PHYSICAL REVIEW E 92, 012317 (2015)

Suppression of Ostwald ripening in active emulsions

David Zwicker,1,* Anthony A. Hyman,2 and Frank Julicher1,†1Max Planck Institute for the Physics of Complex Systems, 01187 Dresden, Germany

2Max Planck Institute of Molecular Cell Biology and Genetics, 01307 Dresden, Germany(Received 3 February 2015; published 22 July 2015)

Emulsions consisting of droplets immersed in a fluid are typically unstable since they coarsen over time.One important coarsening process is Ostwald ripening, which is driven by the surface tension of the droplets.Stability of emulsions is relevant not only in complex fluids but also in biological cells, which contain liquidlikecompartments, e.g., germ granules, Cajal bodies, and centrosomes. Such cellular systems are driven away fromequilibrium, e.g., by chemical reactions, and thus can be called active emulsions. In this paper, we study suchactive emulsions by developing a coarse-grained description of the droplet dynamics, which we analyze for twodifferent chemical reaction schemes. We first consider the simple case of first-order reactions, which leads tostable, monodisperse emulsions in which Ostwald ripening is suppressed within a range of chemical reaction rates.We then consider autocatalytic droplets, which catalyze the production of their own droplet material. Spontaneousnucleation of autocatalytic droplets is strongly suppressed and their emulsions are typically unstable. We showthat autocatalytic droplets can be nucleated reliably and their emulsions stabilized by the help of chemically activecores, which catalyze the production of droplet material. In summary, different reaction schemes and catalyticcores can be used to stabilize emulsions and to control their properties.

DOI: 10.1103/PhysRevE.92.012317 PACS number(s): 82.70.Kj, 87.16.−b, 82.40.−g

I. INTRODUCTION

Emulsions are mixtures of immiscible liquids, in whichdroplet formation is typical. Such systems are important inmany areas such as pharmaceuticals, foods, and cosmetics,where properties like the droplet size distribution and itsstability must be controlled [1]. Besides these technologicalapplications, droplet formation is also important in biologicalsystems. For instance, droplets form compartments in theintracellular fluid [2–6]. Additionally, liquidlike lipid domainsare known to structure the cell membrane [7]. One key aspectof biological systems is that they are active, i.e., they aredriven away from equilibrium. Recently, we have shown thatnonequilibrium chemical reactions could control the formationof centrosomes [5], which are one example for liquidlikesubstructures in cells. In particular, we suggested that twocentrosomes are nucleated by catalytically active cores andgrow by an autocatalytic reaction. We also demonstratedthat the catalytic cores help to suppress Ostwald ripeningand that two coexisting centrosomes can be stable [5]. Thisraises the questions by what mechanisms chemical reactionscould suppress Ostwald ripening in emulsions and under whatconditions chemically active emulsions could be stable.

Stabilizing emulsions over long times is a major challenge[1]. This is because emulsions typically coarsen, i.e., largedroplets grow and small droplets disappear, which is energet-ically favorable. There are two different processes that leadto coarsening: droplet coalescence, which is driven by theBrownian motion of droplets, and Ostwald ripening, which isdriven by diffusive fluxes between droplets [8,9]. Both theseprocesses have to be suppressed to stabilize emulsions. Dropletcoalescence can be suppressed in a number of ways, e.g.,

*Present address: School of Engineering and Applied Sciences,Harvard University, Cambridge, MA 02138, USA.†Corresponding author: julicher@pks.mpg.de

by utilizing surfactants [10,11] or by simply immobilizingthe droplets [12,13]. In contrast, the diffusive flux leadingto Ostwald ripening is more difficult to suppress. It canbe prevented by trapping particles inside droplets that areinsoluble in the surrounding fluid [14–16].

Droplet coarsening due to Ostwald ripening can be sup-pressed by continuous shearing of a system [17,18]. Shearflows break up larger droplets and thus counteract dropletcoarsening. The droplets are in this case typically nonspherical[19,20]. This raises the question of whether other nonequilib-rium conditions could also have an effect on droplet coarseningand the stabilization of emulsions. In fact, chemical reactionsinfluence the phase separation kinetics of mixtures, leading topattern formation and introducing characteristic length scales[21–24]. Furthermore, more complex reaction schemes suchas autocatalytic reactions have been shown to add interestingeffects, including patterns with multiple length scales [25,26].Interestingly, a simple case of first-order reactions in an infinitesystem is formally related to a phase-separating system withlong-ranged repulsive interactions [22]. Therefore, phase sep-aration with long-range repulsion is an interesting precedentfor suppression of Ostwald ripening [27–31], which by formalanalogy provides useful information for the study of dropletswith chemical reactions.

In this paper, we study the role of chemical reactions ondroplet dynamics in systems of coexisting liquid phases. Weintroduce a general framework to study such active emulsionsboth in finite and infinite systems. We use it to show thatOstwald ripening can be suppressed and emulsions aretypically monodisperse when first-order chemical reactionsare considered. We discuss the typical droplet size and thecharacteristic rates of the droplet dynamics and we use theseresults to identify the regions in parameter space where activeemulsions are stable. We also test our results by comparingthem to numerical solutions. In addition to the simple caseof first-order kinetics, we also investigate droplets withautocatalytic reactions, which can describe, for example, the

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DAVID ZWICKER, ANTHONY A. HYMAN, AND FRANK JULICHER PHYSICAL REVIEW E 92, 012317 (2015)

dynamics of centrosomes [5]. These droplets catalyze theirown growth and are best discussed in a ternary fluid. Oneimportant property of autocatalytic droplets is that they arehard to nucleate and tend to be unstable. We discuss nucleationof autocatalytic droplets by active cores that catalyze theproduction of droplet material. These active cores can alsosuppress Ostwald ripening and stabilize autocatalytic droplets.

The paper is organized as follows. In Sec. II, we introducea continuum theory of phase-separating fluids with chemicalreactions and obtain a coarse-grained description of activeemulsions. In Sec. III, we study the simple case of first-order kinetics in a binary fluid. In Sec. IV, we considerautocatalytic droplets in a ternary fluid as an example for amore complex reaction scheme. The results are discussed inSec. V. Throughout the paper, we compare our results to theclassical case of passive droplets without chemical reactions.

II. THEORETICAL DESCRIPTIONOF ACTIVE EMULSIONS

We study droplets of a liquid phase that coexists withthe surrounding fluid. We consider chemical reactions thatcan convert soluble components A in the fluid into dropletmaterial B that phase separates from the fluid. We first discussa continuum theory and then derive dynamic equations fordroplet sizes in the emulsion.

A. Continuum theory of ternary fluids

We consider a ternary fluid consisting of soluble buildingblocks A, droplet material B, and components of the back-ground fluid C, which do not participate in the chemical reac-tions. Volume fractions of components A and B are denotedφA(r,t) and φB(r,t), respectively; the volume fraction φC

obeys φC = 1 − φA − φB . For an incompressible, isothermalfluid with constant molecular volumes, the free energy densityf depends on the two volume fractions φA and φB . Forsimplicity, we use the form

f (φA,φB) = a

2(φA)2 + b

2ψ2(φB)2(ψ − φB)2, (1)

where the positive parameters a and b characterize molecularinteractions and entropic contributions. The double-well po-tential described by the second term accounts for the separationinto two phases, a C-rich phase with φB = 0 and a B-rich phasewith φB = ψ , while A distributes freely between both phases[32]. Here, we consider the strong segregation limit, where thevolume fraction φB can vanish in one of the phases.

We are interested in heterogeneous systems where many B

droplets coexist with the C phase, see Fig. 1(a). Such systemsare globally out of equilibrium but in a local equilibriumapproximation the free energy density can be defined ineach local volume element. The state of the system is thencharacterized by the total free energy

F =∫ [

f (φA,φB) + κ

2(∇φA)2 + κ

2(∇φB)2

]d3r, (2)

where the integral is over the system volume. Here the termsproportional to κ and κ penalize strong gradients, whichin particular occur in the interface region between phases

(b)(a) Continuous System Coarse-grained Description

FIG. 1. (Color online) Schematic representations of emulsionswith droplets of enriched B components (dark orange) in abackground of A components (light blue). (a) Full spatiotemporaldescription of the volume fractions φA and φB including diffuseinterfaces. (b) Simplified description in terms of the droplet radii Ri

and the average volume fractions φA0 and φB

0 in the background fluid.

[33]. The width of such interfaces can be discussed byconsidering a flat interface between two phases with volumefractions φB = 0 and φB = ψ far from the interface. Inthis case, F is minimized by φB(x) = 1

2ψ[1 + tanh(x/w)],where w = 2(κ/b)1/2 is the interface width and x a coordinatenormal to the interface [33]. The free energy associatedwith the interface per unit area yields the surface tensionγ = 1

6ψ2(bκ)1/2 [32].These expressions for w and γ also hold in the case of

curved interfaces, as long as the radius of curvature is largecompared to w. Because the surface tension is positive, the freeenergy F is minimized by droplet configurations with minimalinterface area [34]. In emulsions, larger droplets, which have asmaller surface-to-volume ratio, are thus energetically favoredover multiple smaller droplets with the same volume. Thisminimization of the interfacial area drives Ostwald ripening inpassive fluids [8].

The dynamics of the concentration fields are described bythe equations [22,23]

∂tφA = mA∇2 δF [φA,φB]

δφA− s(φA,φB), (3a)

∂tφB = mB∇2 δF [φA,φB]

δφB+ s(φA,φB), (3b)

where the first terms describe particle-number-conservingdiffusive movements and chemical reactions are introduced bya source and sink term s. Here mA and mB denote mobilitiesof the components and we neglect hydrodynamic effects forsimplicity [32].

B. Chemical reaction schemes

We discuss chemical reactions in which soluble buildingblocks A are transformed into droplet material B, while thecomponents C are not changed by the chemical reactions.The reaction flux introduces a source s in Eq. (3). We considerthe form

s = kfφA − kbφ

B + kφAφB + φA

N∑i=1

Qiδ(r − r i). (4)

Here the first-order reactions A → B and A ← B havethe rates kf and kb, respectively, and the rate k describesthe autocatalytic reaction A + B → 2B. We also introducecatalytically active cores that locally catalyze the reaction

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(b)(a)

A B

A B

B A

B A

BA B2+

A

B

Droplet with First-order Reactions Autocatalytic Droplet

FIG. 2. (Color online) Chemical reaction schemes of buildingblocks A (light blue) and droplet material B (medium orange)considered in this paper. (a) First-order reactions. (b) Autocatalyticdroplets where B is produced by the reaction A + B → 2B and atthe catalytic cores (dark green), such that the production is effectivelyrestricted to inside the droplets. Droplet material B is converted backto A by a simple first-order reaction.

A → B and that therefore can nucleate droplets. The catalyticactivity of the core numbered i at position r i is denoted by Qi ,see Fig. 2.

Equations (3) and (4) are completed by no-flux boundaryconditions. For the reaction schemes considered here, the totalamount of components A and B is conserved, and thus the totalvolume fraction φ = φA + φB is time independent where

φi = 1

Vs

∫φi d3r (5)

denotes the average volume fraction of component i in thesystem, for i = A,B. Here the integral is over the entire systemof volume Vs. Note that when chemical reactions are absent(s = 0), components A and B are not converted into each otherand the volume fractions φA and φB are conserved separately.

When expressing the reaction rates by Eq. (4), we havenot imposed a detailed balance condition for A � B [35,36].This implicit breaking of detailed balance is important inthe following as it gives rise to the rich physics of activedroplets developed here. This is motivated in particular bybiological systems, where detailed balance is broken, e.g.,by the supply of a chemical fuel such as adenosine triphos-phate [37]. That breaking of detailed balance is importantfor droplets to be stable can be seen as follows. If thechemical reactions would obey detailed balance, the systemcould relax to a thermodynamic equilibrium state for whichthe free energy F is minimal. Since the components A andB can be converted into each other, there are no separateconstraints on the numbers of A and B molecules. As aconsequence, the volume fractions φA and φB could relaxto the values at which the free energy density f has aminimum for a given total molecule number. This is in generala homogeneous, mixed state without droplets [35]. A state withdroplets could always lower its free energy by shrinking thedroplets and thereby removing the free energy associated withsurface tension. Therefore, in the following we only considernonequilibrium situations in which detailed balance of thechemical reactions is broken and droplets can be stable.

C. Dynamics of active emulsions

We first consider the dynamics of an individual droplet withradius R. Two different volume fractions φB

− and φB+ coexist

inside and outside of the interface, respectively. The conditionsof coexistence are governed by local thermodynamic equilib-rium at the interface. This local equilibrium condition holds ifchemical reactions rates are sufficiently small. The chemicalpotentials of the droplet material are equal on both sides ofthe interface and the pressure difference between the insideand outside of the droplet is given by the Laplace pressure2γR−1. The equilibrium volume fractions inside and outsideof the interface can be determined from these thermodynamicprinciples, see Appendix A. They are given by

φB− ≈ ψ and φB

+(R) ≈ γβ

R, (6)

for small surface tension, γ � ψRβ−1. Here the coefficientβ = 2/(bψ) describes the effect of Laplace pressure on thevolume fraction at the interface. Note that γβ defines a lengthscale, which is related to the interface width by γβ = wψ/6.Additionally, the coexistence at the interface requires that φA

is continuous across the interface, see Appendix A, whileφC = 1 − φA − φB exhibits a discontinuity at the interface.

The volume fractions φA and φB vary on length scalesthat are large compared to the interface width w. We thusapproximate the Eqs. (3) by reaction-diffusion equations insideand outside the droplet,

∂tφA = DA∇2φA − s(φA,φB), (7a)

∂tφB = DB∇2φB + s(φA,φB), (7b)

where DA = mAa and DB = mBb are diffusivities.The volume fraction profile φB within a spherical droplet

can be discussed using Eqs. (7). The radial profile φB(r)varies on a characteristic length scale l = [DB/(kf + kb)]1/2,where we have used Eqs. (7) without the contributions ofthe autocatalytic reaction, because it does not lead to stronggradients inside the droplet, see Appendix B. In the following,we focus on the simple case where l is large compared tothe droplet radius R, i.e., chemical reactions rates are smallcompared to DBR−2. In this case droplets are homogeneousand we thus have φB � ψ inside the droplet, set by theequilibrium solution at the interface, see Eq. (6).

The volume fraction φB outside of droplets can also bediscussed for a spherical droplet. Here φB(r) exhibits thesame characteristic length scale l given above, since φB

is typically small and the contribution of the autocatalyticreaction is thus negligible. The total volume flux of compo-nent B passing a spherical shell at distance r is given byJ (r) = −4πr2DB∂rφ

B(r). In particular, we are interested inthe flux J = limε→0 J (R + ε) of droplet material outside ofthe interface. In a steady state, where φB(r) approaches thevolume fraction φB

0 at large r , we obtain

J = 4πRDB(1 + Rl−1)[φB

+(R) − φB0

], (8)

see Appendix C. Note that the term Rl−1 can be neglectedfor small droplets or small reaction rates (R � l). In thiscase and for DA ≈ DB , the volume fraction φA(r) will beapproximately homogeneous throughout the system and wethus approximate φA = φA

0 . These arguments also show thatdroplets are inhomogeneous if chemical reactions are fast

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DAVID ZWICKER, ANTHONY A. HYMAN, AND FRANK JULICHER PHYSICAL REVIEW E 92, 012317 (2015)

(R � l). However, as shown below these droplets are typicallyunstable.

We can now discuss the dynamics of N dropletsi = 1, . . . ,N of radii Ri that are sufficiently far from eachother that they do not interact directly. Droplet growth is drivenby the addition of B components to droplets. Changes in thedroplet volume Vi = 4π

3 R3i can thus be related to the flux Ji

and the chemical reactions inside the droplet. Combining thesetwo effects, the volume growth rate can be expressed as

ψdVi

dt= (

kfφA0 − kbψ + kφA

0 ψ)Vi + Qiφ

A0 − Ji, (9)

where we again consider small droplets or small reaction rates(0 < Ri � l). Here the term proportional to Vi accounts for thereactions in the bulk of the droplet and the term proportional toQi describes the activity of a catalytic core inside the droplet.The dynamics of the average volume fraction of B in thebackground fluid is given by

dφB0

dt≈ kfφ

A0 − kbφ

B0 + kφA

0 φB0 + 1

Vs − Vtot

N∑i=1

Ji, (10)

where

Ji = 4πRiDB(1 + Ril−1)

[φB

+(Ri) − φB0

], (11)

and Vtot = ∑i Vi is the total volume of all droplets. Here we

have neglected spatial correlations between φA and φB in theautocatalytic term. Note that the average volume fraction φA

0of building blocks reads

φA0 = φ − Vs − Vtot

VsφB

0 − Vtot

Vsψ, (12)

which results from the mass conservation given in Eq. (5).We thus arrive at an effective description of dynamic

droplets in the presence of chemical reactions. The state ofthe system is characterized by the droplet volumes Vi and thevolume fraction φB

0 in the background fluid. Their dynamicsare governed by Eqs. (10)–(12). The case of conventionalemulsions without chemical reactions is found in the limitkf = 0, kb = 0, k = 0, and Q = 0. This case, which we call thepassive case, has been well characterized and exhibits Ostwaldripening [38,39]. In the next two sections, we focus on theeffects of first-order and autocatalytic reactions on the dropletbehavior and compare these active cases to the passive one.

III. DROPLETS WITH FIRST-ORDER REACTIONS

Droplets under the influence of first-order reactions[kf > 0, kb > 0, k = 0, Qi = 0, see Fig. 2(a)] can be most eas-ily discussed in the case of a binary fluid. Therefore, we removecomponent C for this system, such that φA = 1 − φB . That is,we consider a system where components A and B undergochemical reactions and phase separate from each other. Theassociated free energy density is f (φB) = b

2 (φB)2(ψ − φB)2.The dynamics of the system is of the form of Eq. (3b), withφA = 1 − φB and s given in Eq. (4).

An emulsion of these droplets can be described by thedroplet radii Ri and the average volume fraction φB

0 in thebackground fluid. For the binary fluid, we can write the droplet

growth rate given by Eq. (9) as

dRi

dt= DB

(φB

0

Ri

− γβ

R2i

)− kbRi

3, (13)

where we consider small droplets (0 < Ri � l) that form bystrong phase separation, such that the volume fraction of A

components in the droplet is negligible (ψ = 1). The volumefraction φB

0 changes with a rate

dφB0

dt= kf

(1 − φB

0

) − kbφB0 + 1

Vs

N∑i=1

Ji (14)

if the total droplet volume is small compared to the systemvolume, Vtot � Vs, see Eq. (10). In the following, we analyzethese equations for one, two, and more droplets.

A. Dynamics of a single droplet

1. Large system size

We first investigate a single droplet in the thermodynamiclimit of a large system. Because of the large system size, thesingle droplet does not influence the volume fraction φB

0 inthe background fluid and φB

0 is thus constant. The growthdynamics of the single droplet can then be discussed byconsidering Eq. (13) only.

In a passive system where chemical reactions are absent(kf = kb = 0), the average volume fraction of component B

is conserved and φB0 is thus set by the initial condition. In

this case, a droplet grows indefinitely if it is larger than acritical size Rcrit, see the orange line in Fig. 3(a). This criticalradius corresponds to the single steady state of Eq. (13), whereR = Rcrit with Rcrit = γβ/φB

0 . This steady state is unstable,such that droplets with a radius larger than Rcrit grow, whilesmaller droplets shrink and disappear.

In the case of first-order reactions (kf > 0, kb > 0), theaverage fraction φB

0 in the background fluid is set by thebalance of the chemical reactions. Thus, we have

φB0 = kf/(kf + kb), (15)

which is the steady-state solution of Eq. (14) for large Vs. Inthis case, there exists a critical radius that is similar to thepassive case discussed above, see Fig. 3(a). In particular, theassociated steady state of Eq. (13) at R = Rcrit is given byRcrit ≈ γβ/φB

0 for small reaction rates. However, in contrastto the passive case, large droplets shrink in the presence ofchemical reactions according to Eq. (13). Consequently, thereexists a stable steady state with a droplet radius R larger thanRcrit, see Fig. 3(a). For large systems, the stable radius of asingle droplet reads

R ≈[

3DBkf

(kf + kb)kb

] 12

, (16)

which follows from Eq. (13) in steady state for small surfacetension, γ � RφB

0 /β. The radius R diverges for vanishingchemical reactions, such that a droplet in a passive systemgrows indefinitely, as discussed above. Conversely, R is finitein the presence of chemical reactions and faster reaction ratesgenerally lead to smaller droplets, see Fig. 3(b). Importantly,Eq. (16) is only valid if the droplet radius R is larger than the

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SUPPRESSION OF OSTWALD RIPENING IN ACTIVE . . . PHYSICAL REVIEW E 92, 012317 (2015)

(a) (b)

(c) (d)

FIG. 3. (Color online) Behavior of a single droplet with first-order reactions. (a) Growth rate R = dR/dt given in Eq. (13)as a function of the droplet radius R in an infinite system. Thepassive system (light orange, kf = kb = 0, φB = 1/11) is comparedto droplets with first-order kinetics (dark blue, kf/k0 = 3.6 × 10−5,kb/k0 = 3.6 × 10−4). The critical radius Rcrit and the stable radiusR are indicated. (b) Bifurcation diagram of the droplet radius as afunction of the rate constant kf with kb = 10 kf . R (solid line) andRcrit (dotted line) were determined by solving Eq. (13) numerically.The saddle-node bifurcation point given in Eq. (17) is marked bya blue dot. The triangle indicates the scaling predicted by Eq. (16).[(c) and (d)] Same plots as in (a) and (b) for a finite system withvolume Vs = 1.8 × 106 w3. The maximal droplet volume is Vsφ

B inthe passive system. The gray area in (d) denotes the region wheresystems are relatively small, Vs � (4π/3)[DB/(kf + kb)]3/2. In allpanels, ψ = 1 and quantities are normalized to the length scalew = 6γβ generated by the surface tension γ and the associated ratek0 = DB/w2.

critical radius Rcrit. This condition is violated if the reactionrates are too fast and droplets will shrink and disappear inthis case. Consequently, there exist upper bounds for the rateconstants above which droplets are not stable. To illustrate this,we vary the forward rate constant kf while keeping the ratiokf/kb (and thus also the average volume fraction φB

0 ) fixed. Inthis case, droplets do not form if kf exceeds the critical valuekc

f given by

kcf = 4DB

(3γβ)2

(φB

0

)4

1 − φB0

, (17)

which follows from considering Eq. (13) in steady state.Equation (13) is valid for droplets that are small com-

pared to the characteristic length l = [DB/(kf + kb)]1/2, asdiscussed above in Sec. II C. Therefore, Eq. (16) is onlyvalid if R/ l = (3kf/kb)

12 � 1 or, alternatively, φB

0 � 1/4.This condition corresponds to the case of B-rich droplets inan A-rich background. In the opposite case of a large averagevolume fraction of B, φB

0 � 3/4, the system forms A-richdroplets in a B-rich background. These A-rich droplets aredescribed by the same theory and thus have the size given byEq. (16) but with kf and kb interchanged. In the intermediatecase, where components A and B exist in approximately equal

average volume fractions (kf ≈ kb, φB0 ≈ 1/2), droplets may

not form and the effective description of emulsions is notapplicable. It has been reported that bicontinuous structuresare then prevalent in the solutions to Eq. (3b) [22,23]. Sincewe are interested in the dynamics of emulsions, we only discussthe case where B-rich droplets form (3kf � kb, φB

0 � 1/4).Taken together, droplets will grow indefinitely in pas-

sive, thermodynamically large systems. Conversely, first-orderchemical reactions can stabilize single droplets if kf < kc

f (φB0 )

and φB0 � 1/4. Combining these two necessary conditions,

the rate constant kf must be much smaller than the ratek0 = DB/w2 of diffusion across the interface for a singledroplet to be stable.

2. Finite systems

We next discuss the case of a single droplet in a systemof finite volume Vs. Here the average volume fraction φB

0 inthe background fluid changes with time, which influences thedroplet growth. In fact, the droplet volume must be smallerthan the volume Vsφ

B of B components in the system.In the passive case (kf = kb = 0), a single droplet grows

until an equilibrium with the background fluid is achieved.For a finite system, this occurs at a finite droplet radius R, seeFig. 3(c). In the limit of small surface tension and strong phaseseparation, the actual steady-state volume V is very close tothe upper bound and V ≈ Vsφ

B is thus a good approximation.Note that the average volume fraction φB of B in the systemis constant in passive systems.

In the case with chemical reactions (kf > 0, kb > 0), thereare two regimes for the droplet size, depending on how thesystem size Rs = (3Vs/4π )1/3 compares to the characteristiclength scale l generated by the reaction-diffusion system. Inlarge systems (Rs � l), a droplet grows until it reaches theradius R given by Eq. (16). Conversely, in small systems(Rs � l), the droplet depletes the background fluid of B

components significantly and its volume can be approximatedby V ≈ Vsφ

B , where φB = kf/(kf + kb). These two regimesfor the droplet size are observed in numerical solutions of thesteady states, see Fig. 3(d).

B. Dynamics of multiple droplets

We next study the dynamics of systems containing severaldroplets. The dynamics of these droplets is coupled becausethey compete for the material in the background fluid.

1. Dynamics of a droplet pair in a finite system

We first discuss the qualitative behavior of a droplet pair in afinite system. Starting with two droplets of generally differentvolume, the initial dynamics are such that the combined dropletvolume Vtot and the volume fractions in the background fluidapproach their steady-state values quickly, see Appendix D.Apart from small corrections due to surface tension effects,Vtot is then given by the stationary volume that a single dropletwould reach in the same system, which we discussed in theprevious section. Droplet dynamics on longer times involvesdiffusive transport of material between the droplets, as inconventional emulsions [39].

In the passive case (kf = kb = 0), the material exchangebetween droplets is driven by diffusive fluxes which originate

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DAVID ZWICKER, ANTHONY A. HYMAN, AND FRANK JULICHER PHYSICAL REVIEW E 92, 012317 (2015)

(a) (b)

FIG. 4. (Color online) Behavior of two droplets as a function oftheir radii R1 and R2. The black arrows indicate the temporal evolutionof the state variables R1 and R2 following from Eq. (13), with φB

0 givenby Eq. (14) in steady state. The medium blue and light orange linesare the nullclines, which indicate where the growth rate of droplets1 and 2 vanish, respectively. Their intersections are stable (disks) orunstable fixed points (open circles). (a) Passive system, kf = kb = 0,with φB = 1/11. (b) First-order reactions, kf/k0 = 1.4 × 10−4 andkb = 10 kf . Model parameters are ψ = 1 and Vs/w

3 = 5.8 × 104,where w = 6γβ is the length scale generated by the surface tension γ .

from the differences in the volume fractions φB+ = γβ/R on the

surface outside the droplets. In particular, φB+ is bigger for small

droplets than for large droplets. Consequently, the diffusivematerial flux is directed from smaller droplets toward largerones and thus amplifies initial size differences of droplets, aphenomenon called Ostwald ripening [8].

This behavior can be discussed most clearly by consideringa droplet pair in the case where the background fluid describedby Eq. (14) has reached a steady state. Figure 4 represents thedynamics of two droplet radii described by Eq. (13). Stableand unstable steady states are shown as full and open circles,respectively. An unstable steady state with two large dropletsof equal size exists. In the vicinity of this state, one dropletgrows at the expense of the other one. Additionally, there arethree stable steady states, which have at most a single droplet,and there are three unstable steady states with small droplets,which are related to the critical radius Rcrit. Passive dropletscan thus grow if they start with a radius larger than Rcrit, butat long times only a single droplet can be stable because ofOstwald ripening.

Importantly, Ostwald ripening can be suppressed if chem-ical reactions are occurring (kf > 0, kb > 0). In this case, thesteady state with two coexisting large droplets of equal sizecan become stable as shown in Fig. 4(b). In the vicinity of thissteady state, the smaller droplet grows at the expense of thelarger one. In the case where Ostwald ripening is suppressedby chemical reactions, there are still stable steady states with asingle droplet or even no droplets, see Fig. 4(b). Droplets haveto overcome a critical radius in order to be able to grow. Notethat the critical radius is similar to the one in the passive case.This is consistent with our result obtained for single droplets,see Fig. 3.

2. Stability conditions for multiple droplets

In order to discuss the stability condition for coexistingdroplets of equal size, we consider the general case of N

droplets with same stationary radius R. The value of R dependson the number of droplet, the system size, and other parametersas discussed below. Starting from such a steady state, thestability of this state is governed by the slow exchange ofmaterial between the droplets, while the total droplet volumeand the background fluid quickly reach their stationary state,see Appendix D. The slowest rate of relaxation to the steadystate is given by

λ = DBγβ

R3− 2kb

3, (18)

see Appendix D. Note that λ is independent of the dropletcount, because our effective theory only captures the mean-field coupling of the droplet dynamics. Importantly, λ canbe either positive or negative. For λ > 0, the steady state isunstable and larger droplets grow at the expense of smallerdroplets. In this case, the system exhibits Ostwald ripeningand coarsens over time. For λ < 0, the steady state is stableand Ostwald ripening is suppressed.

In the passive case (kf = kb = 0), the characteristic ratesimplifies to λ = DBγβR−3 [14]. Importantly, λ is positiveand Ostwald ripening always occurs. Furthermore, systemswith larger average droplet size coarsen slower, which isrelated to the well-known Lifshitz-Slyozov kinetics of dropletcoarsening [39]. In order to verify Eq. (18), we simulatedroplets by solving Eqs. (3) numerically and measure theircoarsening rates, see Fig. 5(a) and Appendix E. Figure 5(b)shows that the analytical expression of λ can account for theobserved coarsening in passive systems.

In the case of first-order reactions (kf > 0, kb > 0), thecharacteristic rate λ can become negative and the steady statewith multiple droplets can be stable. In particular, λ is negativeif the radius of the droplets is larger then the threshold value

Rstab =(

3DBγβ

2kb

) 13

, (19)

which follows from Eq. (18). Therefore, multiple droplets canbe stable only if the stationary radius R of the individualdroplets is larger than Rstab. Whether this is the case dependson the droplet number density n = N/Vs.

In a state with low droplet density, n � l−3, droplets donot influence the volume fractions in the background fluidsignificantly, and the droplet radius R is given by Eq. (16).In this case, R is indeed larger than Rstab, because kf must besmaller than the critical value given in Eq. (17) for droplets toexist. Consequently, droplets are stable and Ostwald ripeningis suppressed in this case.

In the case of a large droplet number density, n � l−3, thestationary size of individual droplets is smaller as comparedto the low-density system. This is because the total dropletvolume is limited by the total volume Vsφ

B of droplet materialin the system, where φB = kf/(kf + kb). Using the estimateVsφ

B for the volume of a single droplet in a small system andconsidering that this volume must be distributed among the N

droplets, we have

V ≈ kfVs

(kf + kb)N. (20)

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SUPPRESSION OF OSTWALD RIPENING IN ACTIVE . . . PHYSICAL REVIEW E 92, 012317 (2015)

104 105 106

Norm. Average Volume–V /w3

10−6

10−5

10−4

10−3

NumericsEq. (16)

10−5 10−4 10−3

Normalized Backward Rate kb/k0

−2

−1

0

NumericsEq. (16)–(17)

0 5000102

103

1

1

(c) (d)

0 5000102

103

0 50000

2

4

6

0 50000

1

2

3

(a) (b)

FIG. 5. (Color online) Numerical simulations of the dynamics oftwo droplets. (a) Illustration of the numerical procedure for a passivesystem (kf = kb = 0, φB = 1/11). We numerically solve Eq. (3) ina cylindrical geometry as described in Appendix E. Snapshots (toppanel) are used to measure the droplet volumes Vi as a function oftime t (lower left panel). The rate λ is determined by a linear fitto the logarithm of the volume difference �V (lower right panel).(b) Coarsening rate λ as a function of the average droplet volumeV for kf = kb = 0 with φB = 1/11. The analytical prediction (solidline) given by Eq. (18) is compared to numerical results (symbols)for Vs/w

3 = 1.5 × 105 (discs) and Vs/w3 = 1.2 × 106 (squares).

(c) Same as (a) for kb/k0 = 5 × 10−3 and kf/k0 = 5 × 10−4. (d) Rateλ as a function of the backward rate constant kb with kf = 0.1 kb.The analytical prediction given by combining Eqs. (18) and (20)(solid line) is compared to numerical results (orange disks) forVs/w

3 = 1.5 × 105.

The steady state of N droplets of size R = [3V /(4π )]1/3

is stable if the corresponding relaxation rate λ given byEq. (18) is negative. This is the case for sufficiently large rateconstant kb.

This change of stability as a function of the reaction ratesis shown in Figs. 5(c) and 5(d), which display simulationresults for two droplets with first-order reactions. Thesesimulations show that the measured relaxation rates agreewith the analytical expression of λ given in Eq. (18). Wecan thus use Eq. (18) to determine parameter values at whichmultiple droplets are stable. This can be summarized in a singledimensionless stability number,

χ = Vs

2πDBγβN· kfkb

kf + kb, (21)

which is defined such that χ > 1 corresponds to λ < 0 forthe droplet volume given by Eq. (20). Consequently, a steadystate with N droplets is stable if χ is larger than 1. Equation(21) thus allows us to estimate the maximal number Nmax ofdroplets that can be stable at given parameter values. Generally,more droplets can be stable for faster chemical reactions andthe estimate following from the condition χ > 1 agrees wellwith numerical results, see Fig. 6(a). However, droplets cannotbe stable for rate constants kf above the maximal rate kc

f givenin Eq. (17). The value of Nmax thus vanishes as this point isapproached, see Fig. 6(a).

10−3 10−2 10−1

Norm. Forward Rate kf/k0

10−3

10−2

10−1

10−4

10−3

10−2

10−5 10−4 10−3

Normalized Forward Rate kf/k0

1

10

Nm

ax

(a)

1

1

(b)

hom

ogen

eous

stat

e un

stable

homogeneous,A-rich system

homogeneous,B-rich system

B-droplets

A-d

ropl

ets

FIG. 6. (Color online) Stability of multiple droplets with first-order reactions. (a) The maximal number Nmax of stable droplets(blue line) as a function of the forward reaction rate constant kf

with kb = 10 kf in a finite system of volume Vs = 2.3 × 105 w3. Nmax

is obtained from a numerical linear stability analysis of Eq. (13).The theoretical stability boundaries χ = 1 (left black line) givenby Eq. (21) and kf = kc

f (right black line) given by Eq. (17) areindicated. (b) Stability of possible states in an infinite system as afunction of the reaction rate constants kf and kb. The homogeneousstate φB (r) = φB

0 with φB0 = kf/(kf + kb) is unstable and bicontinous

structures typically form in the orange hatched region bounded bythe stability condition given in Appendix F (dashed line). Dropletsenriched in B components are stable in the upper blue region, whichis defined by 3kf < kb < 16k0(φB

0 )3, following from Eq. (17). Thecolor indicates the maximal droplet density nc, which follows fromEq. (21) for χ = 1. Correspondingly, A-rich droplets are stable in thelower blue region. The homogeneous state φB (r) = φB

0 is the onlystable state in the white region.

3. Stability of multiple droplets in large systems

In passive systems, droplets undergo Ostwald ripening withthe typical Lifshitz-Slyozov kinetics, where the mean dropletradius 〈R〉 evolves as 〈R〉 ∝ (tDB/w2)1/3 [39]. In the casewhere first-order reactions are occurring, we find that thebehavior depends on the number density n = N/Vs of dropletsin the system. In the case where n is larger than a critical valuenc following from Eq. (21) for χ = 1, large droplets growand smaller ones disappear until n falls below nc. If n < nc,the system evolves toward a stable steady state in which alldroplets have the same size. The size of these stationarydroplets depends on the number density n. If the density islow, n � l−3, the size of each droplet is given by Eq. (16). Forlarger number densities, l−3 � n < nc, droplets are smallerthan in the case of low n and their size is given by Eq. (20).Note that faster chemical reactions typically lead to a larger nc,but droplets cannot be stable beyond the critical reaction rategiven by Eq. (17). This is illustrated in Fig. 6(b), which showsthe parameter region in which B-rich droplets are stable. Thefigure also shows the corresponding region of stable, A-richdroplets. Between these two regions, the homogeneous stateis unstable, see Appendix F. Consequently, structures formspontaneously and critical droplet radii do not exist. In fact,bicontinous structures have been reported for the symmetriccase kf = kb [22,23].

IV. AUTOCATALYTIC DROPLETS

So far, we considered droplet dynamics under the influenceof first-order reactions. In the context of biology, chemicalreactions are usually more complex. In this section, westudy an autocatalytic system in which the droplet materialserves as a catalyst for its own production. One interesting

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DAVID ZWICKER, ANTHONY A. HYMAN, AND FRANK JULICHER PHYSICAL REVIEW E 92, 012317 (2015)

aspect of autocatalytic systems is that droplets are difficult tonucleate, because droplet material can be produced only frompreexisting droplet material. We therefore introduce activecores that catalyze the production of droplet material and cannucleate autocatalytic droplets. Such a system can serve as amodel for the growth of centrosomes [5].

Autocatalytic systems are typically unstable when binaryfluids are considered. This is because the entire system can beturned into droplet material by the autocatalytic reaction. Wethus discuss autocatalytic droplets in the framework of ternaryfluids with a nonreactive component C introduced in Sec. II.

For an autocatalytic droplet of radius Ri , the growth rate ofthe droplet volume Vi reads

dVi

dt= 4πDB

ψ

(φB

0 Ri − γβ) + Vi

(ksφ

A0

ψ− kb

)+ Qiφ

A0

ψ,

(22)

where ks = kf + kψ , see Eq. (9). This equation is accompaniedby Eqs. (10) and (12), which describe the dynamics of thevolume fractions φA

0 and φB0 outside of droplets.

A. Dynamics of a single autocatalytic droplet

We first investigate a single, autocatalytic droplet (kf = 0,kb > 0, k > 0, Q = 0), see Fig. 2(b). In systems with a finitevolume Vs, such a droplet can reach a stationary state with avolume V . This volume is smaller than the system size andreads

V ≈ Vs

ψ

(φ − kb

k

), (23)

which is valid for small surface tension, see Appendix B.This expression shows that a sufficiently fast autocatalyticrate is required (k > kb/φ) for droplets to grow. Note thatthe stationary volume V scales with the system size and thatthe autocatalytic reaction does not set a characteristic dropletssize, in contrast to the first-order reactions discussed above.Autocatalytic droplets thus behave more like passive droplets,which also grow up to a certain fraction of the system size.

Similarly to the previously discussed systems, an autocat-alytic droplet only grows if it is larger than a critical radiusRcrit, see Fig. 7(a). This critical radius can be estimated byRcrit ≈ γβ/φB

0 , where the fraction φB0 in the background fluid

is given by φB0 = φ − kb/k. Thus, the critical radius diverges

if k = kb/φ, where droplets no longer grow. In an autocatalyticsystem, the volume fraction φB

0 of droplet material B is lowerthan in a passive system and the critical radius is larger.Droplets in autocatalytic systems are thus harder to nucleatethan in passive systems.

Interestingly, nucleation can be facilitated and even trig-gered reliably by an active core, which catalyzes the productionof droplet material B. The critical radius vanishes for alarge-enough catalytic activity Q, see Fig. 7(a). Consequently,nucleation around an active core is guaranteed if Q is largerthen a critical value Qc. Considering Eq. (22) in the steadystate for small droplets, we obtain Qc ≈ 4πDBγβ/φA

0 withφA

0 = kb/k following from Eq. (10) for kf = 0. Importantly,catalytically active cores strongly influence the nucleationbehavior of autocatalytic droplets, but their effect on the steadystate size is small, see Fig. 7 and Appendix B.

(a) (b)

FIG. 7. (Color online) Behavior of a single autocatalytic dropletin an infinite system (orange; Q1 = 0), in a finite system (blue;Vs/w

3 = 6.9 × 107, Q1 = 0), and in a finite system with a cat-alytically active core (green; Vs/w

3 = 6.9 × 107, Q1 = 300 w3k0).(a) Growth rate R = dR/dt derived from Eq. (22) as a function ofthe droplet radius R for φ = 0.02. The critical radius Rcrit and thestable radius R derived from Eq. (23) are indicated. (b) Bifurcationdiagram of the droplet radius as a function of the total volume fractionφ of material. Dotted lines indicate unstable steady states. Modelparameters are k/k0 = 0.02, kb/k0 = 2 × 10−4, and ψ = 0.1, withk0 = DB/w2.

B. Dynamics of multiple autocatalytic droplets

We next study the dynamics of systems containing severalautocatalytic droplets. In the simple case without catalyticcores (Qi = 0), large droplets grow at the expense of smallerdroplets and states with multiple droplets are unstable. This canbe seen explicitly by considering two autocatalytic droplets ina finite system.

This system possesses a steady state with two large dropletsof the same size, which is unstable, see Fig. 8(a). In fact, atmost a single droplet can be stable in this system. Furthermore,there are critical radii, which droplets have to overcome inorder to grow. Consequently, the dynamics of two autocatalyticdroplets is qualitatively very similar to two droplets in apassive system, compare Fig. 4(a) and Fig. 8(a). In contrast

0 50 1000

50

100

0 50 1000

50

100

(a) (b)Without Catalytically Active Cores With Catalytically Active Cores

FIG. 8. (Color online) Behavior of two autocatalytic droplets asa function of their radii R1 and R2. The black arrows indicatethe temporal evolution of the state variables R1 and R2 followingfrom Eq. (22). The medium blue and light orange lines are thenullclines, which indicate where the growth rate of droplets 1 and 2vanish, respectively. Their intersections are stable (disks) or unstablefixed points (open circles). (a) No cores, Qi = 0. (b) Catalyticallyactive cores, Qi = 300 w3k0. The volume fractions φA

0 and φB0

are fixed to their stationary state value and model parameters arekf = 0, kb/k0 = 2 × 10−4, k/k0 = 0.02, ψ = 0.1, φ = 0.02, andVs/w

3 = 6.9 × 107.

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SUPPRESSION OF OSTWALD RIPENING IN ACTIVE . . . PHYSICAL REVIEW E 92, 012317 (2015)

to the reactions with first-order kinetics discussed above, theautocatalytic reaction thus does not suppress Ostwald ripening.

Interestingly, the two autocatalytic droplets can be stabi-lized by catalytically active cores (Qi > 0), see Fig. 8(b). In theshown case of equal catalytic activity, Q1 = Q2 = 300w3k0,the smaller droplet grows at the expense of the larger dropletuntil both droplets reached the same size. This is because thematerial influx caused by an active core is more importantfor small droplets than for large ones and thus promotes thegrowth of small droplets. Thus, for sufficiently large Q, thesmall droplet can grow at the expense of the large dropletuntil both have the same stable size. Interestingly, there are nocritical radii in this case and the single stable state is reachedfrom all initial conditions. This is because the catalyticallyactive cores not only stabilize the two droplets but they alsotrigger their nucleation, as discussed above. Consequently, thenumber of stable droplets will be set by the number of activecores.

These results can be generalized to multiple dropletswith catalytic cores of equal activity (kf � 0, kb � 0, k � 0,Qi = Q � 0). A linear stability analysis of Eq. (22) revealsthat the total droplet volume Vtot and the volume fractions φA

0and φB

0 in the background fluid quickly reach their steady-statevalues, while droplets exchange material on a slower timescale, similarly to the case of a binary fluid discussed above.The rate λ at which droplets exchange material reads

λ ≈ 4πDBγβ

3ψV+ 2J

3ψV− Q

V k

(kb + J

ψV

), (24)

where J is given by Eq. (11) for Ri = R. The rate λ

corresponds to the slowest relaxation time to reach a steadystate with multiple droplets of equal volume V for fixed φA

0and φB

0 , see Appendix G. A steady state with multiple dropletsis stable if λ is negative, which is the case either for a largeflux of B components into the droplets, J < 0, or for largeQ. These cases correspond to fast first-order reactions andstrong active cores, respectively. In particular, we find Eq. (18)for ψ = 1 in the case of first-order reactions (kf > 0, kb > 0,k = 0, Q = 0), where J = −kbψV .

In the case of an autocatalytic reaction (kf = 0, kb > 0, k > 0,Q � 0) droplet material B is predominately produced insidedroplets. In the stationary state, this production is balanced by asmall efflux J > 0 of droplet material. This droplet material isthen recycled into soluble building blocks outside of droplets.This process leads to a small volume fraction φB

0 < φB+ , where

we have φB0 ≈ φB

+ in small systems (Vs/N � l3) and φB0 ≈ 0

in large systems (Vs/N � l3), with l = (DB/kb)1/2. Becauseof the constant efflux of droplet material, autocatalytic dropletswithout a catalytic core (Q = 0) will coarsen even fasterthan droplets in a passive system, compare Eqs. (24) and(18). In particular, the coarsening should be fastest in largesystems, where J is maximal. In order to test this prediction,we measure λ from numerical solutions of Eqs. (3), seeFig. 9(a). Indeed, both smaller droplets and larger systemslead to faster coarsening and Eq. (24) gives a good estimateof λ. Importantly, the rate λ is always positive, indicatingthat autocatalytic droplets are unstable and exhibit Ostwaldripening.

0 0.01 0.02

Total Volume Fraction φ

0

100

200

300

0

1

2

3

4

Dro

ple

tCou

nt

Nm

ax

104 105

Norm. Average Volume–V /w3

10−5

10−4

10−3 NumericsEq. (21)

(a) (b)

1

1

multipledroplets

singledroplet

nodroplets

FIG. 9. (Color online) Behavior of emulsions of autocatalyticdroplets. (a) Rate λ as a function of the average droplet volumeV . The analytical prediction given by Eq. (24) for small systems(φB

0 = φB+ , lower blue line) and large systems (φB

0 = 0, upper blueline) is compared to numerical results for small (orange disks,Vs/w

3 = 1.5 × 105) and large (orange squares, Vs/w3 = 8.8 × 105)

systems, which were obtained as described in Fig. 5. Model param-eters are ψ = 0.1, k/k0 = 100, kb/k0 = 0.01. (b) Maximal numberNmax of stable droplets as a function of the total volume fraction φ ofmaterial and the catalytic activity Qi of cores within the droplets. Thered dashed line denotes the approximate threshold value of Qi givenin Eq. (25) with φB

0 ≈ 0 valid for large systems. Model parametersare ψ = 0.1, k/k0 = 0.02, kb/k0 = 2 × 10−4, Vs/w

3 = 6.9 × 107.

Ostwald ripening of autocatalytic droplets can be sup-pressed by catalytically active core (Q > 0), see Fig. 8(b).In particular, autocatalytic droplets are stable if Q exceeds thethreshold value

Qstab ≈ 4πDBksR(3φB

+ − 2φB0

)3ψkb

, (25)

which follows from Eq. (24). Note that Qstab becomesindependent of the droplet radius R in the case of large systemswhere φB

0 ≈ 0. Figure 9(b) shows that there is a critical valueQstab above which multiple droplets are stable. Equation (25)gives a good estimate for this threshold if the average fractionφ is large enough. Generally, the stability threshold Qstab islarger than the critical activity Qc that is required for dropletnucleation. This can be seen by considering Eq. (25) in thelimit φB

0 ≈ 0 in the parameter regime where droplets form(k > kb/φ). Consequently, catalytically active cores that arestrong enough can both nucleate and stabilize autocatalyticdroplets.

Finally, we consider autocatalytic droplets with cores ofunequal catalytic activities (kf = 0, kb > 0, k > 0, Qi > 0).In this case, the droplet around the core with stronger catalyticactivity grows faster, see Eq. (22). This difference in the growthrates can be observed in the simple case of two droplets, seeFig. 10(a). The figure shows that both catalytic cores nucleatedroplets and that the droplet with larger core activity growsmore quickly. Similarly to the two droplets with equal catalyticcores shown in Fig. 8(b), there is a single stable steady state,which is reached from all initial conditions. However, contraryto the symmetric system, the steady-state droplet volumes arenot equal, but the droplet with the core with higher activitybecomes larger. This can be understood by considering thegrowth rates given in Eq. (22) in the simple case of vanishingsurface tension γ . In this case, the fraction φB

0 of dropletmaterial in the background fluids vanishes, see Eq. (10), andthe ratio of the two droplet volumes is determined by theratio of their catalytic activities. Note that if only one of the

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DAVID ZWICKER, ANTHONY A. HYMAN, AND FRANK JULICHER PHYSICAL REVIEW E 92, 012317 (2015)

0 200 400 600 800

Catalytic Activity Q1/(w3k0)

0

2

4

6

V1 + V2

V1

V2

0 50 1000

50

100

(a) (b)

FIG. 10. (Color online) Behavior of two autocatalytic dropletswith cores of unequal activity Qi . (a) Temporal evolution of thedroplet radii R1 and R2 (black lines) following from Eq. (22) forQ1 = 1000 w3k0 and Q2 = 300 w3k0. The medium blue and lightorange lines are the nullclines, which indicate where the growth rateof droplets 1 and 2 vanish, respectively. Their intersection (disk)is the only fixed point, which is stable. (b) Droplet volumes Vi insteady state as a function of the catalytic activity Q1 of one core forQ2 = 300 w3k0. In both panels, φA

0 and φB0 are fixed to their stationary

state values and model parameters are kf = 0, kb/k0 = 2 × 10−4,k/k0 = 0.02, ψ = 0.1, φ = 0.02, and Vs/w

3 = 6.9 × 107.

catalytic activities is larger than Qstab while the other one issmaller, only a single droplet is stable, see Fig. 10(b). Thecatalytic cores thus can determine how the droplet material isdistributed among the droplets, but they only have a weak effecton the total droplet volume, which is still well approximatedby Eq. (23).

This behavior has been observed in experiments on cen-trosomes, which can be described as autocatalytic dropletswith active cores [5]. In this case two cores of equal catalyticactivities occur. However, experiments in which one coreis perturbed lead to unequal centrosomes which can beunderstood as a result of unequal catalytic activities [5].

V. DISCUSSION

In practical terms Ostwald ripening inhibits the use ofliquid droplets in many circumstances. This is because largerdroplets tend to grow at the expense of smaller droplets. Asa result, droplets size varies and is hard to control. Therefore,it is very important to find mechanisms that suppress theripening of such liquid emulsions. In this work we demonstratethat emulsions can be stabilized if the constituents of thephase separating fluids are converted into each other bynonequilibrium chemical reactions.

The stability of active emulsions depends on the rate of thechemical reactions. In the limit of vanishing reaction rates,Eq. (3b) reduces to the Cahn-Hilliard equation, such that theactive emulsions behave like their classical, passive counter-parts and undergo Ostwald ripening [33,38]. For intermediatereaction rates, Ostwald ripening is typically suppressed andmultiple droplets are stable. Large reaction rates lead to thebreakup of droplets.

In the simple case of first-order reactions, droplet material isproduced outside of droplets and converted back into solublebuilding blocks inside the droplets. This causes a materialinflux that scales with the droplet radius R, see Eq. (11),while the efflux scales with the droplet volume V ∝ R3, see

Eq. (9). Consequently, large droplets have a tendency to shrink,because the material efflux dominates the influx. The influxand efflux are balanced at a specific droplet size at whichdroplets are stationary. This stationary droplet size depends onthe chemical reaction rates.

Interestingly, we show in this case that the droplet radiusabove which multiple droplets are stable is proportionalto k

−1/3b , see Eq. (19). Note that in the symmetric case

kf = kb the same scaling was also found for the length scaleof bicontinuous structures [22] and for concentric circularpatterns [28]. We find an additional scaling regime in thecase of small droplet densities, where the stable radius isproportional to k

−1/2b for constant kf/kb, see Eq. (16). In both

scaling regimes, the droplet size in the active emulsion canbe controlled by adjusting the rate constants of the first-orderreactions.

Our theory can also be applied to systems with morecomplex chemical reactions. As an example, we consider thecase of second-order autocatalytic reactions, where dropletsproduce their own material. These autocatalytic droplets cangrow at rapidly increasing rates until they have used up thematerial provided in their environment. In the steady state,the material fluxes are then opposite to those in the case offirst-order reactions. Because of this, Ostwald ripening occursat rates that can be even larger than in systems without chemicalreactions. Furthermore, autocatalytic droplets are difficult tonucleate, because they require preexisting droplet material togrow.

Our analysis shows that Ostwald ripening can be suppressedin autocatalytic droplets if catalytic cores are introduced.This is because the catalytic activity facilitates the growthof small droplets while having little effect on large droplets.Active cores therefore allow the control of stability, nucleationbehavior, and the relative size of autocatalytic droplets, whilethe sum of all droplet volumes is mainly determined by theconserved total volume of A and B components in the system.

Our work shows that first-order chemical reactions orsecond-order reactions with active cores allow for emulsions inwhich several droplets are stable. The concept of autocatalyticdroplets with catalytic cores has been exploited to accountfor the growth behaviors of centrosomes [5]. Centrosomes arecollections of proteins that are required for cell division. It isimportant for correct cell division that each cell contains twocentrosomes of the same size. Although it has long been knownthat centrosomes grow from centrioles, it has been unclear howcentrioles control nucleation and growth of centrosomes. Ourwork suggests that centrioles act as active cores nucleatingand stabilizing the growth of centrosomes, thereby suppressingOstwald ripening and leading to two centrosome droplets ofequal size in the cell that coexist during cell division.

More generally, recent work has shown that cells containchemically distinct compartments that form and are main-tained by liquid-liquid demixing [4]. Examples are P granulesand nucleoli among others, which can be viewed as liquidlikedroplets in the cell [40]. In each of these cases the numberand size of droplets is probably controlled by the cell, whichcould be done using the effects of chemical reactions discussedhere. Furthermore, droplet dynamics associated with sizecontrol and the turnover of the constituents may play a rolein preventing droplets to turn into stable protein aggregates,

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SUPPRESSION OF OSTWALD RIPENING IN ACTIVE . . . PHYSICAL REVIEW E 92, 012317 (2015)

which are often linked to diseases [41]. It will be interestingto see whether the principles enunciated in this paper apply tothese and other liquidlike compartments in cells.

Controlling the properties of emulsions is also important inmany technological applications. It would thus be interestingto conduct experiments where Ostwald ripening is suppressedby chemical reactions. As we show here, these reactionsmust be driven away from equilibrium, which could beachieved, for example, by driving photochemical reactionsby light [21,42]. Alternatively, a system could be coupled viaa semipermeable membrane to a source of fast-diffusing fuelmolecules that drive the chemical reactions between the dropletcomponents.

Active emulsions stabilized by chemical reactions are anexample of pattern formation far from equilibrium. Thereis some similarity with the formation of Turing patterns inreaction-diffusion systems [43]. Note, however, that our workgeneralizes the concept of Turing patterns by combining con-ventional phase coexistence with chemical pattern formation.In this case, spatial patterns stemming from reactions anddiffusion of chemical species are combined with the formationof distinct phases by physical interactions between molecules.The interfaces of chemically active droplets are thus relatedto thermodynamics and very different from interface-likestructures in Turing patterns. As a consequence, the systemsdescribed in this manuscript can exhibit stable emulsions inparameter regions in which conventional reaction-diffusionsystems would not form patterns.

Ostwald ripening can also be suppressed by other nonequi-librium mechanisms like shearing the fluid continuously[17,18]. However, while shearing drives the system from theboundary, our system breaks detailed-balance locally and thusbelongs to the class of active matter. The advantage of oursystem is that it reaches a stationary state with sphericaldroplets of a well-defined size. Such uniform emulsions canalso be stabilized by inclusion of additional components insidedroplets, which are insoluble in the background fluid [16].However, large droplets, which could, for instance, form dueto droplet coalescence, would still grow and Ostwald ripeningmight still happen in such systems. In contrast, our theorypredicts that all large droplets are unstable and shrink backto the single stable size. In our model, emulsions thus can bestable despite large fluctuations caused by droplet coalescenceand droplet nucleation. These effects will merely broaden thedroplet size distribution.

The precise influence of droplet coalescence in our modelcan only be studied if it is extended to take into accounthydrodynamic effects, which are known to influence the dy-namics of reactive binary mixtures [44,45]. Such an extendedtheory may also be necessary to compare predicted dropletsizes and rates of growth dynamics to experiments, wherefluid convection and Brownian motion of droplets are hard toeliminate. Moreover, adding hydrodynamics would allow usto study the interaction of active droplets.

ACKNOWLEDGMENTS

We thank Christoph Weber and Rabea Seyboldt for helpfuldiscussions and a critical reading of the manuscript. F.J. andA.A.H. acknowledge funding from the European Commu-

nity’s Seventh Framework Programme (FP7/2007-2013) grantagreement 241548 (MitoSys Project).

APPENDIX A: COEXISTENCE IN A TERNARY FLUID

The conditions for equilibrium at the interface between twocoexisting phases read

0 = μA(φA−,φB

−) − μA(φA+,φB

+), (A1a)

0 = μB(φA−,φB

−) − μB(φA+,φB

+), (A1b)

0 = (φA− − φA

+)μA(φA−,φB

−) + (φB− − φB

+)μB(φA−,φB

−)

+ f (φA+,φB

+) − f (φA−,φB

−) − 2γH , (A1c)

where φi− and φi

+ for i = A,B denote the volume fractions atthe interface inside and outside the droplet, respectively. Hereμi(φA,φB) = ∂f (φA,φB)/∂φi , γ denotes the surface tension,and H is the mean curvature of the interface, with H = R−1

for a sphere of radius R. For the free energy density given inEq. (1), Eq. (A1a) yields φA

+ = φA−. The volume fractions of

B, which result from Eqs. (A1), can be expanded to first orderin H :

φB− ≈ ψ + βγH and φB

+ ≈ βγH, (A2)

where β = 2/(bψ). Equation (A2) also holds for binary fluids(φC = 0), where the volume fraction of building blocks isgiven by φA

± = 1 − φB± .

APPENDIX B: AUTOCATALYTIC DROPLETSAT STEADY STATE

We consider a single, autocatalytic droplet of radius R withan infinitely thin interface (k > 0, kb > 0, kf = 0, Qi = 0).Away from the interface, the volume fractions φA(r) andφB(r) are governed by the reaction-diffusion system given inEqs. (7). For simplicity, we consider a spherically symmetricsystem around the droplet. Inside the droplet (r < R), we havethe boundary conditions φA(R) = φA

0 and φB(R) = ψ , seeEq. (6), and the steady-state solution to Eqs. (7) thus readsφA(r) = kb/k and φB(r) = ψ in this region. Outside of thedroplet (r > R), we have φA(r) = kb/k and φB(r) = γβ/R.Importantly, the volume fraction φB(r) is homogeneous insidethe droplet. Note that this argument also holds for states withmultiple droplets and the contribution of the autocatalyticreaction to the typical length scale l thus can generally beneglected.

We next discuss the steady state of a single autocatalyticdroplet including surface tension effects, γ > 0, and catalyt-ically active cores, Q > 0. For small droplets (R � l), thedynamics of a single autocatalytic droplet in a finite systemare described by Eq. (22) together with Eqs. (10) and (12). Inthe steady state, the droplet absorbs almost all droplet materialB, such that the average volume fraction φB

0 in the backgroundfluid is negligible, φB

0 = 0. In this case, we can solve Eq. (12)for φA

0 . Inserting this solution into Eq. (22) in steady state leadsto a quadratic equation for the stationary droplet volume. Onlykeeping the linear contributions of the catalytic activity Q andthe surface tension γ , the solution corresponding to the stable

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DAVID ZWICKER, ANTHONY A. HYMAN, AND FRANK JULICHER PHYSICAL REVIEW E 92, 012317 (2015)

droplet reads

V ≈ Vs

ψ

(φ − kb

k

)+ Qk2

b − 4πDBk2φγβ

(kφ − kb)kkbψ. (B1)

The second solution to the quadratic equation is unstable andcorresponds to the critical radius.

APPENDIX C: SURFACE FLUXESOF DROPLET MATERIAL

We can obtain a simplified expression for the fluxJ = −4πR2DBφB′(R) of droplet material at the dropletinterface for steady states of the volume fraction φB(r) ina spherical geometry. At the droplet interface at r = R, wehave φB(R) = φB

+(R). For large r � R, φB(r) approaches thebulk volume fraction φB

0 . Using Eq. (7b) with a vanishing timederivative, we have

φB(r) = φB0 + (

φB+ − φB

0

)R

re(R−r)/l, (C1)

where l = [DB/(kf + kb)]1/2 is a characteristic length and weneglected the autocatalytic reaction outside the droplet whereφB is small. The flux J at the interface is then given by Eq. (11).

APPENDIX D: STABILITY ANALYSISOF BINARY EMULSIONS

The dynamics of an emulsion of N droplets with first-orderkinetics are given by Eqs. (13) and (14). These equations have asteady state where all droplets have the same radius R and thevolume fraction of droplet material in the background fluidis given by φB

0 . The stability of this state can be investi-gated by considering small perturbations Ri(t) = R + Ri(t)and φB

0 (t) = φB0 + φB

0 (t) with Ri � R and φB0 � φB

0 . Theassociated dynamics to linear order read

dRi

dt= DB

(φB

0

R− φB

0 Ri

R2+ 2γβRi

R3

)− kbRi

3, (D1)

dφB0

dt= −(kf + kb)φB

0 − 4πDB

Vs

N∑i=1

(RφB

0 + φB0 Ri

). (D2)

Defining a vector X with X0 = φB0 and Xi = Ri for

i = 1, . . . ,N , this can be written as dXi/dt = ∑j JijXj ,

where the components of the Jacobian J are

J00 = −kf − kb − 4πDBNRV −1s , (D3a)

J0i = −4πDBNφB0 V −1

s , (D3b)

Ji0 = DBR−1, (D3c)

Jii = DB

(2γβ

R3− φB

0

R2

)− kb

3, (D3d)

Jij = 0, (D3e)

where i �= j and i,j � 1. The eigenvalues e of J read

e± = 12 [J00 + Jii ±

√(J00 − Jii)2 + 4NJ0iJi0], (D4a)

e = Jii , (D4b)

where the eigenvalue e has multiplicity N − 1 and onlyexist for systems with multiple droplets, N � 2. The eigen-values are ordered, e− < e+ < e, since J0iJi0 < 0 andJ00 < Jii . This can be seen by rewriting Eq. (D3d) asJii = γβDBR−3 − 2kb/3, where we used φB

0 resulting fromsolving Eq. (13) in steady state. Consequently, a system withmultiple droplets can be stable only if e < 0 and it is thussufficient to discuss the value of Jii , see Eq. (18). Note thatthe eigenspace associated with e describes dynamics whereboth φB

0 and Vtot are constant and droplets merely exchangedmaterial with each other.

APPENDIX E: NUMERICAL CALCULATIONS

We study the dynamics of two droplets in a finite system bynumerically solving Eq. (3) in a three-dimensional, cylindricalgeometry. Starting with an initial configuration that representstwo droplets of slightly different volume, we determinedthe temporal dynamics using the XMDS2 framework [46]with an adaptive Runge-Kutta-Fehlberg time stepper and animplementation of the spatial derivatives using a spectralmethod with 128 and 32 support points in the axial andradial direction, respectively. The coarsening rate λ is thendetermined by a linear fit to the logarithm of the differenceof the two droplet volumes, which are determined from thevolume fraction fields.

APPENDIX F: STABILITY OF THEHOMOGENEOUS STATE

The dynamics of the volume fraction φB(r,t) of a binaryfluid with first-order chemical reactions (kf > 0, kb > 0, k = 0,Qi = 0) are given by Eq. (3b) together with the free energydensity f (φB) = b

2 (φB)2(1 − φB)2. The only homogeneousstationary state is φB(r) = φB

0 with φB0 = kf/(kf + kb). Per-

turbations of this state grow with a rate

λ(q) = 2DBq2

[4kfkb − k2

f − k2b

(kf + kb)2− w2

8q2

]− kf − kb, (F1)

where q denotes the wave vector of the perturbation mode.The state φB(r) = φB

0 is unstable if there exists a q for whichλ > 0. This is the case if

|kf − kb| <kf + kb√

3

√√√√1 − w

√kf + kb

DB

. (F2)

This equation only has solutions for kf + kb < DB/w2, whichthus gives an upper bound for the reactions rates, above whichthe homogeneous state is always stable and droplets do notform.

APPENDIX G: STABILITY ANALYSISOF TERNARY EMULSIONS

The growth rate of droplets in a ternary fluid is given inEq. (22). There exist stationary states with multiple dropletsof equal radii Ri = R, in which the volume fractions in thebackground fluid are given by φA

0 = φA0 and φB

0 = φB0 . The

stability of such states can be determined from a linear stabilityanalysis at fixed φA

0 and φB0 , which yields the perturbation

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SUPPRESSION OF OSTWALD RIPENING IN ACTIVE . . . PHYSICAL REVIEW E 92, 012317 (2015)

growth rate

λ = DBφB0

ψR2+ ksφ

A0

ψ− kb. (G1)

Here φA0 can be expressed as

φA0 ≈ kbψV + 4πDB

(γβ − φB

0 R)

ksV

[1 − Q

ksV

], (G2)

which follows from Eq. (22) in steady state for weak catalyticcores, Q � ksV , and V = 4π

3 R3. Additionally,

φB0 ≈ φB

+(R) − J

4πRDB

, (G3)

where J is given by Eq. (11) for Ri = R in the case R � l.Inserting these two expressions into Eq. (G1) results inEq. (24).

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