Flow-induced phase separation of active particles is …PHYSICS Flow-induced phase separation of...

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PHYSICS Flow-induced phase separation of active particles is controlled by boundary conditions Shashi Thutupalli a,b,c,d,1 , Delphine Geyer c , Rajesh Singh e,f , Ronojoy Adhikari e,f , and Howard A. Stone c a Simons Centre for the Study of Living Machines, National Centre for Biological Sciences, Bangalore 560065, India; b International Centre for Theoretical Sciences, Tata Institute of Fundamental Research, Bangalore 560012, India; c Department of Mechanical and Aerospace Engineering, Princeton University, Princeton, NJ 08544; d Joseph Henry Laboratories of Physics, Princeton University, Princeton, NJ 08544; e Department of Theoretical Physics, The Institute of Mathematical Sciences–Homi Bhabha National Institute, Chennai 600113, India; and f Department of Applied Mathematics and Theoretical Physics, Centre for Mathematical Sciences, University of Cambridge, Cambridge CB3 0WA, United Kingdom Edited by David A. Weitz, Harvard University, Cambridge, MA, and approved April 11, 2018 (received for review October 27, 2017) Active particles, including swimming microorganisms, autophore- tic 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 pos- tulated as a possible mechanism underlying this phenomenon. Here, combining experiments, theory, and numerical simulations, we demonstrate the validity of this flow-induced ordering mech- anism 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 crystal- lites 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 exper- imental 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 simu- lations 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 flu- ids, particularly active colloidal suspensions, and demonstrates its control by boundaries, suggesting routes to geometric and topological phenomena in an active matter. active matter | phase separation | hydrodynamics | boundary effects T here are many instances, drawn from biological, physico- chemical, 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 mecha- nisms, 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) pro- vide 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 pos- sible, 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 par- ticles 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 bound- aries 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 col- lective 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 50 μm of the emul- sion droplets and their typical self-propulsion speed vs 5 μm/s implies that the Reynolds number Re = vs b 10 -4 in a fluid with the kinematic viscosity ν of water. Fluid inertia is negligible at such small Re , and the flow is described by the Stokes equa- tion. 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 dis- tinct 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 Significance Active particle suspensions comprise energy-consuming and hydrodynamically interacting units, such as swimming micro- organisms, 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 sus- pension of active emulsion droplets. Our results establish a widely applicable paradigm for flow-induced phase separa- tion in active fluids and offer routes to manipulating their microstructure. Author contributions: S.T., R.A., and H.A.S. designed research; S.T., D.G., and R.S. per- formed research; S.T., R.S., and R.A. contributed new reagents/analytic tools; S.T. and R.S. analyzed data; and S.T., R.S., R.A., and H.A.S. wrote the paper. The authors declare no conflict of interest. This article is a PNAS Direct Submission. Published under the PNAS license. 1 To whom correspondence should be addressed. Email: [email protected]. This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10. 1073/pnas.1718807115/-/DCSupplemental. Published online May 7, 2018. www.pnas.org/cgi/doi/10.1073/pnas.1718807115 PNAS | May 22, 2018 | vol. 115 | no. 21 | 5403–5408 Downloaded by guest on December 12, 2020

Transcript of Flow-induced phase separation of active particles is …PHYSICS Flow-induced phase separation of...

Page 1: Flow-induced phase separation of active particles is …PHYSICS Flow-induced phase separation of active particles is controlled by boundary conditions Shashi Thutupallia,b,c,d,1, Delphine

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Flow-induced phase separation of active particles iscontrolled by boundary conditionsShashi Thutupallia,b,c,d,1, Delphine Geyerc, Rajesh Singhe,f, Ronojoy Adhikarie,f, and Howard A. Stonec

aSimons Centre for the Study of Living Machines, National Centre for Biological Sciences, Bangalore 560065, India; bInternational Centre for TheoreticalSciences, Tata Institute of Fundamental Research, Bangalore 560012, India; cDepartment of Mechanical and Aerospace Engineering, Princeton University,Princeton, NJ 08544; dJoseph Henry Laboratories of Physics, Princeton University, Princeton, NJ 08544; eDepartment of Theoretical Physics, The Institute ofMathematical Sciences–Homi Bhabha National Institute, Chennai 600113, India; and fDepartment of Applied Mathematics and Theoretical Physics, Centrefor Mathematical Sciences, University of Cambridge, Cambridge CB3 0WA, United Kingdom

Edited by David A. Weitz, Harvard University, Cambridge, MA, and approved April 11, 2018 (received for review October 27, 2017)

Active particles, including swimming microorganisms, autophore-tic colloids, and droplets, are known to self-organize into orderedstructures at fluid–solid boundaries. The entrainment of particlesin the attractive parts of their spontaneous flows has been pos-tulated as a possible mechanism underlying this phenomenon.Here, combining experiments, theory, and numerical simulations,we demonstrate the validity of this flow-induced ordering mech-anism in a suspension of active emulsion droplets. We show thatthe mechanism can be controlled, with a variety of resultantordered structures, by simply altering hydrodynamic boundaryconditions. Thus, for flow in Hele–Shaw cells, metastable linesor stable traveling bands can be obtained by varying the cellheight. Similarly, for flow bounded by a plane, dynamic crystal-lites are formed. At a no-slip wall, the crystallites are characterizedby a continuous out-of-plane flux of particles that circulate andre-enter at the crystallite edges, thereby stabilizing them. At aninterface where the tangential stress vanishes, the crystallites arestrictly 2D, with no out-of-plane flux. We rationalize these exper-imental results by calculating, in each case, the slow viscous flowproduced by the droplets and the long-ranged, many-body activeforces and torques between them. The results of numerical simu-lations of motion under the action of the active forces and torquesare in excellent agreement with experiments. Our work elucidatesthe mechanism of flow-induced phase separation in active flu-ids, particularly active colloidal suspensions, and demonstratesits control by boundaries, suggesting routes to geometric andtopological phenomena in an active matter.

active matter | phase separation | hydrodynamics | boundary effects

There are many instances, drawn from biological, physico-chemical, and technological contexts, in which microscopic

particles produce spontaneous flow in a viscous fluid. The energynecessary to maintain this flow is supplied by a variety of mecha-nisms, of which there are a wide variety, at the interface betweenthe particles and the fluid. The ciliary layer in cells (1), thechemically reacting boundary layer in autophoretic colloids (2),and the dissolution layer in auto-osmophoretic drops (3) pro-vide three distinct examples. In each case, the activity within thelayer drives the exterior fluid into motion, which appears as ifit were a spontaneous fluid flow around the particles. It is pos-sible, though not necessary, for the particles to translate and/orrotate in response to the spontaneous flow. Irrespective of theproperty of self-propulsion and/or self-rotation, such active par-ticles in a suspension will each produce a spontaneous flow inwhich other particles will be entrained. This mutual entrainment,if sufficiently strong, can produce states of organization with noanalogue 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 ofthis flow-induced phase separation (FIPS) mechanism is stilllacking. If bulk hydrodynamic flow is the principal cause of

self-organization, any alteration of the flow should manifestitself in altered states of self-organization. The simplest way ofaltering the bulk flow, keeping other experimental conditionsconstant, is to vary the hydrodynamic conditions at the bound-aries of the flow. If this produces correspondingly distinct statesof self-organization, both the FIPS mechanism and the role ofboundaries in controlling it are, thereby, established.

Here, we use a suspension of active, self-propelled emulsiondroplets to investigate the role of hydrodynamics on their col-lective behavior. The system has been used previously (3, 7–10)to provide insights into out-of-equilibrium phenomena relevantto self-organization in both natural (6, 11, 12) and syntheticactive particle settings. The typical size b∼ 50 µm of the emul-sion droplets and their typical self-propulsion speed vs ∼ 5 µm/simplies that the Reynolds number Re = vsb/ν∼ 10−4 in a fluidwith the kinematic viscosity ν of water. Fluid inertia is negligibleat such small Re , and the flow is described by the Stokes equa-tion. It is then possible to exploit the linearity of the governingequations and use Green’s function techniques to compute theflow, the stress in the fluid, and the forces and torques betweenthe droplets for a variety of boundary conditions. We find dis-tinct states of aggregation as the boundary conditions are altered,which both validates the conjectured hydrodynamic mechanismand opens up a route to its manipulation and control.

We should note the effect of steric confinement of active fluidson self-organization has been studied thoroughly in the past (13–19). This knowledge has been exploited in applications such as

Significance

Active particle suspensions comprise energy-consuming andhydrodynamically interacting units, such as swimming micro-organisms, autophoretic colloids, and active droplets. Thoughit is now recognized that emergent self-organization in suchsystems is driven by their spontaneous hydrodynamic flow,it is still not well-understood how the modification of thisflow by confining boundaries impacts self-organization. Herewe combine experiments, theory, and simulations to elucidatethe effect of boundaries on the spontaneous flow in a sus-pension of active emulsion droplets. Our results establish awidely applicable paradigm for flow-induced phase separa-tion in active fluids and offer routes to manipulating theirmicrostructure.

Author contributions: S.T., R.A., and H.A.S. designed research; S.T., D.G., and R.S. per-formed research; S.T., R.S., and R.A. contributed new reagents/analytic tools; S.T. and R.S.analyzed data; and S.T., R.S., R.A., and H.A.S. wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.

Published under the PNAS license.1 To whom correspondence should be addressed. Email: [email protected].

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1718807115/-/DCSupplemental.

Published online May 7, 2018.

www.pnas.org/cgi/doi/10.1073/pnas.1718807115 PNAS | May 22, 2018 | vol. 115 | no. 21 | 5403–5408

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self-assembly, meta-material synthesis, and active fluid compu-tation (20–22). However, the resultant hydrodynamic effects ofconfinement can be distinct due to differences, for instance, inthe slip properties of the boundaries even when the geometry ofthe confinement is identical. Our work suggests an independentroute to ordered states of active matter by using boundaries toalter the hydrodynamic interactions in the system, rather than tomodify its geometric confinement. With this remark, we now turnto our results.

Experimental System and Theoretical ModelOur experimental system is an active emulsion of monodispersedroplets of liquid crystal (5CB) in water whose source of activityis droplet dissolution (8). Surface tension gradients at the inter-face of the droplet, sustained by the free energy of dissolution,produce active hydrodynamic flows both within and external tothe droplet, leading to droplet motion. While this self-propulsiondoes not rely on the liquid crystallinity of the droplet, the nematicstate of 5CB within it enables the internal velocity field to beinferred (Fig. 1A and Movie S1). The presence of surfactantat the interface makes it energetically favorable for the rod-like 5CB molecules to orient normal to it. Such a homeotropicboundary condition on the director field enforces a point defect,which is located at the center of the droplet (Fig. 1B) whensurface tension gradients (and hence fluid flow) are negligible.When observed in polarized light between cross-polarizers, thedroplet shows a four-lobed pattern (Fig. 1A, Leftmost) reflect-ing the symmetric orientation of the director field about thedroplet center. When surface tension gradients become appre-ciable, 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 displacementof the point defect along the axis of droplet motion, and the cen-trosymmetric four-lobed pattern is distorted to another with areduced symmetry, now only about the propulsion axis (Fig. 1 Aand B).

Each droplet propels in a random direction set by its owninternal spontaneously broken symmetry, and these directionsare distributed isotropically (shown in Fig. 1C for the case of aquasi-2D Hele–Shaw cell). The droplet speed (Fig. 1D) is setby its size and the concentration of the surfactant in the exter-nal phase (8). In such a configuration, individual droplets exhibitrandom, diffusive-like motion due to the fluctuations in the self-propulsion mechanism and due to interactions with the otherdroplets (Fig. 1E). The balance between viscous and nematicstresses within the droplet tends to align the velocity field withthe director field (Fig. 1F) (23), resulting in an asymmetry in thecirculatory flow inside the droplets (Movie S2), with a stagnationpoint close to the point defect. This asymmetry also appears inthe external flow generated by the droplets, as can be seen in thevelocity field (Fig. 1G) due to a droplet moving with speed vs . Itis 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 ofradius b with an active slip prescribed at its surface. As our pri-mary interest is in the external flow, we assume the internal flowto be a rigid body motion. The fluid velocity on the boundary ofthe i -th sphere, then, is

v(Ri +ρi)=Vi +Ωi ×ρi + vAi (ρi), [1]

Fig. 1. Active droplets made with nematic liquid crystals. (A) Cross-polarization microscopy images showing spontaneous symmetry breaking and propulsionof an active droplet. The red dot marks the initial position of the active droplet. Each frame is 4 s apart. (Scale bar, 50 µm.) (B) A sketch of the nematicdirector field (black lines) inside the droplet due to homeotropic anchoring conditions at the interface before (hedgehog defect in the center) and afterthe symmetry breaking (escaped radial configuration). (C) Active droplets in a Hele–Shaw cell (height 50 µm, which is also the diameter of the droplets;the lateral dimensions of the cell extend beyond the field of view and are 4 cm × 3 cm. (Scale bar, 500 µm.) The red arrows indicate the instantaneousvelocities of the droplets. (D) The probability distribution, using ∼ 107 measurements, of the velocity vectors (vx , vy ) for individual droplets in a Hele–Shawgeometry. Color bar represents the normalized probability. (E) The mean squared displacements calculated from the trajectories of a dilute suspension ofactive droplets (areal fraction of droplets is < 0.5 %). (Inset) A superposition of the trajectories of the droplets with their point of origin aligned (markedby 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 insidethe droplet. (G) Experimentally measured external fluid flow for an active droplet (marked by black circle) moving at a fixed velocity. (Scale bar is 50 µm.)(H) The theoretical flow from a truncated spherical harmonic expansion, with the expansion coefficients estimated from the experimental flow in G. Thedroplet diameter in F and G is 50 µm, and the color bars represent the normalized logarithm of the local flow speed.

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where Ri is the center of the sphere, ρi is a point on itssurface with respect to the center, and Vi and Ωi are, respec-tively, its linear and angular velocity. The active slip, vAi (ρi),is taken to be the most general surface vector field consistentwith incompressibility. Neither axial symmetry of the slip aboutthe orientation axis, pi , of the sphere nor flow purely tangen-tial to the interface is assumed. These assumptions distinguishour model (4, 24–26) from that of the classical squirmer (27,28). The translational and rotational velocities of the spheresare not known a priori but must be determined in terms of theslip velocities from a balance of all forces and torques actingon them.

Slip induces flow in the exterior fluid, and the stresses thus pro-duced act back on the sphere surface with a force per unit area f .Then, FH

i =∫

f dSi and THi =

∫ρi × f dSi are the net hydrody-

namic force and torque on sphere i , which include contributionsfrom the usual Stokes drag, proportional to Vi and Ωi , and fromthe active stresses, proportional to vAi . The force per unit areais computed from the solution of the Stokes equation satisfyingEq. 1 at the sphere surfaces and the appropriate hydrodynamicboundary conditions at the exterior boundaries.

For our purpose, the boundary integral representation ofStokes flow is most suited for obtaining the force per unit area,as the hydrodynamic boundary conditions can be directly appliedby choosing a suitable Green’s function. We use this approachhere and analytically solve the resulting integral equations forthe force per unit area, to leading order in sphere separation, ina basis of tensorial spherical harmonics with appropriate Green’sfunctions. The forces and torques thus obtained are inserted intoforce and torque balance equations that are integrated numer-ically to obtain the translational and rotational motions of thespheres (further details of the model and simulations are in SIAppendix).

The free parameters in our model, the sphere radius bi and theslip velocity vAi , are determined as follows: We set bi ∼ 50 µm,which is the measured radius of an undissoluted droplet. Thereis less than 1% change to this value during the course of theexperiment. We use the exterior flow of a single droplet to deter-mine the slip, as the two are uniquely related for any given

hydrodynamic boundary condition. We parametrize vAi in termsof its first three tensorial harmonic coefficients, as these fullyaccount for the long-ranged components of the exterior flow. Wethen estimate the coefficients by minimizing the square deviationbetween the experimentally measured flow and the three-modeexpansion. The exterior flow thus obtained (Fig. 1H) is in goodagreement with the experimentally measured flow (Fig. 1G).We note that the flow in the Hele–Shaw cell is used for thisestimation and comparison.

To emphasize, we fit the one-body exterior flow to estimate theactive slip and then use it to predict the many-body exterior flowand the many-body forces and torques for any given boundarycondition. We note that the three-mode expansion is not a lim-itation of the theoretical model, which accommodates as manymodes as may be necessary to represent the exterior flow to thedesired level of accuracy.

Self-Organization and Boundary ConditionsWe now present the main results on the correspondence betweenself-organization of active particles and hydrodynamic boundaryconditions. Our boundary conditions are (i) a plane channel flowin a Hele–Shaw cell where the channel width H is approximatelyone particle diameter (Fig. 2A), (ii) a plane channel flow wherethe channel width varies between several particle diameters (H ∼6− 10b; Fig. 2B), (iii) flow bounded by a plane wall where theflow 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; theonly change is in the exterior boundary conditions.

Snapshots from the experiments (Fig. 2, Bottom) point todistinct signatures of the boundary conditions on the resultantself-organization of the droplets. In the Hele–Shaw cell, H ∼ 2b,droplets spontaneously form metastable lines, which curve intheir direction of motion and eventually break up after traversinga few droplet diameters (Fig. 2A and Movie S3). Increasing thechannel width, H ∼ 8b, transforms these lines into surprisinglystable bands that travel through each other even as they collide(Fig. 2B and Movie S4). In contrast, at a plane wall, dropletsform crystallites parallel to the plane. Droplets comprising the

Fig. 2. The role of boundaries in determining thecollective behavior of active droplets (particles).(Top) Schematic of confinement. (Upper Middle) Theexterior flow field produced by the active particles ineach boundary condition considered. Lower Middleand Bottom contain snapshots from simulations andexperiments, respectively. Traveling lines of activedroplets can be formed in a Hele–Shaw cell. Theselines are metastable; that is, they translate a fewdroplet diameters before breaking up, if the sepa-ration H of the cell is approximately equal to thedroplet diameter (A, H/b∼ 2), while these lines arestable when the separation is a few droplet diame-ters (B, H/b∼ 8). Aggregation of the droplets, lead-ing to crystallization, is observed at a plane wall (C,cell depth H/b∼ 400) and at the air–water interface(D, cell depth H/b∼ 400). At a plane wall, dropletsare expunged from the crystalline core, while thecrystal is stabilized by the recirculation of the fluidflow. At the air–water interface, on the other hand,there is no out-of-plane motion. Here τ = b/vs is thetime in which the active droplet moves a distanceequal to its radius.

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crystallite are constantly expelled from the center of the aggre-gate only to rejoin it at the edges. The recirculating flows ensurea balanced in- and out-flux of the droplets and thereby main-tain a constant mean droplet number within these aggregates(vortex-stabilized crystallites, Fig. 2C and Movie S5). When theplane wall is replaced by a plane interface, the previous inflowand outflow is suppressed, and the droplets form 2D crystallineaggregates. These aggregates are maintained in a steady state bya continuous coagulation and fragmentation of the crystallites(Fig. 2D and Movie S6).

A qualitative understanding of these states of self-organizationis obtained from the one-body external flow of the particle ineach of the four boundary conditions (corresponding panels ofFig. 2). We emphasize, once again, that in this calculation theactive slip is estimated from flow in the Hele–Shaw cell but usedto predict flow for the three remaining boundary conditions.Operationally, the latter only requires the use of the appropriateGreen’s function. In the Hele–Shaw cell, the net flow is parallelto the walls and has an inflowing component perpendicular to thedirection of motion. Entrainment in this inflow leads to the for-mation of metastable lines and stable bands. At both the planewall and the plane interface, the flow has a cylindrical symmetrywhen the propulsion axis is perpendicular to the plane. In the firstcase, the flow has a strong circulation in which entrained parti-cles are drawn inwards along the plane but then expelled normalto it. In the second case, the circulatory component is compara-tively 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 fragment-ing 2D crystallites. The qualitative agreement of the simulations(Fig. 2, Lower Middle) obtained from the numerical integrationof the force and torque balance equations with the experiment(Fig. 2, Bottom) is excellent. This agreement between experi-ment and theory that disregards the internal flow confirms, aposteriori, our hypothesis that entrainment in the external flowis primarily responsible for self-organization.

A quantitative understanding of the states of self-organizationrequires an accurate estimate of the forces and torques betweenparticles. We calculate the components of the active pair forceparallel and perpendicular to their separation vector, as a func-tion of separation distance, for each of the four boundary con-ditions considered (Fig. 3, Upper Middle and Lower Middle). Ineach of the cases, the component of the active force parallel tothe separation vector is negative, and it is this attractive com-ponent of the active force that leads to aggregation. However,there is a considerable variation in the component of the activeforce perpendicular to the separation vector, and it is the sen-sitivity of this component to boundary conditions that accountsfor the variety of the aggregated states. In the Hele–Shaw cell,the perpendicular force is positive, which in our sign conventionmeans that it is directed along the direction of motion. Since, toleading order, hydrodynamic forces are pair-wise additive, thisimplies that the net force on particles at the center of a mov-ing line are greater than those at the edges. Therefore, theytend to move faster, creating a curvature of the line and even-tually its break up. In contrast, the perpendicular force in theplane channel is an order of magnitude smaller (as shown inthe corresponding Inset of Fig. 3B for clarity), and the break-upmechanism has a negligible contribution, which gives the travel-ing bands their surprisingly stability. At a no-slip wall (Fig. 3C),the perpendicular force is negative at large distances but positiveat short distances. This drives particles into the wall when theyare well-separated but away from the wall when they are closeby. Combined with the parallel component of the flow, which isalways attractive, this leads to the expulsion and recirculation ofparticles in the crystalline aggregate. In contrast, the perpendic-ular force at an interface (Fig. 3D) is positive but an order ofmagnitude smaller (again, shown in the Inset for clarity), andthe dominant motion is due to the attractive component of theparallel force. Thus, expulsion is suppressed, and the result isthe formation of 2D crystallites. Similar estimates for the torqueprovide an understanding of the orientational dynamics, which

Fig. 3. Active forces on the active particles are modified by the presence of boundaries. (Top) Schematic of confinement. (Upper Middle) Attractive parallelforces 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 perpendicularforce 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 formore details). Insets show close-up and FA = 6πηbvs. (Bottom) State diagrams in terms of the strengths of the slip modes: V (2s)

0 , symmetric dipole; V (3t)0 ,

vector quadrupole; and V (4t)0 , degenerate octupole. Each dot represents a simulation, while the star denotes the values used in the above rows.

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is relatively unimportant in our case as Brownian reorientationis negligible and the hydrodynamic torques are one power ofseparation smaller than the corresponding forces.

Since each irreducible mode of the slip is independent ofthe 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 momentumindex l =0, 1, 2, . . . and a spin index σ= s, a, t corresponding tothe symmetric, antisymmetric, and pure trace irreducible com-ponents of each mode. Using the abbreviation lσ to indicatea mode with angular momentum index l and spin index σ, the3t mode produces self-propulsion (and a degenerate quadrupo-lar flow), while the 2s mode produces inflow and outflow alongmutually perpendicular axes (and a dipolar flow). The latter isthe stresslet mode, and its instantaneous sign determines if thefluid is being expelled (“pusher”) or ingested (“puller”) along thepropulsion axis. We are then able to construct a state diagramin the V

(2s)0 −V

(3t)0 plane that demarcates regions of stabil-

ity of the principal states of aggregation found for each of thefour boundary conditions studied (Fig. 3, Bottom). In Hele–Shawflow, the stresslet mode promotes stability, but in flows boundedby a plane wall, the perpendicular component of the force isenhanced in proportion to the magnitude of the stresslet, whichleads from 2D crystallites to vortex-stabilized crystallites andfinally to instabilities. Thus, by suitably choosing the strengthof the stresslet mode, it is possible to select either a state of2D crystals or vortex-stabilized crystals. For flow bounded by aninterface, an enhanced stresslet mode does not lead to explu-sion of the droplets out-of-plane but does lead directly to aninstability (unstable crystallites).

Our results above provide convincing evidence of the mech-anism of hydrodynamic entrainment in the spontaneous exte-rior flow as the dominant mechanism for self-organization inactive particles. The active parts of the hydrodynamic forcesand torques that result from this entrainment provide both aqualitative and quantitative explanation of the states of self-organization. These hydrodynamic forces and torques dependon the distance between the particles, their orientation, and themagnitudes of the modes of the slip on each active particle. Itcan be directly verified from the explicit forms of the forces andtorques that they cannot be obtained as gradients of potentials(20). Further, their dependence on the modes of the slip velocityindicates that they have odd parity under time reversal, signal-ing their explicitly dissipative nature (4, 26). Thus, we have asituation in which long-range, dissipative forces and torques pro-mote self-organization. States of self-organization maintainedby 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 tobe an example of a dissipative structure but one in which thedissipative mechanism and the resultant forces and torques areunambiguously identified.

Kinetics of FIPSThe self-organization presented above can be viewed, from thepoint of view of statistical physics, as a phase separation phe-nomenon driven by dissipative forces, rather than the usualconservative forces derived from a potential. The study of suchFIPSs presents several challenges, one of which is to determinea quantity that can serve as the analogue of a thermodynamicpotential. It appears from recent theoretical work that large-deviation results for the stationary distribution of Markov pro-cesses violating detailed balance may provide a tractable routefor answering questions of stability (30). Here we draw attentionto 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 phaseseparation at a plane wall and at a plane interface (Fig. 4 andMovie S7). At a plane wall, there is both an enhanced mixingwithin the crystal plane and an exchange of neighbors due to theclosed streamlines (Fig. 2C) and higher values of perpendicularforces (Fig. 3C) compared with the plane interface. In a biologi-cal context, such hydrodynamic bound structures and associatedkinetics could influence the encounter rate of individuals inaggregates (6, 31).

DiscussionThe FIPS mechanism established here and conjectured earlier(4) has several distinguishing features that bear pointing outand contrasting with other mechanisms. First, self-propulsionand/or self-rotation are not necessary for its operation; it isonly necessary that the particles produce a long-ranged exte-rior hydrodynamic flow. The experimentally observed states ofself-organization persist even when the self-propulsion parame-ter V (3t)

0 vanishes (state diagrams; Fig. 3, Bottom). Second, in theabsence of thermal fluctuations, the suspension is mechanicallyunstable to aggregation at any positive value of the density andany finite amount of activity, however small. It is plausible thatin the presence of thermal fluctuations, finite values of densityand activity are required to overwhelm the loss in entropy due toaggregation. A careful study of aggregation at different tempera-tures, densities, and activities is needed to establish this quan-titatively. Third, FIPS includes, as a special case, aggregationin driven colloidal systems, which are nonetheless force- and

Fig. 4. Kinetics of aggregation of active particlesat a plane wall (Top) and at a plane interface (Bot-tom). The colors are used to indicate and track theparticles based on their initial positions. The instan-taneous snapshots show that there is a faster mixingof particles and exchange of neighbors at the planewall due to the circulatory flow streamlines (Fig. 2C)and higher values of perpendicular forces (Fig. 3C).

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torque-free (32, 33). Fourth, the long-lived, stable travelingbands that we have shown here are qualitatively different com-pared with other dynamic behavior seen in active systems (20,34, 35) or in the emergence of the orientational order in flock-ing 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 activesystems that is complementary to motility-induced phase separa-tion (MIPS) (37). The latter has a kinematic character, in whichthe flux of particles is a prescribed functional of density, reflect-ing the tendency of active particles to slow down or speed upin regions of, respectively, higher or lower density. The physicsunderlying this tendency is left implicit and may plausibly beattributed to nonhydrodynamic interactions, like contact, com-pression, or jamming. In contrast, FIPS has an explicitly dynamiccharacter, as forces and torques of hydrodynamic origin are iden-tified in causing the aggregation. It is entirely conceivable thata state of aggregation like hexagonal crystallites can be formedfrom 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 wallwith no slip or an interface with vanishing tangential stress,aggregate structures that remain stable away from such bound-aries would point to a mechanism such as MIPS as the sourceof stability. Finally, though our experiments are performed withdroplets, the theory and simulation correspond to arbitrary activeparticles, and our flow boundary conditions are generic to manynatural and engineered settings. Together, these underscore thepotential significance of our findings to a wide class of active flu-ids, in particular active colloids, and to the study and control ofgeometric and topological phenomena in active matter.

Materials and MethodsExperiments. Droplets were produced using microfluidic devices that werefabricated from polydimethylsiloxane (Sylgard 184; Dow Corning) usingstandard soft lithographic protocols and bonded to glass slides. The carrier

fluid was comprised of an aqueous solution of 0.25 wt% of the surfac-tant SDS (Sigma-Aldrich). The droplet phase was the liquid crystalline oil4-pentyl-4′-cyano-biphenyl (5CB, Frinton Laboratories Inc.). The flow rateswere volume-controlled using syringe pumps (Harvard Apparatus). A fewmillion droplets were produced and stored in the aqueous mixture of sur-factant. For the experiments, a suitable amount of the droplets were thenresuspended in an aqueous mixture of 10 to 25% by weight of SDS, which setthe droplets spontaneously into motion. The Hele–Shaw cell for the exper-iments was constructed using microscope slides separated by double-sidedsticky tape of precise thickness (the thickness of the Hele–Shaw gap variedfrom 50 µm to 2 cm), which was cut into a chamber of a desired shape(5 cm× 4 cm) using a standard plotter cutter. The microscope slides wererendered hydrophilic by plasma treatment just before assembly.

Image Processing and Analysis. The experiments were recorded using ansCMOS camera (Hamamatsu) at frame rates of 1− 4 fps in a region of∼ 2.2 mm× 2.2 mm close to the center of the Hele–Shaw chamber. Forthe particle image velocimetry (PIV) measurements, the aqueous phase wasseeded with 200 nm red fluorescent polystyrene beads (Thermo Scientific),and movies were recorded at 100 fps. Droplet tracking and all other analysessubsequently were performed using custom-written Matlab code. The flowfield analysis was performed using PIVLab, an open source Matlab code.

Simulations. The simulations were performed by numerical integration ofthe force and torque balance equations (for each boundary condition) usingPyStokes, a Cython library for computing hydrodynamic interactions, withan adaptive time-step integrator. A random packing of hard spheres is usedas the initial distribution of particles in all of the simulations. The particle–particle and the particle–wall repulsive interaction is modeled using theshort-ranged repulsive Weeks–Chandler–Andersen potential, which is given

as U(r) = ε(

rminr

)12− 2ε

(rmin

r

)6+ ε, for r< rmin and zero otherwise, where

ε is the potential strength.

ACKNOWLEDGMENTS. We thank T. Tlusty, M. E. Cates, and R. E. Goldsteinfor discussions. R.S. and R.A. acknowledge the Institute of MathematicalSciences for computing resources on the Nandadevi clusters. S.T. acknowl-edges the Human Frontier Science Program (Cross Disciplinary Fellowship)for funding. R.A. thanks the Indo-US Science and Technology Forum forsupporting a sabbatical visit to Princeton University. R.S. is funded by RoyalSociety-SERB Newton International Fellowship.

1. Brennen C, Winet H (1977) Fluid mechanics of propulsion by cilia and flagella. AnnuRev Fluid Mech 9:339–398.

2. Ebbens SJ, Howse JR (2010) In pursuit of propulsion at the nanoscale. Soft Matter6:726–738.

3. Thutupalli S, Seemann R, Herminghaus S (2011) Swarming behavior of simple modelsquirmers. New J Phys 13:073021.

4. Singh R, Adhikari R (2016) Universal hydrodynamic mechanisms for crystallization inactive colloidal suspensions. Phys Rev Lett 117:228002.

5. Palacci J, Sacanna S, Steinberg AP, Pine DJ, Chaikin PM (2013) Living crystals of light-activated colloidal surfers. Science 339:936–940.

6. Petroff AP, Wu X-L, Libchaber A (2015) Fast-moving bacteria self-organize into activetwo-dimensional crystals of rotating cells. Phys Rev Lett 114:158102.

7. Thutupalli S, Herminghaus S (2013) Tuning active emulsion dynamics via surfactantsand topology. Eur Phys J E Soft Matter 36:91.

8. Herminghaus S, et al. (2014) Interfacial mechanisms in active emulsions. Soft Matter10:7008–7022.

9. Izri Z, Linden MNvd, Michelin S, Dauchot O (2014) Self-propulsion of pure waterdroplets by spontaneous marangoni-stress-driven motion. Phys Rev Lett 113:248302.

10. Shani I, Beatus T, Bar-Ziv RH, Tlusty T (2014) Long-range orientational order in two-dimensional microfluidic dipoles. Nat Phys 10:140–144.

11. Cavagna A, Giardina I (2014) Bird flocks as condensed matter. Annu Rev CondensMatter Phys 5:183–207.

12. Sokolov A, Aranson I, Kessler J, Goldstein R (2007) Concentration dependence of thecollective dynamics of swimming bacteria. Phys Rev Lett 98:158102.

13. Woodhouse FG, Goldstein RE (2012) Spontaneous circulation of confined activesuspensions. Phys Rev Lett 109:168105.

14. Wioland H, Woodhouse FG, Dunkel J, Kessler JO, Goldstein RE (2013) Confinementstabilizes a bacterial suspension into a spiral vortex. Phys Rev Lett 110:268102.

15. Bricard A, et al. (2015) Emergent vortices in populations of colloidal rollers. NatCommun 6:7470.

16. Solon AP, et al. (2015) Pressure is not a state function for generic active fluids. NatPhys 11:673–678.

17. Fily Y, Baskaran A, Hagan M (2014) Dynamics of self-propelled particles under strongconfinement. Soft Matter 10:5609–5617.

18. Bechinger C, et al. (2016) Active particles in complex and crowded environments. RevMod Phys 88:045006.

19. Takatori SC, Yan W, Brady JF (2014) Swim pressure: Stress generation in active matter.Phys Rev Lett 113:028103.

20. Niu R, Palberg T, Speck T (2017) Self-assembly of colloidal molecules due toself-generated flow. Phys Rev Lett 119:028001.

21. Souslov A, van Zuiden, BC, Bartolo D, Vitelli V (2017) Topological sound in active-liquid metamaterials. Nat Phys 13:1091–1094.

22. Woodhouse FG, Dunkel J (2017) Active matter logic for autonomous microfluidics.Nat Commun 8:15169.

23. Prishchepa OO, Shabanov AV, Zyryanov VY (2005) Director configurations in nematicdroplets with inhomogeneous boundary conditions. Phys Rev E 72:031712.

24. Ghose S, Adhikari R (2014) Irreducible representations of oscillatory and swirlingflows in active soft matter. Phys Rev Lett 112:118102.

25. Singh R, Ghose S, Adhikari R (2015) Many-body microhydrodynamics of particleparticles with active boundary layers. J Stat Mech 2015:P06017.

26. Singh R, Adhikari R (2018) Generalized Stokes laws for active particles and theirapplications. J Phys Commun 2:025025.

27. Lighthill JM (1952) On the squirming motion of nearly spherical deformable bod-ies through liquids at very small Reynold number. Comm Pure Appl Maths 5:108–118.

28. Blake JR (1971) A spherical envelope approach to ciliary propulsion. J Fluid Mech46:199–208.

29. Kondepudi D, Prigogine I (1998) Modern Thermodynamics: From Heat Engines toDissipative Structures (Wiley, New York).

30. Touchette H (2009) The large deviation approach to statistical mechanics. Phys Rep478:1–69.

31. Drescher K, et al. (2009) Dancing volvox: Hydrodynamic bound states of swimmingalgae. Phys Rev Lett 102, 168101.

32. Negi AS, Sengupta K, Sood AK (2005) Frequency-dependent shape changes ofcolloidal clusters under transverse electric field. Langmuir 21:11623–11627.

33. Sapozhnikov MV, et al. (2003) Dynamic self-assembly and patterns in electrostaticallydriven granular media. Phys Rev Lett 90, 114301.

34. Schaller V, Weber C, Semmrich C, Frey E, Bausch AR (2010) Polar patterns of drivenfilaments. Nature 467:73–77.

35. Ohta T, Yamanaka S (2014) Traveling bands in self-propelled soft particles. Eur Phys JSpec Top 223:1279–1291.

36. Solon AP, Chate H, Tailleur J (2015) From phase to microphase separation in flock-ing models: The essential role of nonequilibrium fluctuations. Phys Rev Lett 114:068101.

37. Cates ME, Tailleur J (2015) Motility-induced phase separation. Annu Rev CondensMatter Phys 6:219–244.

5408 | www.pnas.org/cgi/doi/10.1073/pnas.1718807115 Thutupalli et al.

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