrm{\mu }$m/s implies that the Reynolds number $Re=v_{s}b/\nu \sim 10^{-4}$ in a fluid with the kinematic viscosity $\nu$ of water. Fluid inertia is negligible at such small $Re$, and the flow is described by the Stokes equation. It is then possible to exploit the linearity of the governing equations and use Green’s function techniques to compute the flow, the stress in the fluid, and the forces and torques between the droplets for a variety of boundary conditions. We find distinct states of aggregation as the boundary conditions are altered, which both validates the conjectured hydrodynamic mechanism and opens up a route to its manipulation and control.

We should note the effect of steric confinement of active fluids on self-organization has been studied thoroughly in the past (13–19). This knowledge has been exploited in applications such as self-assembly, meta-material synthesis, and active fluid computation (20–22). However, the resultant hydrodynamic effects of confinement can be distinct due to differences, for instance, in the slip properties of the boundaries even when the geometry of the confinement is identical. Our work suggests an independent route to ordered states of active matter by using boundaries to alter the hydrodynamic interactions in the system, rather than to modify its geometric confinement. With this remark, we now turn to our results.

Experimental System and Theoretical Model

Our experimental system is an active emulsion of monodisperse droplets of liquid crystal (5CB) in water whose source of activity is droplet dissolution (8). Surface tension gradients at the interface of the droplet, sustained by the free energy of dissolution, produce active hydrodynamic flows both within and external to the droplet, leading to droplet motion. While this self-propulsion does not rely on the liquid crystallinity of the droplet, the nematic state of 5CB within it enables the internal velocity field to be inferred (Fig. 1A and Movie S1). The presence of surfactant at the interface makes it energetically favorable for the rod-like 5CB molecules to orient normal to it. Such a homeotropic boundary condition on the director field enforces a point defect, which is located at the center of the droplet (Fig. 1B) when surface tension gradients (and hence fluid flow) are negligible. When observed in polarized light between cross-polarizers, the droplet shows a four-lobed pattern (Fig. 1A, Leftmost) reflecting the symmetric orientation of the director field about the droplet center. When surface tension gradients become appreciable, viscous stresses and fluid flow are induced in the bulk, and so the nematic stress within the droplet must be redistributed. The resulting reorientation of the director causes a displacement of the point defect along the axis of droplet motion, and the centrosymmetric four-lobed pattern is distorted to another with a reduced symmetry, now only about the propulsion axis (Fig. 1 A and B).

Fig. 1.

Active droplets made with nematic liquid crystals. (A) Cross-polarization microscopy images showing spontaneous symmetry breaking and propulsion of an active droplet. The red dot marks the initial position of the active droplet. Each frame is $4$ s apart. (Scale bar, $50\hspace{0.17em}\mathrm{\mu }$m.) (B) A sketch of the nematic director field (black lines) inside the droplet due to homeotropic anchoring conditions at the interface before (hedgehog defect in the center) and after the symmetry breaking (escaped radial configuration). (C) Active droplets in a Hele–Shaw cell (height $50\hspace{0.17em}\mathrm{\mu }$m, which is also the diameter of the droplets; the lateral dimensions of the cell extend beyond the field of view and are $4$ cm $\times$ $3$ cm. (Scale bar, $500\hspace{0.17em}\mathrm{\mu }$m.) The red arrows indicate the instantaneous velocities of the droplets. (D) The probability distribution, using $\sim 10^7$ measurements, of the velocity vectors ($v_x,v_y$) for individual droplets in a Hele–Shaw geometry. Color bar represents the normalized probability. (E) The mean squared displacements calculated from the trajectories of a dilute suspension of active droplets (areal fraction of droplets is $<0.5\hspace{0.17em}\%$). (Inset) A superposition of the trajectories of the droplets with their point of origin aligned (marked by red spot). (Scale bar, $1$ mm.) (F) The rearrangement of the director field inside a droplet swimming to the right is caused by a spontaneous flow inside the droplet. (G) Experimentally measured external fluid flow for an active droplet (marked by black circle) moving at a fixed velocity. (Scale bar is $50\hspace{0.17em}\mathrm{\mu }$m.) (H) The theoretical flow from a truncated spherical harmonic expansion, with the expansion coefficients estimated from the experimental flow in G. The droplet diameter in F and G is $50\hspace{0.17em}\mathrm{\mu }$m, and the color bars represent the normalized logarithm of the local flow speed.

Each droplet propels in a random direction set by its own internal spontaneously broken symmetry, and these directions are distributed isotropically (shown in Fig. 1C for the case of a quasi-2D Hele–Shaw cell). The droplet speed (Fig. 1D) is set by its size and the concentration of the surfactant in the external phase (8). In such a configuration, individual droplets exhibit random, diffusive-like motion due to the fluctuations in the self-propulsion mechanism and due to interactions with the other droplets (Fig. 1E). The balance between viscous and nematic stresses within the droplet tends to align the velocity field with the director field (Fig. 1F) (23), resulting in an asymmetry in the circulatory flow inside the droplets (Movie S2), with a stagnation point close to the point defect. This asymmetry also appears in the external flow generated by the droplets, as can be seen in the velocity field (Fig. 1G) due to a droplet moving with speed $v_s$. It is in this external flow that other particles are entrained and is, therefore, the focus of our theoretical model.

Our theoretical model for an active particle is a sphere of radius $b$ with an active slip prescribed at its surface. As our primary interest is in the external flow, we assume the internal flow to be a rigid body motion. The fluid velocity on the boundary of the $i$-th sphere, then, is

$$\mathit{v}\left({\mathit{R}}_i+{\mathit{\rho }}_i\right)={\mathbf{\text{{V}}}}_i+{\mathbf{\Omega }}_i\times {\mathit{\rho }}_i+{\mathit{v}}_i^{\mathcal{A}}\left({\mathit{\rho }}_i\right),$$ [1]

where ${\mathit{R}}_i$ is the center of the sphere, ${\mathit{\rho }}_i$ is a point on its surface with respect to the center, and ${\mathbf{V}}_i$ and ${\mathbf{\Omega }}_i$ are, respectively, its linear and angular velocity. The active slip, ${\mathit{v}}_i^{\mathcal{A}}\left({\mathit{\rho }}_i\right)$, is taken to be the most general surface vector field consistent with incompressibility. Neither axial symmetry of the slip about the orientation axis, ${\mathit{p}}_i$, of the sphere nor flow purely tangential to the interface is assumed. These assumptions distinguish our model (4, 24–26) from that of the classical squirmer (27, 28). The translational and rotational velocities of the spheres are not known a priori but must be determined in terms of the slip velocities from a balance of all forces and torques acting on them.

Slip induces flow in the exterior fluid, and the stresses thus produced act back on the sphere surface with a force per unit area $\mathit{f}$. Then, ${\mathbf{F}}_i^H=\int \mathit{f}\hspace{0.17em}d{\text{S}}_i$ and ${\mathbf{T}}_i^H=\int {\mathit{\rho }}_i\times \mathit{f}\hspace{0.17em}d{\text{S}}_i$ are the net hydrodynamic force and torque on sphere $i$, which include contributions from the usual Stokes drag, proportional to ${\mathbf{V}}_i$ and ${\mathbf{\Omega }}_i$, and from the active stresses, proportional to ${\mathit{v}}_i^{\mathcal{A}}$. The force per unit area is computed from the solution of the Stokes equation satisfying Eq. 1 at the sphere surfaces and the appropriate hydrodynamic boundary conditions at the exterior boundaries.

For our purpose, the boundary integral representation of Stokes flow is most suited for obtaining the force per unit area, as the hydrodynamic boundary conditions can be directly applied by choosing a suitable Green’s function. We use this approach here and analytically solve the resulting integral equations for the force per unit area, to leading order in sphere separation, in a basis of tensorial spherical harmonics with appropriate Green’s functions. The forces and torques thus obtained are inserted into force and torque balance equations that are integrated numerically to obtain the translational and rotational motions of the spheres (further details of the model and simulations are in SI Appendix).

The free parameters in our model, the sphere radius $b_i$ and the slip velocity ${\mathit{v}}_i^{\mathcal{A}}$, are determined as follows: We set $b_i\sim 50\hspace{0.17em}\mathrm{\mu }$m, which is the measured radius of an undissoluted droplet. There is less than $1\%$ change to this value during the course of the experiment. We use the exterior flow of a single droplet to determine the slip, as the two are uniquely related for any given hydrodynamic boundary condition. We parametrize ${\mathit{v}}_i^{\mathcal{A}}$ in terms of its first three tensorial harmonic coefficients, as these fully account for the long-ranged components of the exterior flow. We then estimate the coefficients by minimizing the square deviation between the experimentally measured flow and the three-mode expansion. The exterior flow thus obtained (Fig. 1H) is in good agreement with the experimentally measured flow (Fig. 1G). We note that the flow in the Hele–Shaw cell is used for this estimation and comparison.

To emphasize, we fit the one-body exterior flow to estimate the active slip and then use it to predict the many-body exterior flow and the many-body forces and torques for any given boundary condition. We note that the three-mode expansion is not a limitation of the theoretical model, which accommodates as many modes as may be necessary to represent the exterior flow to the desired level of accuracy.

Self-Organization and Boundary Conditions

We now present the main results on the correspondence between self-organization of active particles and hydrodynamic boundary conditions. Our boundary conditions are (i) a plane channel flow in a Hele–Shaw cell where the channel width $H$ is approximately one particle diameter (Fig. 2A), (ii) a plane channel flow where the channel width varies between several particle diameters ($H\sim 6-10b$; Fig. 2B), (iii) flow bounded by a plane wall where the flow vanishes (Fig. 2C), and (iv) flow bounded by a plane air–water interface where the tangential stress vanishes (Fig. 2D). In every case, the emulsion parameters are kept unchanged; the only change is in the exterior boundary conditions.

Fig. 2.

The role of boundaries in determining the collective behavior of active droplets (particles). (Top) Schematic of confinement. (Upper Middle) The exterior flow field produced by the active particles in each boundary condition considered. Lower Middle and Bottom contain snapshots from simulations and experiments, respectively. Traveling lines of active droplets can be formed in a Hele–Shaw cell. These lines are metastable; that is, they translate a few droplet diameters before breaking up, if the separation $H$ of the cell is approximately equal to the droplet diameter (A, $H/b\sim 2$), while these lines are stable when the separation is a few droplet diameters (B, $H/b\sim 8$). Aggregation of the droplets, leading to crystallization, is observed at a plane wall (C, cell depth $H/b\sim 400$) and at the air–water interface (D, cell depth $H/b\sim 400$). At a plane wall, droplets are expunged from the crystalline core, while the crystal is stabilized by the recirculation of the fluid flow. At the air–water interface, on the other hand, there is no out-of-plane motion. Here $\tau =b/v_s$ is the time in which the active droplet moves a distance equal to its radius.

Snapshots from the experiments (Fig. 2, Bottom) point to distinct signatures of the boundary conditions on the resultant self-organization of the droplets. In the Hele–Shaw cell, $H\sim 2b$, droplets spontaneously form metastable lines, which curve in their direction of motion and eventually break up after traversing a few droplet diameters (Fig. 2A and Movie S3). Increasing the channel width, $H\sim 8b$, transforms these lines into surprisingly stable bands that travel through each other even as they collide (Fig. 2B and Movie S4). In contrast, at a plane wall, droplets form crystallites parallel to the plane. Droplets comprising the crystallite are constantly expelled from the center of the aggregate only to rejoin it at the edges. The recirculating flows ensure a balanced in- and out-flux of the droplets and thereby maintain a constant mean droplet number within these aggregates (vortex-stabilized crystallites, Fig. 2C and Movie S5). When the plane wall is replaced by a plane interface, the previous inflow and outflow is suppressed, and the droplets form 2D crystalline aggregates. These aggregates are maintained in a steady state by a continuous coagulation and fragmentation of the crystallites (Fig. 2D and Movie S6).

A qualitative understanding of these states of self-organization is obtained from the one-body external flow of the particle in each of the four boundary conditions (corresponding panels of Fig. 2). We emphasize, once again, that in this calculation the active slip is estimated from flow in the Hele–Shaw cell but used to predict flow for the three remaining boundary conditions. Operationally, the latter only requires the use of the appropriate Green’s function. In the Hele–Shaw cell, the net flow is parallel to the walls and has an inflowing component perpendicular to the direction of motion. Entrainment in this inflow leads to the formation of metastable lines and stable bands. At both the plane wall and the plane interface, the flow has a cylindrical symmetry when the propulsion axis is perpendicular to the plane. In the first case, the flow has a strong circulation in which entrained particles are drawn inwards along the plane but then expelled normal to it. In the second case, the circulatory component is comparatively weak, and entrained particles are primarily drawn inwards. Entrainment in this flow leads to the formation, respectively, of vortex-stabilized crystallites and of coagulating and fragmenting 2D crystallites. The qualitative agreement of the simulations (Fig. 2, Lower Middle) obtained from the numerical integration of the force and torque balance equations with the experiment (Fig. 2, Bottom) is excellent. This agreement between experiment and theory that disregards the internal flow confirms, a posteriori, our hypothesis that entrainment in the external flow is primarily responsible for self-organization.

A quantitative understanding of the states of self-organization requires an accurate estimate of the forces and torques between particles. We calculate the components of the active pair force parallel and perpendicular to their separation vector, as a function of separation distance, for each of the four boundary conditions considered (Fig. 3, Upper Middle and Lower Middle). In each of the cases, the component of the active force parallel to the separation vector is negative, and it is this attractive component of the active force that leads to aggregation. However, there is a considerable variation in the component of the active force perpendicular to the separation vector, and it is the sensitivity of this component to boundary conditions that accounts for the variety of the aggregated states. In the Hele–Shaw cell, the perpendicular force is positive, which in our sign convention means that it is directed along the direction of motion. Since, to leading order, hydrodynamic forces are pair-wise additive, this implies that the net force on particles at the center of a moving line are greater than those at the edges. Therefore, they tend to move faster, creating a curvature of the line and eventually its break up. In contrast, the perpendicular force in the plane channel is an order of magnitude smaller (as shown in the corresponding Inset of Fig. 3B for clarity), and the break-up mechanism has a negligible contribution, which gives the traveling bands their surprisingly stability. At a no-slip wall (Fig. 3C), the perpendicular force is negative at large distances but positive at short distances. This drives particles into the wall when they are well-separated but away from the wall when they are close by. Combined with the parallel component of the flow, which is always attractive, this leads to the expulsion and recirculation of particles in the crystalline aggregate. In contrast, the perpendicular force at an interface (Fig. 3D) is positive but an order of magnitude smaller (again, shown in the Inset for clarity), and the dominant motion is due to the attractive component of the parallel force. Thus, expulsion is suppressed, and the result is the formation of 2D crystallites. Similar estimates for the torque provide an understanding of the orientational dynamics, which is relatively unimportant in our case as Brownian reorientation is negligible and the hydrodynamic torques are one power of separation smaller than the corresponding forces.

Fig. 3.

Active forces on the active particles are modified by the presence of boundaries. (Top) Schematic of confinement. (Upper Middle) Attractive parallel forces lead to the formation of traveling lines in a Hele–Shaw cell (A and B), and aggregation in the plane of the wall (C) and the interface (D). (Lower Middle) The perpendicular forces in a Hele–Shaw cell are an order of magnitude larger for A and account for the metastable lines in this case. The perpendicular force at the plane wall is 10 times larger than the corresponding force at the interface and results in the circulatory motion of active particles (see text for more details). Insets show close-up and $F_{\mathcal{A}}=6\pi \eta b{v}_s.$ (Bottom) State diagrams in terms of the strengths of the slip modes: $V_0^{\left(2s\right)}$, symmetric dipole; $V_0^{\left(3t\right)}$, vector quadrupole; and $V_0^{\left(4t\right)}$, degenerate octupole. Each dot represents a simulation, while the star denotes the values used in the above rows.

Since each irreducible mode of the slip is independent of the others and produces flow of distinct multipolar symmetry, it is possible to isolate the effect of each mode on the self-organization. The modes are labeled by an angular momentum index $l=\mathrm{0,1,2},\dots$ and a spin index $\sigma =s,a,t$ corresponding to the symmetric, antisymmetric, and pure trace irreducible components of each mode. Using the abbreviation $l\sigma$ to indicate a mode with angular momentum index $l$ and spin index $\sigma$, the $3t$ mode produces self-propulsion (and a degenerate quadrupolar flow), while the $2s$ mode produces inflow and outflow along mutually perpendicular axes (and a dipolar flow). The latter is the stresslet mode, and its instantaneous sign determines if the fluid is being expelled (“pusher”) or ingested (“puller”) along the propulsion axis. We are then able to construct a state diagram in the $V_0^{\left(2s\right)}-V_0^{\left(3t\right)}$ plane that demarcates regions of stability of the principal states of aggregation found for each of the four boundary conditions studied (Fig. 3, Bottom). In Hele–Shaw flow, the stresslet mode promotes stability, but in flows bounded by a plane wall, the perpendicular component of the force is enhanced in proportion to the magnitude of the stresslet, which leads from 2D crystallites to vortex-stabilized crystallites and finally to instabilities. Thus, by suitably choosing the strength of the stresslet mode, it is possible to select either a state of 2D crystals or vortex-stabilized crystals. For flow bounded by an interface, an enhanced stresslet mode does not lead to explusion of the droplets out-of-plane but does lead directly to an instability (unstable crystallites).

Our results above provide convincing evidence of the mechanism of hydrodynamic entrainment in the spontaneous exterior flow as the dominant mechanism for self-organization in active particles. The active parts of the hydrodynamic forces and torques that result from this entrainment provide both a qualitative and quantitative explanation of the states of self-organization. These hydrodynamic forces and torques depend on the distance between the particles, their orientation, and the magnitudes of the modes of the slip on each active particle. It can be directly verified from the explicit forms of the forces and torques that they cannot be obtained as gradients of potentials (20). Further, their dependence on the modes of the slip velocity indicates that they have odd parity under time reversal, signaling their explicitly dissipative nature (4, 26). Thus, we have a situation in which long-range, dissipative forces and torques promote self-organization. States of self-organization maintained by entropy production were studied in the past by the Brussels school and were given the name “dissipative structures” (29). Self-organization in active particles, as shown here, appears to be an example of a dissipative structure but one in which the dissipative mechanism and the resultant forces and torques are unambiguously identified.

Kinetics of FIPS

The self-organization presented above can be viewed, from the point of view of statistical physics, as a phase separation phenomenon driven by dissipative forces, rather than the usual conservative forces derived from a potential. The study of such FIPSs presents several challenges, one of which is to determine a quantity that can serve as the analogue of a thermodynamic potential. It appears from recent theoretical work that large-deviation results for the stationary distribution of Markov processes violating detailed balance may provide a tractable route for answering questions of stability (30). Here we draw attention to the fact that kinetic routes to similar flow-induced phase-separated states may vary depending on the boundary condition. This is borne out in the differences between the kinetics of phase separation at a plane wall and at a plane interface (Fig. 4 and Movie S7). At a plane wall, there is both an enhanced mixing within the crystal plane and an exchange of neighbors due to the closed streamlines (Fig. 2C) and higher values of perpendicular forces (Fig. 3C) compared with the plane interface. In a biological context, such hydrodynamic bound structures and associated kinetics could influence the encounter rate of individuals in aggregates (6, 31).

Fig. 4.

Kinetics of aggregation of active particles at a plane wall (Top) and at a plane interface (Bottom). The colors are used to indicate and track the particles based on their initial positions. The instantaneous snapshots show that there is a faster mixing of particles and exchange of neighbors at the plane wall due to the circulatory flow streamlines (Fig. 2C) and higher values of perpendicular forces (Fig. 3C).

Discussion

The FIPS mechanism established here and conjectured earlier (4) has several distinguishing features that bear pointing out and contrasting with other mechanisms. First, self-propulsion and/or self-rotation are not necessary for its operation; it is only necessary that the particles produce a long-ranged exterior hydrodynamic flow. The experimentally observed states of self-organization persist even when the self-propulsion parameter $V_0^{\left(3t\right)}$ vanishes (state diagrams; Fig. 3, Bottom). Second, in the absence of thermal fluctuations, the suspension is mechanically unstable to aggregation at any positive value of the density and any finite amount of activity, however small. It is plausible that in the presence of thermal fluctuations, finite values of density and activity are required to overwhelm the loss in entropy due to aggregation. A careful study of aggregation at different temperatures, densities, and activities is needed to establish this quantitatively. Third, FIPS includes, as a special case, aggregation in driven colloidal systems, which are nonetheless force- and torque-free (32, 33). Fourth, the long-lived, stable traveling bands that we have shown here are qualitatively different compared with other dynamic behavior seen in active systems (20, 34, 35) or in the emergence of the orientational order in flocking models (36). Despite the similarity in the aggregated states, the hydrodynamic mechanism identified here is distinct. Fifth, our work provides an understanding of phase separation in active systems that is complementary to motility-induced phase separation (MIPS) (37). The latter has a kinematic character, in which the flux of particles is a prescribed functional of density, reflecting the tendency of active particles to slow down or speed up in regions of, respectively, higher or lower density. The physics underlying this tendency is left implicit and may plausibly be attributed to nonhydrodynamic interactions, like contact, compression, or jamming. In contrast, FIPS has an explicitly dynamic character, as forces and torques of hydrodynamic origin are identified in causing the aggregation. It is entirely conceivable that a state of aggregation like hexagonal crystallites can be formed from either of these mechanisms. On the other hand, vortex-stabilized crystallites appear difficult to explain within MIPS, whereas they appear naturally in FIPS. Since FIPS needs a wall with no slip or an interface with vanishing tangential stress, aggregate structures that remain stable away from such boundaries would point to a mechanism such as MIPS as the source of stability. Finally, though our experiments are performed with droplets, the theory and simulation correspond to arbitrary active particles, and our flow boundary conditions are generic to many natural and engineered settings. Together, these underscore the potential significance of our findings to a wide class of active fluids, in particular active colloids, and to the study and control of geometric and topological phenomena in active matter.

Materials and Methods

Experiments.

Droplets were produced using microfluidic devices that were fabricated from polydimethylsiloxane (Sylgard 184; Dow Corning) using standard soft lithographic protocols and bonded to glass slides. The carrier fluid was comprised of an aqueous solution of $0.25$ wt$\%$ of the surfactant SDS (Sigma-Aldrich). The droplet phase was the liquid crystalline oil $4$-pentyl-$4\prime$-cyano-biphenyl (5CB, Frinton Laboratories Inc.). The flow rates were volume-controlled using syringe pumps (Harvard Apparatus). A few million droplets were produced and stored in the aqueous mixture of surfactant. For the experiments, a suitable amount of the droplets were then resuspended in an aqueous mixture of 10 to 25$\%$ by weight of SDS, which set the droplets spontaneously into motion. The Hele–Shaw cell for the experiments was constructed using microscope slides separated by double-sided sticky tape of precise thickness (the thickness of the Hele–Shaw gap varied from $50\hspace{0.17em}\mathrm{\mu }$m to $2$ cm), which was cut into a chamber of a desired shape ($5$ cm$\hspace{0.17em}\times \hspace{0.17em}4$ cm) using a standard plotter cutter. The microscope slides were rendered hydrophilic by plasma treatment just before assembly.

Image Processing and Analysis.

The experiments were recorded using an sCMOS camera (Hamamatsu) at frame rates of $1-4$ fps in a region of $\sim 2.2$ mm$\times 2.2$ mm close to the center of the Hele–Shaw chamber. For the particle image velocimetry (PIV) measurements, the aqueous phase was seeded with $200$ nm red fluorescent polystyrene beads (Thermo Scientific), and movies were recorded at $100$ fps. Droplet tracking and all other analyses subsequently were performed using custom-written Matlab code. The flow field analysis was performed using PIVLab, an open source Matlab code.

Simulations.

The simulations were performed by numerical integration of the force and torque balance equations (for each boundary condition) using PyStokes, a Cython library for computing hydrodynamic interactions, with an adaptive time-step integrator. A random packing of hard spheres is used as the initial distribution of particles in all of the simulations. The particle–particle and the particle–wall repulsive interaction is modeled using the short-ranged repulsive Weeks–Chandler–Andersen potential, which is given as $U\left(r\right)=ϵ{\left(\frac{r_{min}}{r}\right)}^{12}-2ϵ{\left(\frac{r_{min}}{r}\right)}^6+ϵ,$ for $r<r_{min}$ and zero otherwise, where $ϵ$ is the potential strength.