The vortex-driven dynamics of droplets within droplets

A. Tiribocchi∗,1, 2 A. Montessori,3 M. Lauricella,3 F. Bonaccorso,1, 3 S. Succi,1, 3, 4S. Aime,4, 5 M. Milani,6 and D. A. Weitz4, 7

1Center for Life Nano Science@La Sapienza, Istituto Italiano di Tecnologia, 00161 Roma, Italy2Istituto per le Applicazioni del Calcolo CNR, via dei Taurini 19, 00185 Rome, Italy

adriano.tiribocchi@iit.it3Istituto per le Applicazioni del Calcolo CNR, via dei Taurini 19, 00185 Rome, Italy

4Institute for Applied Computational Science, John A. Paulson School of Engineering and Applied Sciences,Harvard University, Cambridge, Massachusetts 02138, USA

5Matière Molle et Chimie, Ecole Supérieure de Physique et Chimie Industrielles, 75005 Paris, France6Università degli Studi di Milano, via Celoria 16, 20133 Milano, Italy

7Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA

Understanding the fluid-structure interaction is crucial for an optimal design and manufacturingof soft mesoscale materials. Multi-core emulsions are a class of soft fluids assembled from clusterconfigurations of deformable oil-water double droplets (cores), often employed as building-blocks forthe realisation of devices of interest in bio-technology, such as drug-delivery, tissue engineering andregenerative medicine. Here, we study the physics of multi-core emulsions flowing in microfluidicchannels and report numerical evidence of a surprisingly rich variety of driven non-equilibrium states(NES), whose formation is caused by a dipolar fluid vortex triggered by the sheared structure of theflow carrier within the microchannel. The observed dynamic regimes range from long-lived NES atlow core-area fraction, characterised by a planetary-like motion of the internal drops, to short-livedones at high core-area fraction, in which a pre-chaotic motion results from multi-body collisions ofinner drops, as combined with self-consistent hydrodynamic interactions. The onset of pre-chaoticbehavior is marked by transitions of the cores from one vortex to another, a process that we interpretas manifestations of the system to maximize its entropy by filling voids, as they arise dynamicallywithin the capsule.

INTRODUCTION

Recent advances in microfluidics have highlighted thepossibility to design highly ordered, multi-core emul-sions in an unprecedentedly controlled manner [1–12].These emulsions are hierarchical soft fluids, consistingof small drops (often termed “cores”) immersed withinlarger ones, and stabilized over extended periods of timeby a surfactant confined within their interface.

A typical example is a collection of immiscible wa-ter droplets, embedded within a surrounding oil phase[13]. This is usually manufactured in a two-step process,by first emulsifying different acqueous solutions in theoil phase and then encapsulating water drops within thesame oil phase in a second emulsification step [1, 9, 10].

Due to their peculiar tiered architecture, they haveserved as templates to manufacture microcapsules witha core-shell geometry that have found applications in anumber of sectors of modern industry, such as in foodscience for the realisation of low calory dietary prod-ucts and encapsulation of flavours [14–17], in pharmaceu-tics for the delivery and controlled release of substances[18–22], in cosmetics for the production of personal careitems [6, 23–25] and in tissue engineering, as building-blocks for the design of tissue-like soft porous materials[26–28]. More recently, they have also been used as atool to mimic cell-cell interactions within a dynamic en-vironment provided by flowing capsules in microcapillarychannels [29, 30].

Understanding their behavior in the presence of fluidflows, even under controlled experimental conditions, re-mains a crucial requirement for a purposeful design ofsuch functionalized materials. The rate of release of thedrug carried by the cores, for example, is significantly in-fluenced by surfactant concentration and hydrodynamicinteractions [31]. The latter ones, even when moderate,can foster drop collisions as well as shape deformationsthat may ultimately compromise the release and the de-livery towards targeted diseased tissues. Controlling me-chanical properties as well as long-range hydrodynamicinteractions of the liquid film formed among cores is ofparamount importance for ensuring the prolonged stabil-ity of food-grade multiple emulsions [32]. This is essentialin high internal phase emulsions extensively used in tis-sue engineering, where such forces can considerably alterpore size and rate of polydispersity [28, 33], thus jeopar-dizing the structural homogeneity of the material.

Building ad hoc mathematical and computationalmodels is therefore fundamental to make progress alongthis direction. Indeed, in stark contrast with the impres-sive advances in the experimental realisation of multi-core emulsions [9, 10, 34, 35], it is only recently thatefforts have been directed to the theoretical investiga-tion of the rheology resulting from the highly non-linearfluid-structure interactions taking place in such systems[36–38].

Continuum theories, combined with suitable numericalapproaches (such as lattice Boltzmann methods [39–41]

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and boundary integral method [38]) have proven capa-ble to capture characteristic features observed in doubleemulsion experiments, such as their production withinmicrochannels [40], the typical shape deformations ofthe capsule (elliptical and bullet-like) under moderateshear flows [42–44], as well as more complex dynamicbehaviors, such as the breakup of the enveloping shelloccurring under intense fluid flows [45]. However, muchless is known about the dynamics of more sophisticatedsystems, such as multiple emulsions with distinct innercores, theoretically investigated only by a few authors todate [38, 39].

A befitting theoretical framework for describing theirphysics can be built on well-established continuum prin-ciples, whose details are illustrated in the Method Sec-tion. It is essentially based on a phase field approach[39, 46–48], in which a set of passive scalar fields φi(r, t)(i = 1, ..., Nd, where Nd = N + 1 is the total number ofdroplets and N is the number of cores) accounts for thedensity of each droplet, while a vector field v(r, t) repre-sents the global fluid velocity. The dynamics of each fieldφi is governed by a set of Cahn-Hilliard equations, whilethe velocity obeys the Navier-Stokes equations [49, 50].The thermodynamics of the mixture is encoded in a Lan-dau free-energy functional augmented with a term pre-venting the coalescence of the cores, thus capturing therepulsive effect produced by a surfactant adsorbed ontoa droplet interface.

In this paper we employ the aforementioned method tonumerically study the pressure-driven flow of multi-coredroplets confined in a microchannel, following a designdirectly inspired to actual microfluidic devices. Exten-sive lattice Boltzmann simulations provide evidence of arich variety of driven nonequilibrium states (NES), fromlong-lived ones at low droplet area fraction (and low num-ber of inner drops), characterised by a highly correlated,planetary-like motion of the cores, to short-lived ones athigh droplet area fraction (with moderate/high numberof cores), in which multiple collisions and intense fluidflows trigger a chaotic-like dynamics. Central to each ofthese non-equilibrium states is the onset of a vorticitydipole within the capsule, which arises as an inevitableconsequence of the sheared structure of the velocity fieldwithin the microchannel. Such dipole structure naturallyinvites a classification of these states in close analogy withthe statistical mechanics of occupation numbers. Eventhough our system is completely classical, such repre-sentation discloses a transparent and insightful interpre-tation of the transition from periodic to quasi-chaoticsteady-states, in terms of level crossings between the oc-cupation numbers in the two vortex structures. Suchlevel crossings are interpreted as manifestations of thesystem to maximise its entropy by filling voids whicharise dynamically within the multibody structure result-ing from the self-consistent motion of the cores within thecapsule. This is consistent with the notion of entropy as

propensity to motion rather than microscopic disorder[51].

RESULTS

Fluid-structure interaction in a core-free emul-sion. We start by describing droplet shape and patternof the fluid velocity at the steady state in a core-freeemulsion under a Poiseuille flow. Once this is imposed,the droplet, initially at equilibrium (Fig.1a), acquires mo-tion, driven by a constant pressure gradient ∆p appliedacross the longitudinal direction of the microfluidic chan-nel. The resulting fluid flow as well as the shape of theemulsion are controlled by the capillary and the Reynoldsnumbers, defined as Ca = vmaxησ and Re =

ρvmaxDOη . Here

vmax is the maximum value of the droplet speed, η is theshear viscosity of the fluid, σ is the surface tension, ρis the fluid density and DO is the diameter of the shell(taken as characteristic length of the multi-core emul-sion). In our simulations Ca roughly varies between 0.02and 1 and Re may range from 0.02 and 5, hence iner-tial effects are mild and the laminar regime is generallypreserved.

In Fig.1b-c we show an example of shape of a core-freeemulsion at the steady state (see Supplementary Movie1 for the full dynamics). In agreement with previousstudies [52–56], the droplet attaines a bullet-like form,more stretched along the flow direction for higher valuesof Re and Ca (i.e. larger pressure gradients). Such shaperesults from the parabolic structure of the flow profile(see Supplementary Figures 1 and 2 for further detailsabout the steady state velocity profile), moving fasterin the center of the channel and progressively slower to-wards the wall (Fig.1b). If computed with respect tothe droplet frame, the flow field exhibits two symmet-ric counterrotating eddies, resulting from the confininginterface of the capsule and whose direction of rotationis consistent with a droplet moving forward (rightwardshere, see Fig.1c). These structures are due to the gradi-ent of vy along the z direction, a quantity positive withinthe lower half of the emulsion and negative in the upper.As long as the droplet remains core-free, such fluid recir-culations (also observed, for instance, in micro-emulsionspropelled either through Marangoni effect [57–59] or bymeans of an active material, such as actomyosin proteins[60–62], dispersed within) are stable, and their patternremains basically unaltered.

One may wonder whether this picture still holds whensmall cores are encapsulated. This essentially means un-derstanding i) to what extent the dipolar fluid flow struc-ture is stable when the effective area fraction of the inter-nal droplets Ac = NπR

2i

πR2O(where Ri and RO are the radii

of the cores and of the shell) increases, and ii) how thecoupling between the fluid velocity v and a number ofpassive scalar fields φi affects the dynamics of the multi-

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Figure 1. Steady state shapes and velocity field structure of a core-free emulsion under Poiseuille flow. (a)Equilibrium configuration of a core-free emulsion. (b)-(c) Steady state shapes at Re ' 3 and Ca ' 0.8. Red arrows indicatethe direction of the fluid flow computed in the lab frame (b) and with respect to the center of mass of the droplet (c). Here twoeddies rotating clockwise (bottom) and counterclockwise (top) emerge within the droplet. The droplet radius at equilibriumis R = 30 and the color map represents the value of the order parameter φ, ranging between 0 (black) and 2 (yellow). Thisapplies to all figures in the paper.

core emulsion.In the next sections we show that the double eddy

fluid structure is substantially preserved, although mod-ifications to this pattern occur when Ac is larger thanroughly 0.35. More specifically, while in a double emul-sion (N = 1) the stream in the middle of the dropletdrives the core at the front-end of the shell where thecore gets stuck, when N > 1 the vortices trigger and sus-tain a persistent periodic motion of the cores within eachhalf of the emulsion. As long as Ac < 0.35 (achieved withN = 3), cores remain confined within either the upperor the lower part of the emulsion giving rise to long-livednonequilibrium steady states. When Ac > 0.35 (N ≥ 4)droplet crossings between the two regions may occur, andshort-lived aggregates of three or more cores only tem-porarily survive. Hence, the whole emulsion can be effec-tively visualized as a two-state system in which the twoeddies foster the motion of the cores and crucially affectthe duration of the states.

By using the tools of statistical mechanics, we proposea classification of such states in terms of the occupationnumber formalism, where 〈α1, ..., αj |αj+1, ..., αN 〉 repre-sents a state in which j and N − j distinct cores occupy,respectively, the upper and the lower half of the emul-sion, with j = 1, .., N . This is analogous to determinethe number of ways N distinguishable particles can beplaced in two boxes.

Classification of states. In Fig.2(a)-(f), we show theequilibrium configurations of a single core ((a), N = 1), atwo-core ((b), N = 2), a three-core ((c), N = 3), a four-core ((d), N = 4), a five-core ((e), N = 5) and a six-core((f), N = 6) emulsion. Once a Poiseuille flow is applied,the emulsions attain a steady state in which, unlike thecore-free droplet of Fig.1, the dynamics of the cores andthe velocity field are crucially influenced by the effective

area fraction Ac and the number of cores N . In Fig.3 weshow a selection of the nonequilibrium states observed.

Long-lived non equilibrium states. The simplestconfiguration is the one in which N = 1. In this case thecore and the shell are initially advected forward (right-wards in the figure) and, at the steady state, the for-mer gets stuck at the front-end of the latter (see, forexample, Supplementary Movie 2). In Fig.3, two exam-ples for slightly different values of Re and Ca are shown.Note incidentally that, in agreement with previous stud-ies [39, 40, 42, 44, 52, 53, 63], as long as Re ' 1, theinternal core, unlike the interface of the external shell, isonly mildly affected by the fluid flow. This is due to itshigher surface tension induced by the smaller curvatureradius, which prevents relevant shape deformations.

This picture is dramatically altered when the numberof cores increases. If N = 2 (Ac ∼ 0.18), for example,two nonequilibrium long-lived states emerge. In the firstone (Fig.3c and Supplementary Movie 3), the two coresremain locked in the upper (or lower) part of the emul-sion, where the fluid eddy fosters a periodic motion inwhich each core chases the other one in a coupled-dancefashion (see the next section for a detailed description ofthis dynamics). In the second one (Fig.3d and Supple-mentary Movie 4), the two cores remain separately con-fined within the top and the bottom of the emulsion, andmove, weakly, along circular trajectories. By using thestatistics of occupation number, we indicate with 〈1, 2|0〉the state in which cores 1 and 2 are in the upper regionwhile the lower one is empty, and with 〈2|1〉 the statewhere cores 2 and 1 occupy, separately, each half.

More complex effects emerge when Ac is further in-creased. If N = 3 (Ac ∼ 0.27), once again we find twodifferent long-lived nonequilibrium states, namely 〈3|1, 2〉and 〈1, 2, 3|0〉. In the first one (Fig.3e and Supplementary

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Figure 2. Equilibrium states of multi-core emulsions. Equilibrium configurations of a single-core (a), a two-core (b), athree-core (c), a four-core (d), a five-core (e) and a six-core (f) emulsion. Droplet radii are as follows: Ri = 15, RO = 30 (a),Ri = 17, RO = 56 (b)-(f).

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Figure 3. Nonequilibrium states of multi-core emulsions under Poiseuille flow. The dotted black arrow indicates theflow direction (which applies to all cases) while N represents the number of cores encapsulated. (a) is obtained for Re ' 3and Ca ' 0.85, while (b) for Re ' 1.2 and Ca ' 0.35. All the other cases correspond to Re ' 3 and Ca ' 0.85. As long asAc < 0.35, the nonequilibrium states are long-lived (l.l.), while if Ac > 0.35 they turn into short-lived (s.l.). Here, crossings ofcores from one region towards the other one start to occur. The magenta arrow indicates the direction of a crossing. Multiplecrossings are observed as Ac increases. In each snapshot, states are indicated by means of the occupation number classification.

Movie 5), the core 3 is confined at the top of the emulsionand moves following a circular path, while cores 1 and 2remain at the bottom of the emulsion and reproduce thecoupled-dance dynamics observed for N = 2. In the sec-ond one (Fig.3f and Supplementary Movie 6), the threecores, sequestered in the upper region, exhibit a morecomplex three-body periodic motion, whose dynamics isdiscussed later. However, although long-lasting, suchstate may turn unstable due to hydrodynamic interac-tions and to direct collisions with other cores. This isprecisely what happens when N and Ac are further aug-mented.

Short-lived nonequilibrium states. Indeed, whenN = 4 (Ac ∼ 0.37), we find a state in which threecores move in one region and a single core within theother one. This is indicated as 〈1, 2, 3|4〉 (Fig.3g). How-ever, this state lives for a short period of time sincedroplet 3 crosses from the top towards the bottom ofthe emulsion and produces the long-lived nonequilib-rium state 〈1, 2|3, 4〉, in which couples of cores cease-lessly dance within two separate regions (Fig.3e andSupplementary movie 7). Such transition, indicated as〈1, 2, 3|4〉 → 〈1, 2|3, 4〉, occurs due to the high values of

Ac, generally larger than 0.35. This means that, withinhalf of the emulsion, the effective area fraction is evenhigher (more than 0.5), and the three-core state would beunstable to changes of the flow direction and to unavoid-able collisions with the other cores. This is the reasonwhy, for example, the state 〈1, 2, 3, 4|0〉, although realiz-able in principle, has not been observed.

For higher values of N , Ac further increases and mul-tiple crossings occur. If N = 5 (Ac ∼ 0.46) for instance,short-lived dynamical states appear, such as 〈3, 5|1, 2, 4〉or 〈2|1, 3, 4, 5〉, in which four cores are temporarily packedwithin half of the emulsion (Fig.3i-j and SupplementaryMovie 8). However, these states are highly unstablesince cores cross the two regions of the emulsion mul-tiple times. Finally, more complex short-lived states areobserved when N = 6 (Ac ∼ 0.55), such as 〈3, 5, 6|1, 2, 4〉and 〈5, 6|1, 2, 3, 4〉 (Fig.3k-l and Supplementary Movie 9).

In the next section we discuss more specifically the dy-namics of these states focussing, in particular, on how thefluid-structure interaction affects the droplets motion.

Two-core emulsion. We begin from the two-coreemulsion, in which two droplets are initially encapsu-lated as in Fig.2b and attain the state 〈1, 2|0〉. Like the

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single-core case, once the flow is applied both cores andexternal droplet are dragged forward by the fluid. How-ever, while the latter rapidly attains its steady state (seeFig.4a in which an instantaneous configuration is shownfor Re ' 3, Ca ' 0.52), the internal core placed at therear side (core 1) approaches the one at its front (core 2),since it moves slightly faster due to the larger flow fieldin the middle line of the emulsion (see Fig.4, and positionand speed of their center of mass for t < 105 in Fig.5 andFig.6). This results in a dynamic chain-like aggregateof droplets, moving together towards the leading edge ofthe external interface. Note that droplet merging is in-hibited, since the repulsion among cores (mimicking theeffect of a surfactant) is included (see section Methods).

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Figure 4. Onset of periodic motion in a two-core emul-sion. In (a)-(c) Re ' 3 and Ca ' 0.52, while in (b)-(d)Re ' 1.2 and Ca ' 0.2. Red arrows indicate the directionof the fluid flow in the lab frame (a)-(b), and in the externaldroplet frame (c)-(d). Green arrows in (a) and (b) denote thedirection, perpendicular to the flow, along which the droplet1 starts its motion, while white arrows in (c) and (d) bespeakthe direction of the fluid recirculations.

However such aggregate is unstable to weak perturba-tions of the flow field, an effect due to the non-trivialcoupling between the velocity of the fluid and the inter-faces of the cores, and essentially governed by a term ofthe form ∇ · (φiv) (see Equation 2 in section Methods).In Fig.4 we show the typical flow field in the lab frame(a-b) and in the external droplet frame (c-d) at the on-set of this instability for two different values of Re andCa. Where the former looks approximately laminar, thelatter, once again, exhibits two separate counter-rotatingpatterns, in the upper and in the lower part of the emul-sion. Unlike the core-free emulsion, here the flow field

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Figure 5. Time evolution of the displacement ∆zcm and∆ycm of the cores. On the left axis ∆zcm(i) = zcm(i) −zcm(O) (red/plusses and blue/crosses) and on the right axis∆ycm(i) = ycm(i)− ycm(O) (pink/asterisk and cyan/squares)of the two cores. They are calculated with respect to thecenter of mass of the external droplet. The inset shows thetypical trajectories of the cores during a cycle. Red and bluearrows indicate the direction of rotation of droplet 1 and 2,respectively. Snapshots are taken at t = 5.54 × 105 (a), t =6.3 × 105 (b), t = 7.5 × 105 (c) and t = 8.16 × 105 (d).Multiplication by a factor 105 is understood for the simulationtime t on the horizontal axis. This applies to all plots in thepaper.

in the middle departs from the typical homogeneous andunidirectional structure, especially within and in the sur-roundings of the cores where it fosters the motion of thedrops either upwards (a-c) or downwards (b-d).

Once this occurs, the fluid vortices capture the coresand sustain a persistent and periodic circular motion ofboth of them around a common center of mass, confinedwithin the lower or the upper region of the emulsion.Such dynamics is described in Fig.5, where we show thetime evolution of the displacement ∆zcm(i) = zcm(i) −zcm(O) and ∆ycm(i) = ycm(i)− ycm(O) of the cores withrespect to the center of mass of the external droplet, atRe ' 3 and Ca ' 0.52.

After being driven forward in the middle of the emul-sion, the cores acquire motion upwards and backwardsnear the external interface, with the droplet at the rear(1) closely preceding the front one (2). Subsequently, thecircular flow pushes both cores back towards the centerof the emulsion, but while the former (1) moves towardsthe leading edge, the latter (2) follows a shorter circularcounterclockwise trajectory which allows to overtake theother core. The process self-repeats periodically, alter-nating, at each cycle, the core leading the motion.

In Fig.6 we show the y and z components of the cen-ter of mass velocity (in the external droplet frame) ofboth cores. They have a cyclic behavior with the sameperiodicity, although ∆vzcm = vzcm(i) − vzcm(O) shows

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Figure 6. Speed of the cores. Time evolution of the y andz components of the velocity of center of mass of the two corescomputed with respect to that of the external droplet. Theinset highlights the position of the inner drops at the pointsof inversion of motion. Red and blue arrows indicate theirdirection of motion. Snaphots are taken at t = 3.6 × 105 (a)and t = 4.94× 105 (b) timesteps.

an asymmetric pattern. This occurs because, when mov-ing upwards (during the first half of the cycle), the corehas a lower velocity (approximately 5 × 10−4 in simu-lation units) than when moving back towards the bulk(roughly −2 × 10−3). This is not the case of ∆vycm =vzcm(i)−vzcm(O), which, on the contrary, exhibits a sym-metric structure, with maxima and minima attained atthe center of the emulsion and at the top, respectively.

Finally note that the initial location of the cores im-portantly affects the dynamic response of the emulsion.Indeed, if cores 1 and 2 are originally aligned vertically,rather than horizontally, the state 〈2|1〉 is obtained. Oncethe flow is applied, they acquire a circular motion trig-gered by the eddies but, since they are placed within twoseparate sectors (top and bottom of the emulsion) fromthe beginning, they interact only occasionally along themiddle region of the emulsion.

Three-core emulsion. We first discuss the dynamicsof the state 〈3|1, 2〉. In Fig.7 we show the time evolu-tion of the displacement of three cores, initially placedas in Fig.2, under a Poiseuille flow when Re ' 3 andCa ' 0.52. Here cores 1 and 2 are captured by the eddyformed downward and, like the state 〈1, 2|0〉, they rotateperiodically around approximately circular orbits. Onthe other hand, core 3 is locked upward, where it rotatesalong a shorter rounded trajectory at a higher frequency,roughly twice larger than that of cores 1 and 2, but ba-sically at the same speed (O(10−3) in simulation units).This is shown in Fig. 8 a-b-c, where the time evolution ofthe speed of each core (computed with respect to that ofthe external droplet) is reported. Although, at the steadystate, two fluid eddies can be clearly distinguished (seeFig.8d), the pattern of the lower one deviates from thetypical rounded shape due to the presence of the cores.In the next section we will show that such distortions willconsolidate when Ac (and N) increases, and will essen-tially favour crossings of cores between the two regionsof the emulsion.

A dynamics in which two cores rotate in the upper re-gion and the other core in the lower one (such as the state〈1, 2|3〉) can be observed, for instance, for smaller valuesof Re and Ca, and shares essentially the same featureswith the previous case.

Finally if the three cores are initially confined withinthe upper half of the emulsion, al late times the state〈1, 2, 3|0〉 is attained (see Fig.3f and SupplementaryMovie 6.) Once again, the internal cores exhibit aperiodic motion along roughly circular trajectories, al-though the reciprocal interactions complicate the dynam-ics. Shortly, the core at the rear, 1, is initially pushedforward by the flow in the middle of the emulsion, whilecores 2 and 3 get locked upwards and rotate synchronized(like the state 〈1, 2|0〉). Afterwards, core 1 progressivelyshifts towards the upper region, connects with the othertwo to form a three-core chain moving coherently. Suchaggregate is only partially broken when a core approachesthe middle of the emulsion (where an intense flow cur-rent pushes it forward), but is rapidly reshaped when thecore, once again, migrates back upwards.

Such dynamic behavior suggests, once more, that theinitial position of the cores plays a crucial role in drivingthe dynamics of the emulsion towards a targeted steadystate. Indeed, the formation of the state 〈1, 2, 3|0〉, start-ing, for example, from Fig.2c, would require crossings ofcores from the bottom towards the top of the emulsion,a very unlikely event if Ac remains low.

However, this state may turn to unstable and decayinto a long-lived one when Ac augments. This is whathappens if four cores are included.

Four cores: Droplet crossings and short-livedclusters. In Fig.9a we show the time evolution of thedisplacement ∆zcm of four cores originally encapsulatedas in Fig.2d. While droplets 1 and 2, initially carried

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Figure 7. Time evolution of ∆zcm and ∆ycm in a three-core emulsion. Cores 1 and 2 show a periodic motion arounda common center of mass in the bottom region of the emulsionwhile core 3 travels persistently along a circular path confinedin the upper part. The inset shows four instantaneous configu-rations of φi during a periodic cycle. White arrows denote thetrajectories of the cores. Snapshots are taken at t = 6.22×105(a), t = 6.96× 105 (b), t = 7.8× 105 (c), t = 8.42× 105 (d).

rightwards by the fluid, are then driven towards the up-per part of the emulsion where soon acquire the usualdance-like periodic motion, droplet 3 travels towards thebottom driven by an intense, mainly longitudinal, flow(Fig.9b-c), which significantly alters the typical doubleeddy pattern of the velocity fluid (see SupplementaryFigure 3 for a detailed structure of the velocity field).

After the transition a clear fluid recirculation is defini-tively restored and favours, once again, the onset of apersistent coupled motion occurring in a similar manneras the others: each droplet chases the other one and, re-cursively, the droplet at the front (say 4) is pushed back-wards (i.e. towards the leading edge of the emulsion)along a circular trajectory larger than the one coveredby the droplet at the rear (say 3), which, now, leads tomotion (see Supplementary Movie 7).

At the steady-state, the four cores exhibit a long-lived

coupled dance in pairs of two, occurring without furthercrossings and within two separate regions of the emulsion.These results suggest that, although for a short period oftime, the periodicity of the motion can be temporarilylost when Ac is sufficiently high.

This steady-state dynamics looks rather robust, since,unlike the three-core emulsion, occurs even if the coresare initially confined, for example, within the lower halfof the emulsion (see Supplementary Movie 8, where coresare numbered the same as in Fig.2d). Despite the asym-metric starting configuation, the core on top of the others(i.e. 1) and the one on the back (i.e. 3) are quickly driventowards the leading edge of the emulsion by the flow inthe middle and are then captured by the fluid vortex inun upper region. Thus here two cores, rather than one,actually exhibit a crossing, although this occurs soon af-ter the Poiseuille flow sweeps over the droplet. A steadystate of the form 〈1, 3|2, 4〉 emerges and lasts for longperiods of time.

In the next section we show that when Ac attains avalue equal to (or higher than) 0.4, the dynamics of thecores lacks of any periodic regularity and the motionturns to chaotic.

Five and six cores: Onset of non-periodic dy-namics. In Fig. 10 we show the time evolution of thedisplacement ∆zcm in a five (a-e) and six-core (f-k) emul-sion, where Ac ' 0.46 and Ac ' 0.55 respectively (seealso Supplementary Movies 9 and 10).

In such systems, only short-lived states are observed(such as those shown in Fig.3(i)-(j)-(k)-(l)), since multi-ple crossings occur between the top and the bottom ofthe emulsion. An enduring dance-like dynamics, for in-stance, is observed only for pair of cores and lasts for rela-tively shorts times, interrupted by crossings taking placewithin the emulsion. When this event occurs, the incom-ing droplet temporarily binds with the others, yielding toa short-lived nonequilibrium steady state such as thoseshown in Fig.3, in turn destroyed as soon as a further coreapproaches. Importantly, such complex dynamics almostcompletely removes any periodicity of the motion of theinternal droplets. This results from the non-trivial cou-pling between fluid velocity and internal cores (see Sup-plementary Figure 3 for the structure of the flow field):continuous changes of droplet positions lead to signifi-cant variations of the local velocity field which, in turn,further modifies the motion of the cores in a typical self-consistent fluid-structure interaction loop.

Suppression of crossings. Before concluding, weobserve that a viable route to prevent core crossings bykeeping N fixed, can be achieved by reducing the size(i.e. the diameter) of the inner drops, thus diminishingthe area fraction Ac they occupy within the emulsion.

In Supplementary Movie 11 we show, for example, thedynamics under Poiseuille flow of a four-core emulsion inwhich the inner drops, each of radius Ri = 12 lattice sites,are initially located in the lower half of the outer droplet

8

0

-6×10-3

-4×10-3

-2×10-3

2×10-3

1 2 3 4 5 6 7 8 9 10

0

-2×10-3

2×10-3

4×10-3

6×10-3

8×10-3

∆v

z cm

∆v

ycm

t

∆vzcm(1)

∆vycm(1)

0

-6×10-3

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1 2 3 4 5 6 7 8 9 10

0

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2×10-3

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6×10-3

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∆v

z cm

∆v

ycm

t

∆vzcm(2)

∆vycm(2)

0

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2×10-3

1 2 3 4 5 6 7 8 9 10

0

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2×10-3

4×10-3

6×10-3

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∆v

z cm

∆v

ycm

t

∆vzcm(3)

∆vycm(3)

3

1 2

(a) (b) (c) (d)

Figure 8. Speed of the cores and fluid velocity structure of the emulsion. (a)-(b)-(c) Time evolution of ∆vzcm(red/plusses) and ∆vycm (blue/crosses) of the cores, computed with respect to the speed of center of mass of the externaldroplet. (d) Typical pattern of the velocity field calculated with respect to the center of mass velocity of the external droplet.Two large eddies dominate the dynamics although relevant distortions, due to the presence of the cores, are mainly producedin the lower part of the emulsion.

-0.2

-0.1

0

0.1

0.2

0.3

0 2 4 6 8 10

∆z

cm

/Lz

t

∆zcm(1)∆zcm(2)

∆zcm(3)∆zcm(4)

1

3

4

2 1 2

34

(a) (b) (c)

Figure 9. Level crossing in a four-core emulsion. (a) Time evolution of the displacement ∆zcm of four cores. Drops 1and 2 (red/plusses and green/crosses, respectively) exhibit a periodic motion in the upper part of the emulsion whereas cores3 and 4 (blue/asterisk and pink/square) in the lower one, once a crossing has occurred. The black dotted line is a guide for theeye representing the separation between the two regions of the emulsion. Panels (b) and (c) show the positions of the internaldroplets during the crossing of droplet 3. They are taken at t = 2 × 105 (a) and t = 4 × 105 (b), marked by two spots (blackand yellow) in (a). A white arrow indicates the direction of motion of droplet 3.

of radius RO = 56 lattice sites. This sets an area fractionAc ∼ 0.2. Following the scheme proposed in Fig.3, onewould expect the formation of a long-lived steady-statewithout core crossings. Indeed, once the Poiseuille flow isimposed, the cores are soon captured by the fluid vortexin the lower region where they remain confined and movewith a persistent periodic motion. Such state, indicatedas 〈0|1, 2, 3, 4〉 is overall akin to 〈1, 2, 3|0〉 shown in Fig.3f,where three larger cores, with Ac ∼ 0.27, are set intoperiodic motion by the fluid recirculation in the upperregion of the emulsion.

If, unlike the previous case, cores of radius Ri = 12 lat-tice sites are placed symmetrically as in Fig.2d, at latetimes the inner drop initially at the bottom (i.e. 4) iscaptured by the vortex in the lower half of the emulsionwhile those in the middle and at the top (i.e. 1, 2, 3)remain confined in the upper part (see SupplementaryMovie 12). Clearly, this dynamics occurs without cross-ings. The emulsion finally attains a long-lived steady-state of the form 〈1, 2, 3|4〉, analogous to the state 〈3|1, 2〉of Fig.3e, i.e. a configuration in which an isolated core

remains locked in a sector of the emulsion and the re-maining drops move periodically in the other one.

Hence the scheme proposed in Fig. 3 describes ratherwell the formation of these further steady states too, sincethey are long-lived, crossing-free and observed for Ac <0.35.

DISCUSSION

Summarising, we have investigated the physics of amulti-core emulsion within a pressure-driven flow for val-ues of Re and Ca typical of microfluidic experiments.

We have shown that, as long as Ac and the N are keptsufficiently low, the cores exhibit a periodic steady-statedynamics confined within a sector of the emulsion, remi-niscent of a dancing couple, in which each dancer chasesthe partner. Our results strongly suggest that this pecu-liar behaviour is triggered and sustained by the internalvorticity which forms within the external droplet, whoseinterface acts as an effective bag confining the cores. The

9

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m/L

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∆zcm(1)

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m/L

z

t

∆zcm(2)

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m/L

z

t

∆zcm(3)

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m/L

z

t

∆zcm(4)

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m/L

z

t

∆zcm(5)

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m/L

z

t

∆zcm(1)

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m/L

z

t

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m/L

z

t

∆zcm(6)

(a) (b) (c)

(d) (e)

(f) (g) (h)

(i) (j) (k)

Figure 10. Multiple crossings in five and six-core emulsions. (a)-(e) Time evolution of the dislpacement ∆zcm of thecores in a five-core emulsion. Regular periodic motion survives for short periods of time and is temporarily broken by crossingsof the cores towards either the top or the bottom of the emulsion. (f)-(k) Time evolution of ∆zcm in a six-core emulsion.Although some cores travel along approximately similar trajectories (such as 1, red/plusses, and 2, green/crosses), there is noevidence of a persistent coupled periodic motion involving two (or more) drops. The black dotted line is a guide for the eyeindicating the separation between the two regions of the emulsion.

internal vorticity, in turn, is sustained by the heterogene-ity of the micro-confined carrier flow, under the effect ofthe pressure drive.

As the area fraction and the number of cores in-crease, a more complex multi-body dynamics emerges.Due to unavoidable collisions between droplets, as com-bined with self-consistent hydrodynamic interactions,cores may leave the confining vortex and switch to theother one. Whenever this occurs, they either restore the

planetary-like dynamics or temporarily aggregate withother cores to form unstable multi-droplets chains, whichare repeatedly destroyed and re-established, due to thenon-trivial coupling between the flow field and local vari-ations of the phase field.

The jumps from one vortex to another are interpretedas entropic events, expressing the tendency of the systemto maximise its entropy (propensity to motion) by fillingvoids, as they dynamically arise in this complex multi-

10

body fluid-structure interaction.Drawing inspiration from the occupation number for-

malism in statistical mechanics, we propose a classi-fication of the nonequilibrium states that provides atransparent interpretation of their intricate dynamicbehaviour. By denoting each dynamical state as〈α1, ..., αJ |αj+1, ..., αN 〉, in which j and N − j distinctcores occupy the two sectors of the emulsion, respectively,this occupation-number formalism provides an elegantand transparent tool to i) classify the various nonequilib-rium states of the system and ii) to describe the dynamictransitions among them.

On an experimental side, the aforementioned resultscan be realized by using standard microfluidic techniques,such as a glass capillary device with two distinc in-ner channels to form drops of different species withinthe emulsion [10, 64]. Our system, in particular, couldbe mapped onto an emulsion in which cores, of diam-eter Di ∼ 30µm, are immersed in a drop of diameterDO ∼ 100µm, with surface tension equal or higher thanσ ∼ 1mN/m. Provided that the flow is kept within thelaminar regime (to prevent turbulent-like behaviors), andthe channel is sufficiently long (say order of millimiters)and large to minimize the squeezing of the emulsion, thesteady-states discussed in Fig.3 as well as their dynamicsshould be observed.

The insights on the non-equilibrium states deliveredby the present study may prove useful to gain a deeperunderstanding of the role played by hydrodynamic inter-actions in multiple emulsions employed, for example, infields like biology, pharmaceutics and material science.

Recent microfluidic experiments, for instance, havebeen capable of encapsulating cells (the cores in ourmodel) within acqueous droplets in the presence of otherhosts, such as bacteria to study their pathogenicity [65].Pinpointing the effect of the flow in these systems couldshed light on the non-trivial dynamics governing the in-teraction among such biological objects, and could po-tentially suggest how to tune the flow rate to controlparameters of experimental interest, like reciprocal dis-tance and position. Fluid-stucture interactions are alsorelevant in multiple emulsions used as carriers for drugdelivery, since the release of the drug, generally storedwithin the cores, can be significantly influenced by theshape of the emulsion as well as by the structure of thefluid velocity within the shell [31]. The flow could facili-tate, for example, the migration of a core towards regionsexhibiting higher shape deformations, where a faster re-lease is expected to occur [31]. In the context of mate-rial science, multiple emulsions are employed as buildingblocks to design droplet based soft materials (such as tis-sues) with improved mechanical properties [1]. Monitor-ing the fluid flow is crucial here to achieve a uniform andregular arrangement of the droplets (generally requiredfor these materials [34, 35]) and to prevent structuralmodifications jeopardizing the design, an event occur-

ring, for example, in the presence of multiple crossings.Our results support the view that this latter effect canbe significantly mitigate by keeping the area fraction ofthe cores sufficiently low. It would be also of interestto investigate how the dynamics in this regime is influ-enced by a change of the viscoelastic properties of theemulsion, achieved, for example, either by increasing theviscosity of the middle fluid (to harden the emulsion) orby partially covering the interface of the cores with a sur-factant. This would be the case of Janus particles, which,owing to the takeover phenomena discussed in the text,would be periodically exposed to both front1-rear2 andfront2-rear1 contacts.

Finally one could also wonder whether periodic orbitsof the cores as well as their transition may represent apotential mesoscopic-scale analogy with level crossing ofatoms in quantum systems, along the lines pioneered byprevious authors for the case of bouncing droplets on vi-brating baths undergoing a tunnelling effect or orbitingwith quantised diameters [66, 67]. If so, one may evenhope that multi-core emulsions may also provide hydro-dynamic analogues of quantum materials. However atthis stage, this is only a speculation which calls for amuch more detailed and quantitative analysis.

METHODS

Following the approach of Ref. [39, 46], we describethe physics of a multi-core emulsion in terms of (i) aset of scalar phase field variables φi(r, t), i = 1, ..., Nd(where Nd is the total number of droplets) accountingfor the density of each droplet, and (ii) the global fluidvelocity v(r, t). The equilibrium properties are capturedby a coarse-grained free-energy density

f = a4

Nd∑i

φ2i (φi−φ0)2 +k

2

Nd∑i

(∇φi)2 +�∑i,j,i

11

where M is the mobility and µi = ∂f/∂φi−∂αf/∂(∂αφi)is the chemical potential (Greek letters denote Cartesiancomponents).

The fluid velocity v is governed by the Navier-Stokesequation which, in the incompressible limit, reads

ρ

(∂

∂t+ v · ∇

)v = −∇p+ η∇2v−

∑i

φi∇µi. (3)

In Equation 3, ρ is the density of the fluid, p is theisotropic pressure and η is the dynamic viscosity. Equa-tions 2-3 are numerically solved by using a hybrid latticeBoltzmann (LB) approach [68, 69], in which a finite dif-ference scheme, adopted to integrate Equation 2, is cou-pled to a standard LB method employed for Equation3. Further details about numerical implementation andthermodynamic parameters can be found in Supplemen-tary Note 1.

We finally provide an approximate mapping betweenour simulation parameters and real physical values. Sim-ulations are run on a rectangular mesh of size vary-ing from Ly = 600 ÷ 800 (length of the channel) toLz = 100÷170 (height of the channel), in which droplets,of radius ranging from 15 to 56 lattice sites, are included.Lattice spacing and time-step are ∆x = 1 and ∆t = 1.These values would correspond to a microfluidic chan-nel of length ∼ 1mm, in which droplets, with diameterranging between 30µm and 100µm and surface tensionσ ∼ 1mN/m, are set in a fluid of viscosiy ' 10−1 Pa·s.By fixing the length scale, the time scale, and the forcescale as L = 1µm, T = 10µs, and F = 10nN, a velocity of10−2 in simulation units corresponds approximately to adroplet speed of 1 mm/s. The Reynolds number (definedas Re = ρDOvmaxη , where DO is the diameter of the shell)ranges approximately between 1 (∆p = 4 × 10−4 andvmax ' 0.01) and 5 (∆p = 10−3 and vmax ' 0.025), whilethe capillary number (defined as Ca = vmaxησ ) ranges be-tween 0.1 and 1. These values ensure that inertial effectsare mild and are in good agreement with those reportedin previous experiments [1] and simulations [52].

DATA AVAILABILITY

All data are available upon request from the authors.

ACKNOWLEDGMENTS

A. T., A. M., M. L., F. B. and S. S. acknowledge fund-ing from the European Research Council under the Euro-pean Union’s Horizon 2020 Framework Programme (No.FP/2014-2020) ERC Grant Agreement No.739964 (COP-MAT).

AUTHOR CONTRIBUTIONS

A.T., A.M., M.L., F.B., S.S, S.A., M.M. and D.A.W.conceived the research. A.T. and S.S. designed theproject. A.T. run simulations and processed data, andwith A.M. and S.S. analyzed the results. A.T. wrotethe paper with contributions from A.M., M.L, F.B, S.S.,S.A., M.M. and D.A.W.

COMPETING INTERESTS

The authors declare no competing interests.

SUPPLEMENTARY MATERIAL

SUPPLEMENTARY NOTE 1: NUMERICALDETAILS

Here we provide further details about the numericalmethod and the simulation parameters.

The equations for the order parameter φi (Equation 2of the main text) and the Navier-Stokes equation (Equa-tion 3 of the main text) are solved by using a hybridlattice Boltzmann (LB) method [69], in which Equation2 is integrated via a finite-difference predictor-correctoralgorithm and Equation 3 via a standard LB approach.

As reported in the main text, simulations are peformedon a rectangular lattice with size ratio Γ = LzLy rang-ing from 0.16 to 0.22. More specifically, Γ = 0.167(Ly = 600, Lz = 100) for the core-free droplet, Γ = 0.2(Ly = 600, Lz = 120) for the single-core emulsion andΓ = 0.21 (Ly = 800, Lz = 170) for the two-core andhigher complex emulsions. Periodic boundary conditionsare set along the y-axis and two flat walls along the z-axis, placed at z = 0 and z = Lz. Here no-slip conditionshold for the velocity field (i.e. vz(z = 0, z = Lz) = 0) andneutral wetting for the order parameters φi. The latterones are achieved by setting

∂µi∂z

∣∣∣∣∣z=0,z=Lz

= 0 (4)

∂∇2φi∂z

∣∣∣∣∣z=0,z=Lz

= 0. (5)

The first one guarantess density conservation (no massflux through the walls) while the second one imposes thewetting to be neutral.

Like in previous works [46, 47], the pressure gradient∆p producing the Poiseulle flow is modeled through abody force (force per unit density) added to the collisionoperator of the LB equation at each lattice node.

Thermodynamic parameters have been chosen as fol-lows: a = 0.07, k = 0.1, M = 0.1 and � = 0.05 (a value

12

larger than 0.005 is enough to prevent droplet merging).These values fix the surface tension and the interfacewidth to σ =

√8ak

9 ' 0.08 and ξ = 2√

2ka ' 3 − 4,

respectively. Also, the dynamic viscosity η of both fluidcomponents is set equal to 5/3. Such approximation, re-tained for simplicity, may be relaxed by letting η dependson φ [70, 71]. Lattice spacing and integration time-stephave been kept fixed to ∆x = 1 and ∆t = 1, while dropletradii are chosen as follows: R = 30 for the core-freedroplet, Ri = 15 and RO = 30 for a single-core emul-sion, and Ri = 17 and RO = 56 for emulsions containingmore than one core. Here Ri is the radius of the coreswhile RO is the one of the surrounding shell.

SUPPLEMENTARY NOTE 2: VELOCITYPROFILE UNDER POISEUILLE FLOW

In Supplementary Figure 1 we report, for example, thetypical steady-state velocity profile observed in a two-core emulsion for different values of the pressure gradient.They are averaged over space and time, i.e. the channellength and approximately 3×105 time steps at the steadystate. The curves are compatible with a parabolic profileexpected in an isotropic fluid with the same viscosity,and remain essentially unaltered for the other multi-coreemulsions considered in this work.

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0 0.2 0.4 0.6 0.8 1

<v>

s,t

z/Lz

∆p=10-4

∆p=4×10-4

∆p=6×10-4

∆p=8×10-4

∆p=10-3

∆p=1.5×10-3

∆p=2×10-3

∆p=3×10-3

Figure 11. Supplementary Figure 1. AveragedPoiseuille profile. This plot shows the typical steady-statevelocity profile for different values of pressure gradient ∆pin a two-core emulsion. Here 〈v〉 is averaged over space andtime.

However, substantial modifications occur when instan-taneous configurations are considered. In SupplementaryFigure 2 we show, for instance, the instantaneous veloc-ity profile observed in core-free (a), one-core (b), two-core(c) and three-core (d) emulsions calculated along a crosssection of the channel where internal cores temporarilyaccumulate. While in (a) the parabolic profile is only

weakly disturbed by the droplet interface, in (b)-(d) itis significantly modified by local bumps and dips causedby internal cores. Such distortions wash out when theseprofiles are averaged over space and time.

SUPPLEMENTARY NOTE 3: STRUCTURE OFTHE VELOCITY FIELD IN FOUR, FIVE AND

SIX-CORE EMULSIONS

In Supplementary Figure 3 we show the typical velocityfield observed in multiple emulsions containing (a) four,(b) five and (c) six cores. In all cases, the interior struc-ture of the field exhibits significant deviations (heavieras the number of cores increases) from the double-vortexpattern of a core-free emulsion.

As discussed in the paper, under Poiseuille flow a four-core emulsion, originally designed as in Fig.2d of the maintext, only temporarily survives in a state of the form〈1, 2, 3|4〉 (Supplementary Figure 3a), since the effectivearea fraction occupied by three cores is larger than 0.35in half emulsion. This causes a crossing of a drop (3 inSupplementary Figure 3a) driven a heavy flux pushing itdownwards, thus leading to the long-lived nonequilibriumstate 〈1, 2|3, 4〉 (see Fig.3h of the main text). In thisstate, couples of drops display a planetary-like motionwithin each half of the emulsion and no further crossingoccurs.

Increasing the number of inner drops, such as infive and six-core systems (Supplementary Figure 3b,c),favours multiple crossings between the two regions of theemulsions, a process generally driven by a pre-chaoticflows resulting from the complex coupling between veloc-ity and phase field. This is why only short-lived statesare observed in these systems.

REFERENCES

[1] A. S. Utada et al., Monodisperse double emulsions gener-ated from a microcapillary device, Science 308, 537-541(2005).

[2] L. Y. Chu, A. S. Utada, R. K. Shah, J. W. Kim, and D.A. Weitz, Controllable monodisperse multiple emulsions,Angew. Chem., Int. Ed. 46, 8970 (2007).

[3] J. A. Hanson et al., Nanoscale double emulsions stabi-lized by single-component block copolypeptides, Nature455, 85 (2008).

[4] A. R. Abate and D. A. Weitz, High-order multipleemulsions formed in poly(dimethylsiloxane) microflu-idics, Small 5, 20302032 (2009)

[5] W. Wang et al., Controllable microfluidic production ofmulticomponent multiple emulsions, Lab on a Chip 11,1587 (2011).

13

0

0.005

0.01

0.015

0.02

0.025

0 0.2 0.4 0.6 0.8 1

v

z/Lz

0

0.005

0.01

0.015

0.02

0.025

0 0.2 0.4 0.6 0.8 1

v

z/Lz

0

0.005

0.01

0.015

0.02

0.025

0 0.2 0.4 0.6 0.8 1

v

z/Lz

0

0.005

0.01

0.015

0.02

0.025

0 0.2 0.4 0.6 0.8 1

v

z/Lz

(a) (b) (c) (d)

Figure 12. Supplementary Figure 2. Instantaneous Poiseuille profile. These plots show the instantaneous velocityprofile of a core-free (a), one-core (b), two-core (c) and three-core (d) emulsion computed along the cross section of the channel,indicated by the dotted green line. Insets show the corresponding configuration of the emulsion.

5 3

1 42

12

3

4

(a) (b) (c)

3

2 14

65

Figure 13. Supplementary Figure 3. Velocity field. Characteristic structures of the velocity field in (a) four-core (b)five-core and (c) six-core emulsions computed with respect to the external droplet frame. The typical double eddy structureobserved in a core-free emulsion undergoes significant distortions as the number of cores increases. Panel (a) shows the flowfield observed during the crossing of droplet 3 from the top towards the bottom of the emulsion (see also Fig.9 of the maintext). A transition from the state 〈1, 2, 3|4〉 to the state 〈1, 2|3, 4〉 occurs. Panels (b) and (c) show two short-lived states of theform 〈3, 5|1, 2, 4〉 and 〈3, 5, 6|1, 2, 4〉, only temporarily surviving due to multiple crossings.

[6] S. Datta et al., 25th Anniversary Article: Double Emul-sion Templated Solid Microcapsules: Mechanics AndControlled Release, Adv. Mater. 26, 2205 (2014).

[7] P. S. Clegg, J. W. Tavacoli, and P. J. Wilde, One-stepproduction of multiple emulsions: Microfluidic, polymer-stabilized and particle-stabilized approaches, Soft Matter12, 998 (2016).

[8] G. T. Vladisavljevic, Recent advances in the productionof controllable multiple emulsions using microfabricateddevices, Particuology 24, 1 (2016).

[9] T. Y. Lee, T. M. Choi, T. S. Shim, R. A. Frijns, S. H.Kim, Microfluidic production of multiple emulsions andfunctional microcapsules, Lab on a Chip 16, 3415 (2016).

[10] G. T. Vladisavljevic, R. Al Nuumani, and S. A. Nabavi,Microfluidic production of multiple emulsions, Microma-chines 8, 75 (2017).

[11] E. Y. Liu, S. Jung, D. A. Weitz, H. Yi, C. H. Choi, High-throughput double emulsion-based microfluidic produc-tion of hydrogel microspheres with tunable chemical func-tionalities toward biomolecular conjugation, Lab Chip18, 323 (2018).

[12] S. Nawar et al., Parallelizable microfluidic dropmakerswith multilayer geometry for the generation of doubleemulsions. Lab on a Chip 20, 147 (2020).

[13] S. Ding, C.A. Serra, T.F. Vandamme, W. Yu, N. Anton,

Double emulsions prepared by two-step emulsification:History, state-of-the-art and perspective, J. Contr. Deliv.295, 31 (2019).

[14] N. Vasishtha, H. W. Schlameus, Microencapsulation ofFood Ingredients, P. Vilstrup, Ed. (Leatherhead FoodInternational, Leatherhead, UK, 2001).

[15] D. J. McClements, Advances in fabrication of emulsionswith enhanced functionality using structural design prin-ciples. Curr. Opin. Colloid Interface Sci. 17, 235-245(2012).

[16] T. A. Comunian, A. Abbaspourrad, C. S.Favaro-Trindade, D. A.Weitz, Fabrication of solid lipid micro-capsules containing ascorbic acid using a microfluidictechnique, Food Chem. 152, 271 (2014).

[17] G. Muschiolik, E. Dickinson, Double Emulsions Relevantto Food Systems: Preparation, Stability, and Applica-tions, Comprehensive Reviews in Food Science and FoodSafety 16, 532 (2017).

[18] C. Laugel, P. Rafidison, G. Potard, L. Aguadisch, A.Baillet, Modulated release of triterpenic compounds froma O/W/O multiple emulsion formulated with dime-thicones: infrared spectrophotometric and differentialcalorimetric approaches, J. Controlled Release 63, 7(2000).

[19] K. Pays, J. Giermanska-Kahn, B. Pouligny, J. Bibette

14

and F. Leal-Calderon, Double emulsions: how does re-lease occur?, Journ. Control. Rel. 79, 193 (2002).

[20] R. Cortesi, E. Esposito, G. Luca, and C. Nastruzzi,Production of lipospheres as carriers for bioactive com-pounds, Biomaterials 23, 2283 (2002).

[21] E. C. Sela, M. Chorny, N. Koroukhov, H. D. Danen-berg, G. Golomb, A new double emulsion solvent dif-fusion technique for encapsulating hydrophilic moleculesin PLGA nanoparticles, J. Control. Rel. 133, 90 (2009).

[22] McCall, R. L., Sirianni, R. W. PLGA NanoparticlesFormed by Single- or Double-emulsion with VitaminE-TPGS, J. Vis. Exp. 82, e51015, doi:10.3791/51015(2013).

[23] M. H. Lee, S. G. Oh, S. K. Moon, S. Y. Bae, Preparationof Silica Particles Encapsulating Retinol Using O/W/OMultiple Emulsions, J. Colloid Interface Sci. 240, 83(2001).

[24] D. H. Lee et al., Effective Formation of Silicone-in-Fluorocarbon-in-Water Double Emulsions: Studies onDroplet Morphology and Stability, J. Dispersion Sci.Technol. 23, 491 (2002)

[25] J. S. Lee et al., The stabilization of L-ascorbic acid inaqueous solution and water-in-oil-in-water double emul-sion by controlling pH and electrolyte concentration, Int.Jour. Cosm. Sci 26, 217 (2004).

[26] J. Elisseeff, W. McIntosh, K. Fu, T. Blunk, R. Langer,Controlled-release of IGF-I and TGF-β1 in a photopoly-merizing hydrogel for cartilage tissue engineering, Jour.Ortop. Res. 19, 1098 (2001).

[27] B. G. Chung, K. H. Lee, A. Khademhosseini, S. H.Lee, Microfluidic fabrication of microengineered hydro-gels and their application in tissue engineering, Lab on aChip 12, 45 (2012).

[28] M. Costantini et al., Highly ordered and tunable poly-HIPEs by using microfluidics, Journ. Mat. Chem. B 2,2290 (2014).

[29] C. H. Choi et al., One-step generation of cell-laden micro-gels using double emulsion drops with a sacrificial ultra-thin oil shel, Lab on a Chip 16, 1549 (2016).

[30] A. S. Mao, Programmable microencapsulation for en-hanced mesenchymal stem cell persistence and im-munomodulation, Proc. Nat. Am. Sci. 116, 15392 (2019).

[31] G. Pontrelli, E. J. Carr, A. Tiribocchi, S. Succi, Model-ing drug delivery from multiple emulsions, Phys. Rev. E.102, 023114 (2020).

[32] G. Muschiolik, Multiple emulsions for food use, Curr. Op.Coll. and Inter. Sci. 12, 213 (2007).

[33] J. P. Raven, P. Marmottant, Microfluidic Crystals: Dy-namic Interplay between Rearrangement Waves andFlow, Phys. Rev. Lett. 102, 084501 (2009).

[34] P. Garstecki, H. A. Stone, G. M. Whitesides, Mechanismfor Flow-Rate Controlled Breakup in Confined Geome-tries: A Route to Monodisperse Emulsions, Phys. Rev.Lett. 94, 164501 (2005).

[35] J. Guzowski, P. Garstecki, Droplet Clusters: Exploringthe Phase Space of Soft Mesoscale Atoms, Phys. Rev.Lett. 114, 188302 (2015).

[36] R. Pal, Rheology of double emulsions, J. Colloid InterfaceSci. 307, 509 (2007).

[37] R. Pal, Rheology of simple and multiple emulsions, Curr.Opin. Colloid Interface Sci. 16, 41 (2011).

[38] J. Tao, X. Song, J. Liu, J. Wang, Microfluidic rheology ofthe multiple-emulsion globule transiting in a contractiontube through a boundary element method, Chem Enging.

Sci 97, 328 (2013).[39] A. Tiribocchi et al., Novel nonequilibrium steady states

in multiple emulsions, Phys. Fluids 32, 017102 (2020).[40] N. Wang, C. Semprebon, H. Liu, C. Zhang, and H.

Kusumaatmaja, Modelling double emulsion formation inplanar flow-focusing microchannels, J. Fluid Mech. 895,A22 (2020).

[41] A. Montessori, M. Lauricella, N. Tirelli, S. Succi,Mesoscale modelling of near-contact interactions for com-plex flowing interfaces, Journ. Fluid Mech. 872, 327-347(2019).

[42] X. Chen, Y. Liu and M. Shi, Hydrodynamics of doubleemulsion droplet in shear flow, Appl. Phys. Lett. 102,051609 (2013).

[43] J. Wang, J. Liu, J. Han, and J. Guan, Effects of ComplexInternal Structures on Rheology of Multiple EmulsionsParticles in 2D from a Boundary Integral Method, Phys.Rev. Lett. 110, 066001 (2013).

[44] Y. Chen, X. Liu, and Y. Zhao, Deformation dynamics ofdouble emulsion droplet under shear, Appl. Phys. Lett.106, 141601 (2015).

[45] K. A. Smith, J. M. Ottino, and M. Olvera de la Cruz,Encapsulated Drop Breakup in Shear Flow, Phys. Rev.Lett. 93, 204501 (2004).

[46] M. Foglino, A. N. Morozov, O. Henrich, D. Marenduzzo,Flow of deformable droplets: Discontinuous shear thin-ning and velocity oscillations, Phys. Rev. Lett. 119,208002 (2017).

[47] M. Foglino, A. M. Morozov, D. Marenduzzo, Rheologyand microrheology of deformable droplet suspensions,Soft Matter 18, 9361 (2018).

[48] R. Mueller, J. M. Yeomans, A. Doostmohammadi, Emer-gence of Active Nematic Behavior in Monolayers ofIsotropic Cells, Phys. Rev. Lett. 122, 048004 (2019).

[49] S. R. De Groot and P. Mazur, Non-Equilibrium Thermo-dynamics (New York, NY, Dover, 1984).

[50] G. Lebon, D. Jou, J. C. Vazquez, Understanding Non-equilibrium Thermodynamics (Springer-Verlag BerlinHeidelberg, 2008)

[51] R. Piazza, Soft Matter: The stuff that dreams are madeof, Springer, Rotterdam, Netherlands (2012).

[52] S. Guido, V. Preziosi, Droplet deformation under con-fined Poiseuille flow, Adv. Coll. and Int. Sci. 161, 89(2010).

[53] A. Pommella, D. Donnarumma, S. Caserta, S. Guido,Dynamic behaviour of multilamellar vesicles underPoiseuille flow, Soft Matter 13, 6304-6313 (2017).

[54] S. Das, S. Mandal, S. Chakraborty, Effect of transversetemperature gradient on the migration of a deformabledroplet in a Poiseuille flow, Jour. Fluid. Mech. 850, 1142(2018).

[55] D. Abreu, M. Levant, V. Steinberg, U. Seifert, Fluid vesi-cles in flow, Adv. Coll. Int. Sci. 208, 129 (2014).

[56] G. Coupier, A. Farutin, C. Minetti, T. Podgorski, andC. Misbah, Shape Diagram of Vesicles in Poiseuille Flow,Phys. Rev. Lett. 108, 178106 (2012).

[57] C. A. Weber, D. Zwicker, F. Jülicher, C. F. Lee, Physicsof active emulsions, Rep. Prog. Phys. 82, 064601 (2019)

[58] S. Herminghaus et al., Interfacial mechanisms in activeemulsions, Soft Matter 10, 7008 (2014).

[59] F. Fadda, G. Gonnella, A. Lamura, A. Tiribocchi, Lat-tice Boltzmann study of chemically-driven self-propelleddroplets, Eur. Phys. Journ. E 40, 112 (2017

[60] T. Sanchez, D. T. N. Chen, S. J. DeCamp, M. Heymann,

15

Z. Dogic, Spontaneous motion in hierarchically assem-bled active matter, Nature 491, 431 (2012).

[61] E. Tjhung, D. Marenduzzo, M. E. Cates, Spontaneoussymmetry breaking in active droplets provides a genericroute to motility, Proc. Nat. Acad. Sci. USA 109, 12381(2012).

[62] B. V. Hokmabad, K. A. Baldwin, C. Krüger, C. Bahr, C.C. Maass, Topological stabilization and dynamics of self-propelling nematic shells, Phys. Rev. Lett. 123, 178003(2019).

[63] J. M. Rallison, The Deformation of Small Viscous Dropsand Bubbles in Shear Flows, Ann. Rev. Fluid Mech. 16,45 (1984).

[64] L. L. A. Adams et al., Single step emulsification for thegeneration of multi-component double emulsions, SoftMatter 8, 10719-10724 (2012).

[65] T. S. Kaminski, O. Scheler, P. Garstecki, Droplet mi-crofluidics for microbiology: techniques, applications andchallenges, Lab Chip 16, 2168-2187 (2016).

[66] Y. Couder, S. Protiére, E. Fort, A. Boudaoud, Walkingand orbiting droplets, Nature 437, 208 (2005).

[67] J. W. M. Bush, Quantum mechanics writ large, Proc.Nat. Acad. Sci. USA 107, 17455 (2010).

[68] S. Succi, The Lattice Boltzmann Equation: For Com-plex States of Flowing Matter (Oxford University Press,2018).

[69] L. N. Carenza, G. Gonnella, A. Lamura, G. Negro, A.Tiribocchi, Lattice Boltzmann methods and active fluids,Eur. Phys. Jour. E 42, 81 (2019).

[70] K. Langaas, J. M. Yeomans, Lattice Boltzmann simula-tion of a binary fluid with different phase viscosities andits application to fingering in two dimensions, Eur. Phys.Journ. B 15, 133-141 (2000)

[71] E. Tjhung, A. Tiribocchi, D. Marenduzzo, M. E. Cates, Aminimal physical model captures the shapes of crawlingcells, Nat. Comm. 6, 5420 (2015).

The vortex-driven dynamics of droplets within dropletsAbstract Introduction Results Discussion Methods Data availability Acknowledgments Author contributions Competing interests Supplementary Material Supplementary Note 1: Numerical details Supplementary Note 2: Velocity profile under Poiseuille flow Supplementary note 3: Structure of the velocity field in four, five and six-core emulsions References References