Liquid-Liquid Phase Separation in an Elastic Network

Robert W. Style,1 Tianqi Sai,1 Nicolo Fanelli,1 Mahdiye Ijavi,1 Katrina

Smith-Mannschott,1 Qin Xu,1 Lawrence A. Wilen,2 and Eric R. Dufresne1

1Department of Materials, ETH Zurich, Switzerland. ∗

2Yale University, New Haven, CT 06520, USA.(Dated: September 28, 2017)

Living and engineered systems rely on the stable coexistence of two interspersed liquid phases.Yet surface tension drives their complete separation. Here we show that stable droplets of uniformand tuneable size can be produced through arrested phase separation in an elastic matrix. Start-ing with an elastic polymer network swollen by a solvent mixture, we change the temperature orcomposition to drive demixing. Droplets nucleate and grow to a stable size that is tuneable by thenetwork cross-linking density, the cooling rate, and the composition of the solvent mixture. Wediscuss thermodynamic and mechanical constraints on the process. In particular, we show that thethreshold for macroscopic phase separation is altered by the elasticity of the polymer network, andwe highlight the role of internuclear correlations in determining the droplet size and polydispersity.This phenomenon has potential applications ranging from colloid synthesis and structural colour tophase separation in biological cells.

Nucleation and growth of liquid droplets is a ubiqui-tous process. In the sky above us, it underlies the forma-tion of clouds. Inside our own cells, the condensation ofprotein-rich droplets helps to regulate the translation ofRNA, among other cellular phenomena [5, 30]. The es-sential thermodynamics of condensation was worked outin the nineteenth century by Gibbs [17]. The kinetics ofnucleation is more complex and depends sensitively onthe presence of impurities [1, 2, 8, 26].

A growing appreciation for the biological implicationsof the phase separation of proteins within living cells[5, 22, 31, 33, 34, 42, 44] raises a host of questions aboutthe underlying physics [7]. Examples include the role ofactive processes in determining the properties of phase-separated droplets [6] and the combination of phase sep-aration with kinetic arrest [11, 33]. Of particular interesthere is the interplay of the structure of the cytoplasmand phase separation. The physics of droplet growth hasbeen extensively studied when the surrounding matrixis a simple fluid such as a vapour or Newtonian liquid.However a living cell has a complex rheology, includingsignificant elasticity from the cytoskeleton [13, 39].

Here, we investigate nucleation and growth of liquiddroplets inside of a polymer network. We show that theresulting droplets are stable and highly uniform, witha size that can be tuned by the cross-linking density,quench rate, and loading of the minority fluid. Conden-sation in a compliant elastic network is a generic physicalprocess for making monodisperse droplets, and works fora wide variety of chemistries. We demonstrate its effi-cacy for both temperature- and composition-driven con-densation inside of covalently or physically cross-linkedpolymer networks swollen with silicone or aqueous sol-vents. This process may provide a flexible route to thebulk synthesis of monodisperse, polymeric micro- andnano-particles and enable the self-assembly of flexible,structurally-coloured materials [20, 24]. The interaction

of condensation and network elasticity may play a role inthe cellular physiology of phase-separated proteins, andthe physical parameters identified here could be exploitedby living cells to regulate phase separation [11, 12].

Liquid-Liquid Phase Separation in an ElasticNetwork

The stability of a fluid mixture depends on the tem-perature and the concentrations of its components (e.g.[8, 19, 22]). A schematic phase diagram for a typicalmixture of two liquids, A and B, is shown in Figure 1a.Above a critical temperature, any mixture of the two flu-ids is stable. Below the critical temperature, some mix-tures – indicated by the gray region in Figure 1a – areunstable. Now consider a marginally stable fluid mix-ture at the boundary between the stable and unstableregions, as indicated by point (i). When the mixture iscooled rapidly to point (ii) in the unstable region, i.e.‘quenched’, it spontaneously separates into two stablecompositions (Figure 1b,c) . When the quench is nottoo deep, separation happens through the nucleation ofdiscrete droplets rich in A (iii’), which grow until the con-centration of A reaches its saturation value in the contin-uous phase (iii). Since the interface between the dropletand the continuous phase costs energy, the droplets tendto coarsen over time. Just like forced mixtures of oil andwater, droplets continue to grow until there is a singleblob of each phase. This can occur by direct coalescenceor through diffusion of the minority component throughthe continuous phase (i.e. Ostwald Ripening).

We investigate how this classic process is modified bythe presence of a polymer network. Consider a polymernetwork that is swollen by a mixture of two solvents, Aand B, as shown in Figure 1d. Imagine that this solventmixture initially lies at point (i) on the phase diagram

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FIG. 1: Phase separation of a liquid mixture in a polymer net-work. (a-c) Schematic diagram of simple liquid-liquid phaseseparation. Point (i) on the phase diagram in indicates amarginally stable mixture where the majority component, B,is saturated with a dilute component, A. Upon cooling to (ii),the system spontaneously separates into two phases, a contin-uous phase (iii) with a low concentration of A and a dropletphase rich in A (iii’). (d,e) Schematics of (d) a polymer net-work swollen by a mixture of A and B (at point (i) in thephase diagram), and (e) the same system after quenching to(ii) and phase separating. Note that the droplet must deformthe network in order to grow beyond the mesh size.

and is quenched to the unstable point (ii). As before,the fluid mixture needs to separate. For simplicity, let’sassume that the polymer network is excluded by the A-rich droplet phase. Provided that the network is not toodense, initial nucleation and growth will not be affectedby the network. However once the droplet is compara-ble or bigger than the network mesh size, it cannot growwithout deforming the network (Figure 1e). How doesthis impact phase separation? At a minimum, we expectthe network to prevent motion of the droplets, blockingdirect coalescence as a route to coarsening. More intrigu-ingly, the thermodynamic forces which drive condensa-tion could be balanced by elastic forces in the polymernetwork, arresting growth.

When a mixture is quenched into the two-phaseregime, it is imbued with excess free energy that candeform a polymer network. Consider an unconstrainedliquid in equilibrium with a dilute solute. Here the soluteconcentration, c, will take its saturation value, csat(T ).If c is increased beyond csat, the solute will condenseinto the liquid phase until equilibrium is achieved. Thisis driven by an excess free energy per solute moleculegiven by ∆g(T, c) = kBT ln(c/csat), where kB is Boltz-

mann’s constant. Here we have assumed the solute isdilute enough to be ideal. However if the liquid phase isconstrained by the elastic network of a gel, it can have anincreased internal pressure, P . Now the pressure and vol-ume in the droplet will increase as the solute condensesuntil the work done by the condensing solute to grow thedroplet, P∆V = PM/ρlNA, balances ∆g (here ρl is thedensity of the liquid, M is the molar mass of the solute,and NA is Avogadro’s number). Thus, in equilibrium, weexpect

P =ρ`RT

Mln

(c

csat

), (1)

where R = kBNA is the gas constant (cf [9, 40]). Ifthe restraining force of the polymer network exceeds thispressure, it cannot grow. This is perfectly analogous tothe stalling of a processive molecular motor by a suffi-ciently large opposing force [35]. Note that these argu-ments do not rely on phase-separation being driven bychanges in temperature. Thus Equation (1) can also beused to estimate the driving pressure when phase sepa-ration is triggered by other processes, such as changes insolvent or solute composition.

Droplets grow freely when they are smaller than thenetwork mesh size, but to grow beyond the mesh size,they have to deform the network. Applying the classictheory of elastic cavitation [3, 14–16, 25, 43] to dropletgrowth, we expect two regimes delimited by a criticalpressure Pcrit = 5E/6, where E is Young’s modulus ofthe gel. When P < Pcrit, a growing droplet should makeonly modest deformations to the network, and will notgrow much beyond the network mesh size. However whenP > Pcrit, droplet growth cannot be stopped by a linear-elastic (or Neo-Hookean) material (e.g. [15, 41, 43]), andis only limited by the availability of solute. Equivalently,since elastic forces make nucleation and growth energeti-cally unfavourable at small supersaturations, the appar-ent phase boundary for macroscopic phase separation isshifted:

cappsat = csat(T )e5EM6ρlRT . (2)

Simple scaling arguments suggest that the conditionfor solute-limited growth can readily be satisfied in mostsolvent-swollen polymer networks. Ideal rubber elasticitytheory [14] relates Young’s modulus to the structure ofthe polymer network, E ≈ nckT . Here, nc is the numberdensity of crosslinks in the network. Combining this re-sult with Equation (1), we find P/E ≈ nl/nc, where nlis the number density of molecules in the droplet phase(assuming that ln(c/csat) ∼ O(1)). Thus, we find thatdroplets can strongly deform a polymer network when-ever nl & nc, or equivalently when the size of a moleculein the droplet is smaller than the mesh size of the poly-mer network. Polymer networks swollen with a reason-ably supersaturated mixture readily meet this condition.

3

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Droplet radii [µm]

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E=4.2kPa

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E=675kPa

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Young's modulus [Pa]D

rople

t ra

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m]

104

105

106

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5

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15

c/csat=1.37

P=3.4 MPa

c/csat=1.30

P=2.9 MPa

c

FIG. 2: Droplets formed by phase separation of fluorinated oil in silicone gels. a) Examples of droplets formed in a soft (top),and stiff (bottom) gels. Examples in the left column have a lower loading than the right column, as they are saturated at37/44◦C respectively, before being cooled at a rate of 1.5◦C/min to 23◦C. b) Examples of typical droplet distributions indifferent stiffness gels, each saturated at 44◦C. c) The dependence of droplet size on stiffness and saturation temperature. Datapoints are centred on the average value and the error bars indicate the standard deviation. In each panel, P is calculated usingEquation (1)

Thermally-induced phase separation in silicone gels

To demonstrate this process, we grow droplets of fluo-rinated oil in a silicone gel. The silicone gel consists of across-linked silicone polymer network swollen in siliconeoil. By varying the cross-linking density, we can tuneYoung’s modulus, E, from about 5 kPa to 1 MPa (e.g.[36]). We saturate the gels with fluorinated oil at an ele-vated temperature, Te. The saturation concentration ofthe oil in the gel has no significant dependence on thecross-linking density, but increases with temperature inthe range 23 - 55 ◦C, as csat[wt%] = 0.093Te[

◦C] + 3.2(see Supplemental Material). To initiate droplet forma-tion, we cool samples to 23◦C at a controlled rate.

We estimate the pressure available to to deform theelastic network using Equation (1). For fluorinatedoil (ρl/M = 4494 mol/m3), the pressure pre-factorρlRT/M = 11MPa. In these experiments, we can readilyreach c/csat up to 2, corresponding to a driving pressureof 7.7 MPa. This exceeds the anticipated threshold fordroplet growth, even for the stiffest gels we consider here.

Tuning Droplet Size

Over a wide range of experimental conditions weobserved uniform micron-scale spherical droplets dis-tributed homogeneously throughout the sample. Typi-cal microstructures of the resulting droplet dispersionsare shown in Figure 2a, and bright-field optical z-stacksare found in Supplemental Videos 1-4. The dropletsare fixed in position and show no visible changes in theirradii over time scales of hours. At longer time scales,the droplets shrink as the oil evaporates. In contrast,droplets of fluorinated oil nucleating and growing in liq-uid silicone coarsen to the millimetre scale within secondsof cooling. As described below, the size of the dropletsdepends on the cross-linking density, the level of satura-tion, and the quench rate.

The droplet size is primarily controlled by the cross-linking density of the gel. Analysis of the bright-fieldz-stacks allows us to size each droplet and construct sizedistributions, as shown in Figure 2b. In these data, sam-ples are saturated at T=44◦C and quenched to 23◦C at1.5◦C/min. This corresponds to a driving pressure of3.4MPa. We varied the cross-linking density to achieveYoung’s moduli from 4.2 to 840kPa. In all cases, thedroplets have single-peaked size distributions that arevery well approximated by normal distributions with

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Cooling rate [°C/s]

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Po

lyd

isp

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ity [σ

/µ]

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Dro

ple

t ra

diu

s [µ

m]

FIG. 3: Rate dependence of droplet size and polydispersity.(top) The mean droplet radius reduces roughly logarithmi-cally with the cooling rate. Here, the error bars are thestandard deviation, σ, of the droplet size distribution. (bot-tom) The polydispersity (σ/µ) is effectively independent ofthe cooling rate. E = 186kPa, and the sample is saturated at40◦C, so that c/csat = 1.37.

mean µ and standard deviation σ. The mean dropletradius varies from about 5 µm to 14 µm. The decreasein mean droplet size with gel stiffness is shown in Figure2c. The dependence on Young’s modulus is modest, suchthat a 200-fold increase in gel stiffness only decreases thedroplet size by a factor of 2.

Droplet size is also impacted by the level of saturation.By varying the incubation temperature, we varied theloading of the oil from c/csat =1.24 to 1.37 wt%. This10% increase in loading (corresponding to a 50% increasein the supersaturation, c − csat) leads to a 25% increasein droplet radius, as shown in Figure 2c.

We can also control droplet size by controlling the cool-ing rate: droplets decrease in size as the cooling rateincreases, as shown in Figure 3a. Again, we find a mod-est, roughly logarithmic, dependence of droplet radius ofcooling rate. Droplets are slightly more uniform whencooled at a slower rate. We quantify the uniformity withthe polydispersity index (σ/µ), shown in Figure 3b. Overthe range of two decades in cooling rate, the polydisper-sity ranged from 7 to 9 %. Thus, we can control thesize this way with surprisingly little penalty in dropletuniformity as the cooling rate increases.

Optical Properties of Composites with Uniform Droplets

The uniformity of the droplet size is reflected in themacroscopic optical properties of the gel. Figure 4ashows the scattering pattern in transmission for a thinsection of silicone gel containing fluorinated-oil droplets,illuminated with a HeNe laser. There is a clear ring,consistent with the expected Mie scatting pattern for

FIG. 4: Optical properties of self-assembled composites ofuniform fluorinated-oil droplets in a silicone gel matrix. (a)The light-scattering pattern of a Helium-Neon laser by a sam-ple (E = 837kPa silicone gel saturated with fluorinated oilat 40◦C and cooled at 30◦C/min) has a clear ring, indicat-ing droplet uniformity. The centre of the dark ring is at ascattering angle of 6.5◦. (b) Different wavelengths of lightare scattered to different angles, leading to colourful ringswhen a sample (Sylgard 184, E = 40kPa, cooled from 42◦Cat 6◦C/min) is illuminated from behind with a white-lightLED. The scalebar is 10mm long. (c,d) The color pattern isdeformed upon stretching (Sylgard 184, E = 412kPa, cooledfrom 42◦C at 6◦C/min). Here, the direct light from the white-light LED source is blocked by a 2mm-thick rod to bettervisualise the colours. See Supplemental Video 2.

monodisperse spheres [4]. There is no significant con-tribution to the observed scattering pattern from corre-lations in the droplet locations (i.e. structure factor).Since the silicone gels are highly elastic, we can deformthe droplets by stretching the gel [36]. As shown in Sup-plemental Video 5, this yields a corresponding stretch-ing of the scattering pattern in the direction perpendicu-lar to the applied stretch – indicating the coupling of themacroscopic deformation to the shape of the microscopicdroplets. Different wavelengths are scattered to rings ofdifferent radii by the samples, so we observe a pattern ofcolor when a point source of white light is viewed throughthe sample (Figure 4b). This pattern changes as the sam-ple is stretched and the microstructure is deformed, asshown in Figure 4c,d and Supplemental Video 6.

Correlated Nucleation and Growth

As expected for gentle quenches far from the criti-cal point, droplets form by subsequent nucleation andgrowth (e.g. [27, 29]). Supplemental video 7 showsa typical example. Here E = 186kPa and the sample iscooled from 42◦C to 22◦C at 2◦C/min. New droplet nu-clei appear over a period of about 150s, and grow over alonger interval, of about 800s. This suggests that nucle-ation and growth can be thought of as two separate stagesof droplet formation [26]. Here we show that while thesetwo processes are separated in time, they are stronglycoupled in a manner than impacts the droplet size dis-tributions.

Since droplets are trapped in an elastic matrix, nu-cleation positions are given by the final positions of thedroplets. We identified the position of each nucleus using

5

FIG. 5: Correlated nucleation and growth. a) The numberdensity of droplets depends on stiffness, but not saturation.The dashed line is the line of best fit to the data. b) Thepair-correlation function, g(r) of droplet centres for a typicalexperiment, blue crosses, shows that nuclei do not form neareach other. The range of internuclear repulsions is compa-rable to the inter-droplet distance. The pink line shows theg(r) for nuclear positions in our Monte-Carlo growth simula-tion. Note that the experimental measurements of g(r) areonly accurate for r & 5µm, due to inaccuracy of out-of-planetracking of droplet centres. c) Droplet sizes are strongly cor-related with the radius of their voronoi cell. Blue: data fromthe experiment in b). Green and pink points show results ofgrowth simulations with random and correlated nuclear posi-tions. d) Droplet size distributions for the experimental datain b) (blue) and simulations with random (green) or corre-lated (pink) positions.

the same bright-field z-stacks underlying Figure 2. First,we determined the number density of droplets, nd, asa function of cross-linking density and supersaturation.As shown in Figure 5a, nd increases linearly with Young’smodulus, but is independent of the supersaturation. Bya simple application of mass conservation, this variationin nucleation number density fully accounts for the de-pendence of the mean droplet size on the cross-linkingdensity and extent of supersaturation (see Supplement).

In classical theory the interaction of nucleation eventsis typically ignored. However we find that the locationsof nucleation sites are significantly correlated, and cen-tral to the form of the final droplet distribution. Wequantified spatial correlations in the nucleation positionsusing the radial distribution function, g(r), as shownin Figure 5b for typical experimental data. The pair-correlation function compares the probability of observ-ing two objects at a given separation, relative to a casewhere all objects are placed perfectly randomly. We findthat the number density of closely-neighbouring dropletsis strongly reduced over a distance comparable to themean inter-droplet spacing, about 40µm in the example

of Figure 5. This suggests that there is some mechanismthat prevents droplets from nucleating near each other.

Intriguingly, variations in the local spacing of nucleihave a significant impact on the size distribution. Wecharacterised the structure about each nucleus using theVoronoi construction. The Voronoi cell associated witha nucleus is the collection of points that are closer to itthan any other nucleus. In general, this is a complexpolyhedral shape. We reduce it to a single length scale,by taking the cube-root of the cell volume, or Voronoiradius, rv. As shown in Figure 5c, the Voronoi radiusis strongly correlated to the droplet size. Thus dropletsin tight clusters tend to be smaller than more widely-spaced droplets. This effect, if present in conventionalliquid-liquid phase separation, is masked by Brownianmotion and coalescence of droplets.

Monte-Carlo simulations suggest that the correlationof droplet size and nuclear spacing is a natural conse-quence of the diffusion-limited growth of droplets. Us-ing Brownian dynamics, we simulated the diffusion andcapture of solute molecules by a number of fixed nuclei.In one case, the nuclei were positioned completely ran-domly, in the other case, the nuclei positions were se-lected to have the same g(r) as experiments. In bothcases, we find a pronounced correlation between rv anddroplet radii, as shown by the green (random) and pink(correlated) points in Figure 5c, which nicely matchthe experimentally-observed correlation (blue). How-ever, the simulation with correlated nucleation positionsshows a narrower spread in rv and droplet radii, whichclosely matches the experiments, as shown in Figure 5d.Thus long-range interactions of nuclei make the structurearound each nucleus more regular and result in more uni-form droplet radii.

Together, these results suggest that understanding thenucleation process is crucial to gaining a quantitative un-derstanding of droplet size and monodispersity.

Isothermal Phase Separation in Hydrogels

To show the generality of uniform droplet productionby phase separation in a polymer network, we demon-strate its efficacy for a distinct form of phase separationand diverse polymeric networks.

Previously we supersaturated the system by quenchinga binary liquid mixture with a change of temperature.Now we drive phase separation of a solute by changingthe composition of a solvent mixture [26, 38]. This formof phase separation is familiar from aperitifs like ouzoand pastis. In these drinks, fragrant anise oil is solu-bilised in water by a high concentration of ethanol [37].When water is added to the drink, the mixture becomesunstable, and oil-rich droplets nucleate and grow.

We use three hydrogels with distinct chemistries: achemically-crosslinked gel (polydimethylacrylamide or

6

FIG. 6: Arrested isothermal phase-separation in diverse hy-drogels. Here, anise oil droplets nucleated and grew in anethanol-water mixture (i.e. the Ouzo effect). Micrographs ofdispersed droplets in (a) PDMA, a covalently crosslinked gel,(b) gellan gum, a thermally set physically crosslinked gel, and(c) alginate, an ionically crosslinked physical gel. All dropletsare stable against coalescence. Droplets in PDMA and gel-lan are reasonably monodisperse (see Supplement). The scalebars are all 40µm wide.

PDMA), a thermally-set physical gel (gellan gum), andan ionically-crosslinked physical gel (alginate). In eachcase we soak the hydrogel in a large volume of a sta-ble mixture of water/ethanol/anise oil to exchange thesolvent (see Figure 1b). After this we cover the gel withdeionized water, which diffuses inward and initiates phaseseparation. The resulting oil droplets are shown in Fig-ure 6. As with the fluorinated-oil/silicone system, we seerather uniform droplet distributions in the different ma-terials (see Supplement for distributions), especially inthe PDMA and gellan gum samples which we expect tobe much more uniform gels than alginate [18].

Conclusions

We have shown that phase separation of a solvent mix-ture in a polymer network is a simple bulk technique tocreate uniform droplets. The size of the resulting dropletsdepends on a range of factors including the cross-linkingdensity of the polymer network, the supersaturation, andthe quench rate. This process for the production ofmonodisperse droplets in bulk has broad potential appli-cations. For example, it could be used to create compos-ites with well-defined microstructures that enable noveloptical or mechanical properties [11, 36]. Additionally,by using a monomer as the phase-separating component,it can offer a flexible route for the synthesis of uniformmicroparticles.

Our results suggest a role for polymer networks in theregulation of membrane-less organelles within living cells.Consider phase-separating macromolecules with molecu-lar weights in the range of 75-600 kDa. Equation (1)suggests they can exert pressures of order 4 − 33kPa.These pressures are comparable to the stiffness of manycytoskeletal networks, which suggests that living cellsmay be able to regulate the nucleation and growth ofmembrane-less organelles through their mechanical prop-

erties.While we have outlined the essential phenomenology

and highlighted key thermodynamic and mechanical as-pects of the process, much work must be done to yield aquantitative understanding. The central question is howan increase in the cross-linking density leads to an in-crease of nucleation sites. There are two key aspects tonucleation – its spatial distribution, and rate. The spa-tial distribution is likely to be set by the fact that whena droplet nucleates and starts to grow, it will deplete thesurrounding material of solute, making it less likely for anearby nucleus to appear. The control of nucleation rateis more complex, as it will depend on a range of factors,such as cooling rate and the ability of droplets to nucleatewithin the (nanometric-scale) mesh of the polymer gel.However the monotonic increase of nucleation sites withcross-linking density suggests the simple possibility thatthe cross-linking sites themselves serve as heterogeneousnucleation sites.

The mechanical aspects of this process demand furtherexploration. We have shown that the mechanical prop-erties of the gel alters the phase diagram. However thereare number of further interesting questions and possi-bilities. While elastic cavitation theory predicts a bal-ance of elastic forces and condensation pressure only fordroplets that are modestly larger than the mesh size, it isclear that strain-stiffening networks can suppress dropletgrowth and limit droplet size over a wider range of driv-ing pressures. Furthermore, elastic forces could create aelastic Ostwald ripening effect. In classic Ostwald ripen-ing, surface tension favors the collapse of small dropsto feed big ones. Here, we anticipate that elastic forcesin strain-stiffening networks could drive transport of thecondensed phase from big drops to small ones. The bal-ance of these two effects could lead to a mechanicallydefined equilibrium droplet size. However precise quan-titative theories of these effects will be challenging, sincethe deformations are so large – here, the final dropletsare about three orders of magnitude larger than the net-work mesh size. Therefore there must be strong irre-versible deformations – especially in the vicinity of thedroplet – which are bound to be strongly rate-dependent[10, 16, 21, 28].

Materials & Methods

Silicone gels preparation and characterisation

We prepare silicone gels following the recipe of [36], ex-cept where stated otherwise. Stiffness is tuned by varyingthe proportion of crosslinker in the gel between 0.8% and3.3%. We measure the Young’s modulus of each sampleby making an additional bulk sample and performing acompression test. For this, the silicone gel is formed ina cylindrical plastic mould (depth 10mm, radius 15mm),

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and then indented with a 1mm radius, cylindrical inden-ter using a TA.XT plus Texture Analyser with a 500gload cell (Stable Micro Systems).

For some of the light-scattering experiments, we useda different variety of silicone gel (Sylgard 184, Dow Corn-ing), as this is less sticky, and thus easier to stretch man-ually. For these samples, stiffness was again tuned byvarying the ratio of base to crosslinker.

Silicone-gel samples are made by coating the insideof glass-bottomed petri dishes with a thin, O(1mm)-thick film of silicone gel. Silicone gel samples are thensaturated with fluorinated oil (Fluorinert FC770, Fluo-rochem) at an elevated temperature, Te.

The solubility of fluorinated oil in the gels was mea-sured by weighing samples of silicone gels of two differ-ent stiffnesses (E = 20, 840kPa) before and after satu-ration at 23,40,55◦C. No significant change in solubilitywas found with E. See Supplemental Material for furtherdetails.

Growth simulations

We explored the effect of inter-nuclei structural cor-relations on the growth of droplets using Monte-Carlosimulations. In the first step, we positioned 103 nucleiin a cube with periodic boundary conditions, either byplacing the nuclei independently or using the Metropolisalgorithm to generate a pattern of points with the sameg(r) as observed in experiments. In the second step,we released 106 molecules, which underwent a randomwalk until they were captured by a nucleus. By countinghow many molecules were captured by each nucleus, wewere able to determine the final size of each nucleus. Forsimplicity, the target radius for each nucleus was fixedthroughout the simulation so that the volume fraction ofthe targets was 1.25%.

Hydrogel preparation

PDMA samples were fabricated by free-radicalpolymerisation of N-N-dimethylacrylamide (DMA,Sigma-Aldrich) with N,N’-methylenebisacrylamide(MBA, Sigma-Aldrich) as a crosslinker. Ammo-nium persulphate (APS, Fisher-Bio) and N,N,N’,N’-tetramethylethylenediamine (TEMED, Apollo Scientific)were used as the redox/initiator system (e.g. [32]). Allchemicals were used as received, and the reaction wasperformed in a nitrogen glove box to avoid inhibition ofthe reaction by oxygen. We prepared fresh stock solu-tions with deionized water as follows: 30%w/v DMA,2%w/v MBA, 10%w/v APS and 10%w/v TEMED.To make 10mls of PDMA, we combined 2.5ml DMAsolution, 0.8ml MBA solution and 6.64ml deionizedwater before gently mixing. Then we added 0.03ml

TEMED solution, and 0.03ml APS solution, mixing thesample gently after each addition. Finally the samplewas poured into a mould and left to completely cure.

Gellan-gum samples were fabricated from 0.5 wt% gel-lan gum (Apollo Scientific Ltd.), 59.5 wt% deionized wa-ter, and 40 wt% ethanol. The gum powder is mixed withthe water, heated to 90◦C and stirred until fully dis-solved. Ethanol is then added drop-wise while stirringand the sample is removed from the heat. Typically wesee a separation of the sample into an upper, cloudy part,and a lower, transparent part. We pipette the lower, clearphase into petri dishes and allow it to gel. Finally, thegel is soaked in CaCl2 solution (50-500mM) to allow fullcrosslinking by the calcium ions.

Alginate-gel samples were fabricated from 2 wt%sodium alginate (Acros) in deionized water. This is gen-tly placed in contact with 100mM CaCl2 solution and leftfor long enough to allow the calcium ions to diffuse into,and crosslink the alginate solution (e.g. [23]).

Further details

Further information about the controlled cooling appa-ratus, image analysis, and optical experiments are givenin the Supplemental Material.

We acknowledge helpful discussions with Alain Gorielyand Ian Griffiths and funding from the Swiss Na-tional Science Foundation (grants 200021 172824 and200021 172827).

Author contributions

R.W.S., L. A. W. and E. R. D. designed experiments.R.W.S., T. S., N. F., M. I., K. S.-M. and Q. X. per-formed experiments. R.W.S., L.A.W. and E.R.D anal-ysed data. R.W.S, L.A.W. and E.R.D. developed thetheory. R.W.S, and E.R.D. wrote the paper.

Competing financial interests

The authors declare no competing financial interests.

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