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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
[email protected] 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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3
y
z
(c)(b)(a)
0.5
1.5
1
0
2
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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4
3
1 23 4
51 23
4
1 2
1 2
5
3 4
61 1 2
(f)(e)(d)(c)(b)(a)
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).
N=2 N=3
l.l.
l.l.
l.l.
l.l.
l.l.
l.l.
N=1
1
1
12
1
2
12
3
123
(a)
(b)
(c)
(d)
(e)
(f)
N=4 N=5 N=6
s.l.
s.l.
s.l.
s.l.
s.l.
l.l.
crossing
1 2
3
4
4
3 1 2
1 42
5 3
145
23
12 4
63 5
213 456
multiple crossing
(g)
(h)
(i)
(j)
(k)
(l)
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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5
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).
1 2
(a)
(b)
(c)
1 2 1 2
(d)
1 2
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
-0.5
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0 2 4 6 8 10
0
0.06
0.12
0.18
0.24
0.3
0.36
∆z c
m/L
z
∆y
cm/L
y
t
∆zcm(1)
∆zcm(2)
∆ycm(1)
∆ycm(2)
2 1
12
12
2 1
(a)
(b)
(c)
(d)
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
-
6
0
-2.0×10-3
2.0×10-3
4.0×10-3
6.0×10-3
1 2 3 4 5 6 7 8 9 10
∆v
ycm
t
∆vycm(1)
∆vycm(2)
0
-2.0×10-3
-1.0×10-3
1.0×10-3
2.0×10-3
1 2 3 4 5 6 7 8 9 10
∆v
z cm
t
∆vzcm(1)
∆vzcm(2)
21
12
(a) (b)
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
-
7
-0.25
-0.2
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
0.2
0.25
0 2 4 6 8 10
∆z c
m/L
z
t
∆zcm(1)∆zcm(2)∆zcm(3)
-0.04
-0.02
0
0.02
0.04
0.06
0 2 4 6 8 10
∆y
cm
/Ly
t
∆ycm(1)∆ycm(2)∆ycm(3)
3 3
33
12 21
122 1
(a) (c)
(b) (d)
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
-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(2)
∆vycm(2)
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(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
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0 2 4 6 8 10 12 14
∆z c
m/L
z
t
∆zcm(1)
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0 2 4 6 8 10 12 14
∆z c
m/L
z
t
∆zcm(2)
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0 2 4 6 8 10 12 14
∆z c
m/L
z
t
∆zcm(3)
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0 2 4 6 8 10 12 14
∆z c
m/L
z
t
∆zcm(4)
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0 2 4 6 8 10 12 14
∆z c
m/L
z
t
∆zcm(5)
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0 2 4 6 8 10 12 14
∆z c
m/L
z
t
∆zcm(1)
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0 2 4 6 8 10 12 14
∆z c
m/L
z
t
∆zcm(2)
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0 2 4 6 8 10 12 14
∆z c
m/L
zt
∆zcm(3)
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0 2 4 6 8 10 12 14
∆z c
m/L
z
t
∆zcm(4)
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0 2 4 6 8 10 12 14
∆z c
m/L
z
t
∆zcm(5)
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0 2 4 6 8 10 12 14
∆z c
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