Cavitation controls droplet sizes in elastic media
CCavitation controls droplet sizes in elasticmedia
Estefania Vidal-Henriquez and David Zwicker Max-Planck Institute for Dynamics and Self-Organization, AmFaßberg 17, 37077 Göttingen, Germany.
Abstract
Biological cells use droplets to separate components and spatiallycontrol their interior. Experiments demonstrate that the complex,crowded cellular environment affects the droplet arrangement and theirsizes. To understand this behavior, we here construct a theoreticaldescription of droplets growing in an elastic matrix, which is motivatedby experiments in synthetic systems where monodisperse emulsionsform during a temperature decrease. We show that large dropletsonly form when they break the surrounding matrix in a cavitationevent. The energy barrier associated with cavitation stabilizes smalldroplets on the order of the mesh size and diminishes the stochasticeffects of nucleation. Consequently, the cavitated droplets have similarsizes and highly correlated positions. In particular, we predict thedensity of cavitated droplets, which increases with faster cooling, as inthe experiments. Our model also suggests how adjusting the coolingprotocol and the density of nucleation sites affects the droplet sizedistribution. In summary, our theory explains how elastic matricesaffect droplets in the synthetic system and it provides a framework forunderstanding the biological case.
Phase separation has emerged as a powerful concept to explain how bi-ological cells structure their interior [1, 2]. It explains how membrane-lesscompartments with distinct chemical composition, called biomolecular con-densates, form spontaneously. In contrast to classical liquid-liquid phaseseparation, these condensates exist in complex, crowded environments, e.g.,1 a r X i v : . [ c ond - m a t . s o f t ] F e b rovided by the cytoskeleton in the cytosol or the chromatin in the nucleus.This fundamentally affects the behavior of condensates: their coarsening isslowed down by sub-diffusive motion [3], they are supported against gravityby the F-actin network in the nuclei of large cells [4], and their assemblydepends on the stiffness of their surrounding [5, 6]. Another example areartificially induced condensates, which typically appear in soft regions of thechromatin [7]. Taken together, these experiments and recent numerical sim-ulations [8] demonstrate that condensates react to the elastic properties oftheir surrounding [9], but the detailed dynamics are still unclear.The interaction of droplets with soft elastic matrices can be studied indetail in a synthetic system, where oil droplets are induced in a PDMS ma-trix by lowering the temperature [10]. Similar to the biological case, dropletsare biased towards softer regions in this system [11, 12]. This elastic ripen-ing is absent when the elastic properties of the system are homogeneous.Instead, all observable droplets attain similar sizes and their positions arecorrelated [10]. Interestingly, one observes smaller droplets in stiffer sys-tems and at larger cooling rates [10]. This implies that the final state isgoverned by non-equilibrium processes, which is also demonstrated by thebidisperse emulsions that form after increasing the cooling rate during theexperiment [11].Theoretical descriptions of such systems have to describe how the elasticmatrix affects the droplets’ dynamics. In the simplest case, the matrix exertsa pressure onto the droplets proportionally to the local stiffness, which is suf-ficient to explain elastic ripening [13]. Moreover, assuming a strain-stiffeningsurrounding can explain why droplets attain the same size, which decreaseswith stiffness [14–16]. However, these equilibrium models cannot describethe dependence on the cooling rate.In this paper, we present a dynamic theory of droplet formation in elasticmatrices, which is based on the assumption that droplets can break the sur-rounding matrix. We show that in this case some droplets cavitate and growmacroscopically, while a large fraction is restricted to mesh size. The cavi-tated droplets have a similar size, which decreases with larger cooling rate.We motivate our theory by first considering how the elastic matrix affectsa single droplet. We then couple the dynamics of multiple droplets via thediffusion of monomers in the dilute phase. Using numerical simulations andanalytical approximations, we demonstrate that this model can explain allthe experimental observations of the synthetic system.2 xternal pressure governs dynamics of droplets To understand how droplets interact with an elastic matrix, we first considerthe free energy of a single spherical droplet of radius R . Droplet growth isdriven by the differences in chemical potential and osmotic pressure betweenthe droplet and its surrounding. We show in the SI that this can be capturedby a driving strength g , which quantifies the energy gain when the dropletvolume V = (4 π/ R increases. However, when the droplet grows its surfacearea A = 4 πR also increases, which comes at a cost proportional to thesurface energy γ .Moreover, the matrix surrounding the droplet must be displaced, whichwe capture by an elastic energy F E ( V ) . Taken together, the free energy F ofthe entire system reads F = − g V + γA + F E ( V ) , (1)where we for simplicity first consider constant driving strength g and surfaceenergy γ .A droplet will grow spontaneously when the free energy decreases ( ∂F/∂V < ), i.e., if g > P ( R ) with P ( R ) = P γ ( R ) + P E ( R ) , (2)where P γ = 2 γ/R is the Laplace pressure due to the surface tension γ [17,18] and P E = ∂F E /∂V is the pressure exerted by the elastic matrix; see SI.A droplet thus grows when the driving strength g exceeds the pressure P exerted on the droplet. A stationary state with droplet radius R ∗ is reachedwhen g = P ( R ∗ ) , which is stable if ∂ F/∂V > , or, P (cid:48) ( R ∗ ) > . (3)A droplet is thus stable when the exerted pressure increases with its size.Without an elastic matrix, the droplet is only affected by the Laplacepressure P γ ; see Fig. 1A. The corresponding free energy shown in Fig. 1Ddemonstrates that surface tension dominates for small droplets. In particular,droplets can only grow spontaneously ( ∂F/∂V < ) after overcoming a nucle-ation barrier , e.g., by thermal fluctuations (homogeneous nucleation)[19] orthanks to nucleation sites that lower the barrier (heterogeneous nucleation);see SI. Once the droplet is big enough, the energy decreases with increasingradius and the droplet is always unstable ( P (cid:48) γ ( R ) < ). Droplet growth isthen only restricted by the available amount of material.3 roplet radius R Droplet radius R Droplet radius R F r ee e n e r g y F P r e ss u r e P Nucleation barrier Cavitationbarrier
Surface Tension (ST)
ST+ Neo-Hookean (NH) ST + NH + Breakage
A B CD E F
Figure 1:
Breakage implies a cavitation barrier
Example of pressure P = P E + P γ exerted on a droplet during growth (upper panels) and corre-sponding free energy F (lower panels) as a function of the droplet radius R for three different scenarios: Without an elastic mesh ( P E = 0 ), the pressurecurve is monotonously decreasing (Panel A). After crossing a nucleation bar-rier, droplets grow until all material is absorbed (Panel D). An elastic matrixincreases the pressure once the droplet grows beyond mesh size (Panel B).This can lead to an energy minimum where droplets are stable (Panel E). Ifthe mesh can break, the pressure curve exhibits a local maximum (Panel C),which leads to a cavitation barrier (Panel F).4 n elastic matrix restricts droplet growth An elastic matrix surrounding the droplet exerts an additional pressure andthus potentially opposes growth; see Eq. [2]. The pressure exerted by thematrix depends on its elastic response. For small deformations, the responsecan be characterized by the Young’s modulus E . However, droplets can growmuch larger than the mesh size (cid:96) , implying large deformations of the matrix.The simplest model describing such hyperelastic material is the Neo-Hookeanmodel, where the pressure on a spherical cavity of radius R is monotonicallyincreasing ( P (cid:48) E ( R ) > ) and converges at large radii to P E = 5 E/ [20]. If thedriving strength g is lower than the maximal pressure, the system opposesfurther droplet growth and leads to a stable radius R ∗ when g = P ( R ∗ ) ; seeFig. 1B. This steady state corresponds to a minimum in the free energy; seeFig. 1E. Therefore, an elastic mesh providing resistance to droplet growthcan stabilize droplets. Breakage provides a cavitation barrier for droplets
The Neo-Hookean model is often too simple to describe realistic materials, inpart because it does not account for breaking bonds in the elastic mesh. Tocapture breakage, we next consider a stress-strain curve that has a maximalpressure P cav at a finite radius R cav ; see Fig. 1C. Similar to the Neo-Hookeanmodel, we consider an increasing pressure when droplets grow beyond themesh size (cid:96) . However, at the critical radius R cav the mesh cannot sustainthe stress anymore and breaks, resulting in a pressure decrease [21]. Thestability criterion given in Eq. [3] indicates that droplets with R = R cav areunstable and will thus expand rapidly in a cavitation event [22, 23].The non-monotonous stress-strain relation results in a free energy thathas two energy barriers; see Fig. 1F. The first barrier is the familiar nucleationbarrier, while the second one is the cavitation barrier . The local minimumbetween the two barriers corresponds to the stable state described in the caseof the Neo-Hookean model. However, with breakage, droplets can overcomethe second barrier and cavitate if the driving strength g exceeds P cav . Thegrowth of such droplets would then only be limited by the available amountof material, similar to the case without any elastic matrix.5 ultiple droplets grow when temperature is de-creased In the experiments of Style et al. [10], multiple oil droplets appeared simulta-neously when the temperature was lowered. Since lowering the temperaturecorresponds to increasing the driving strength g , droplets appear when g reaches the maximal pressure exerted by the surrounding matrix. If thismaximal pressure increases with the overall stiffness, we predict that lowertemperatures are necessary to create droplets in stiffer systems, which wasindeed observed [11]. However, this qualitative analysis does not distinguishbetween the Neo-Hookean and the Breakage model, since both provide amaximal pressure that explains the simultaneous growth of droplets.To distinguish the Neo-Hookean from the Breakage model, we need to an-alyze the droplet dynamics in detail. Since the elasto-adhesive length scaleof the experimental system is smaller than the droplet size [24], elastic in-teractions of droplets are negligible. In contrast, growing droplets competefor the material dissolved in the dilute phase, which couples their dynamics.We analyze this using a mean-field theory, where we describe a collection ofimmobile droplets by their positions (cid:126)x i and their radii R i together with theconcentration field c ( (cid:126)x ) in the dilute field [13]. For simplicity, we assumethat the concentration c in inside each droplet is constant and that dropletsare in equilibrium with their immediate surrounding, which exerts the pres-sure P ( R ) onto the droplet. This implies that the concentration right outsidethe interface of a droplet is given by [13] c eq ( P, T ) = c sat ( T ) exp (cid:18) Pc in k B T (cid:19) , (4)where c sat is the equilibrium concentration in the absence of an elastic mesh inthe thermodynamic limit, k B is Boltzmann’s constant, and T is the system’stemperature.Droplets grow when their surrounding is supersaturated ( c > c eq ). Thedroplet growth rate reads [13] d R i d t = DR i c in (cid:2) c ( (cid:126)x i ) − c eq (cid:0) P ( R i ) , T (cid:1)(cid:3) , (5)where D is the diffusivity of the droplet material in the dilute phase. The6oncentration in the dilute phase obeys ∂ t c = D ∇ c − c in (cid:88) i d V i d t δ ( (cid:126)x i − (cid:126)x ) , (6)where the last term accounts for material exchange with the droplets [13].Taken together with no-flux conditions at the system’s boundary, Eqs. [5–6]conserve the total amount of droplet material.We simulate the system by mimicking the experimental protocol of Styleet al. [10]. In particular, we consider a linear relation between the saturationconcentration c sat and temperature T together with a constant cooling rate.Consequently, c sat decreases linearly from the initial value c until it reachesthe minimal value c − ∆ c at the final temperature, c sat ( t ) = (cid:40) c − αt t < ∆ cα c − ∆ c otherwise , (7)where α is the rate of the decreases. As c sat is lowered, the equilibriumconcentration c eq also decrease, see Eq. [4], implying a larger supersaturation c − c eq , which corresponds to a higher driving strength g . Starting with ahomogeneous system at high temperature (high c sat ), droplets will nucleateonce g is large enough to cross the nucleation barrier; see Fig. 1D–F. In theexperimental system, homogeneous nucleation is basically impossible anddroplets must thus nucleate heterogeneously at nucleation sites; see SI. Thissuggests that surface tension effects are negligible for small droplets. In fact,surface tension is also negligible for large droplets since Ostwald ripening isslow (see SI), suggesting that the total pressure P is always dominated bythe elastic pressure P E . We thus neglect surface tension for simplicity andrather assume that droplets form quickly at nucleation sites. In particular, weinitialize our simulations with many small droplets with radii on the orderof the mesh size, R i ( t = 0) = (cid:96) , and focus on the subsequent dynamics.The radius at which droplets are initialized is unimportant, since they arerestricted by the elastic matrix to have a small radius R that is governed bythe condition g = P ( R ) . Consequently, many microscopic droplets coexistearly in the simulation.Droplets can grow macroscopically ( R (cid:29) (cid:96) ) when the driving strength g exceeds the pressure exerted by the mesh. Since realistic meshes are hetero-geneous [25], the exerted pressure will vary slightly from droplet to droplet.7o capture such heterogeneity for the Neo-Hookean model (NH), we considervariable mesh sizes (cid:96) i , P NH i ( R ) = E (cid:18) − (cid:96) i R − (cid:96) i R (cid:19) , (8)where E is the macroscopic Young’s modulus of the material. Conversely,in the Breakage model (BR), we choose random cavitation pressures P ( i )cav ,since this parameter dominates the cavitation barrier. We thus consider thesimple form P BR i ( R ) = R < (cid:96)P ( i )cav R − (cid:96)R cav − (cid:96) (cid:96) ≤ R ≤ R cav P ∞ R > R cav , (9)where we keep both (cid:96) and R cav fixed for all droplets, since varying theseparameters does not affect the results significantly; see SI. Eq. [9] impliesthat the external pressure increases linearly when the droplet grows beyondthe mesh size (cid:96) until it reaches the cavitation radius R cav . Beyond thisthreshold, the mesh breaks and provides a constant resistance quantified bya pressure P ∞ < P ( i )cav .Fig. 2 shows typical simulations of Eqs. [4]–[7] for both the Neo-Hookeanmodel (Eq. [8]) and the Breakage model (Eq. [9]). In both cases, macroscopicdroplets appear and they grow with very similar rates. However, the Break-age model additionally exhibits a large number of microscopic droplets, whichapparently do not grow. Since these microscopic droplets are likely not visi-ble in the experiment, both models appear to yield mono-disperse emulsions,although this requires an extremely homogeneous mesh in the Neo-Hookeanmodel. In contrast, the models behave differently when we nucleate newdroplets during the simulation: While all newly nucleated droplets grow inthe Neo-Hookean model, in the Breakage model most droplets are restrictedto microscopic sizes; see SI. Consequently, we expect that the Breakage modelleads to a more uniform size distribution of large droplets in realistic situa-tions.To see which of the two models provide a better explanation of the ex-periments, we next test their predictions quantitatively. Here, we use theexperimentally measured values of D , ∆ c , α , P ∞ , E , and c in , while the val-ues of the mesh size (cid:96) and the cavitation radius R cav are arbitrary and do8 r o p l e t r a d i u s Neo-Hookean Model Breakage Model
A BC D time: 200s time: 800s time: 200s time: 800s
Time t [s] Time t [s] NumericalTheoryFinal temperature 2.55.07.50.0
Figure 2:
Decreasing temperatures cause monodisperse emulsions inthe Neo-Hookean model (NH, Panels A,C) and Breakage model (BR, PanelsB,D). (A-B) 2d projections of typical simulations at two time points. Theheat map indicates the concentration c in the dilute phase and droplets aremarked with disks where color saturation indicates depth. (C-D) Dropletradii R as a function of time t showing that large droplets are monodisperseafter reaching the final temperature at t = ∆ c/α (dotted black line). The reddashed line shows the theoretical prediction given by Eq. [10]. The modelparameters are α = 1.864 · -5 s -1 ν -1 , E =
186 kPa, ∆ c = ν − , D =50 µm s -1 , c in = ν − , and c in k B T =
11 MPa. For the NH model, we sample (cid:96) uniformly between 0.1 µm and 0.102 µm [25] and use m = 7 · -6 µm -3 . Forthe BR model, we have R cav = m = 1.875 · -4 µm -3 , η/m = E µm , P mincav = E , and P ∞ = E .not affect the predictions of the model; see SI. The only relevant parameter,which we adjust to match the experimental data, is η/m quantifying the meshheterogeneity and the density of nucleated droplets. We first focus on theintriguing non-equilibrium effect that larger cooling rates lead to more andsmaller droplets. In the Neo-Hookean model, the average droplet size (cid:104) R (cid:105) is independent of the cooling rate α (see Fig. 3A), while it matches the ex-perimental data in the Breakage model (Fig. 3B), including the dispersionstatistics; see SI. The two models also differ in the spatial distribution oflarge droplets, which we quantify by the pair correlation function, similar tothe experiments [10]. Fig. 3C shows that droplets are uniformly distributedin the Neo-Hookean model since their positions are solely controlled by theirnucleation. In contrast, droplet cavitation seems to be correlated in the9reakage model (Fig. 3D), leading to a low probability of finding two largedroplets close to each other, similar to the experiments [10]. The shown datacollapse suggest that the pair correlation function is scale-free. Moreover,the volume surrounding a droplet, measured from a Voronoi tessellation, isstrongly correlated with its size; see inset of Fig. 3D. The fact that our sim-ulations match the experimental data quantitatively suggests that breakageis a crucial aspect. Large droplets suppress further cavitation by de-pleting their vicinity
To understand how breakage affects the droplets’ dynamics, we next inves-tigate why some droplets cavitate while others remain small; see Fig. 2D.Initially, all droplets are small and grow due to the decreasing saturationconcentration by absorbing the excess material from the dilute phase. Notethat droplets in a softer environment, i.e., with a lower P cav , exhibit a lowerequilibrium concentration c eq , see Eq. [4], and thus grow faster. This ini-tial growth phase continues until the droplet with the lowest P cav reaches itscavitation radius R cav . At this point, the elastic matrix no longer providesenough resistance ( ∂P/∂R < ) and the droplet radius becomes unstable;see Eq. [3]. The droplet thus cavitates by recruiting material from the dilutephase as fast as possible in a diffusion limited process. Such a quickly growingdroplet depletes its surrounding, effectively fixing the local driving strengthto g = P ∞ . Consequently, other droplets in the vicinity cannot cavitate andwill remain small forever. Taken together, the growth of a cavitated dropletprevents the cavitation of other droplets in its surrounding while dropletsfurther away might still grow, which qualitatively explains the observed paircorrelation function; see Fig. 3D.The numerical data shown in Fig. 2D. suggests that all droplets thatbecome large cavitated at very similar times t = t cav and grow with simi-lar rates. To understand the growth dynamics, we first consider cavitateddroplets that are homogeneously distributed with a number density n . As-suming that the cavitated droplets absorb all excess material from the dilutephase, we predict their volume to increase as V ( t ) = V cav + αnc in ( t − t cav ) , (10)10 r o p l e t r a d i u s Neo-Hookean Model Breakage ModelA BC D
ExperimentalNumericalTheory1210864 10 -5 -4 -3 -5 -4 -3 Rate Rate P a i r c o rr e l a t i o n ExperimentalNumericalTheory0.6Droplet separation / mean droplet dist. Droplet separation / mean droplet dist.
Figure 3:
The Breakage model explains the experimental data. (A,B)Comparison of the experimental data (blue dots, [10]), numerical simulations(black symbols), and analytical predictions (red lines) for the radius R ofthe cavitated droplets as a function of the rate α with which the saturationconcentration decreases for the Neo-Hookean model and the Breakage model.(C,D) Scaled pairwise correlation function g ( r ) of the cavitated droplets (seeSI) for three rates α . The inset in panel D shows the correlation between theradius R of droplets and the size R voro of their surrounding, which is obtainedfrom a Voronoi tesselation. (A-D) Model parameters are as in Fig. 2, exceptfor η/m = · E µm .where V cav = (4 π/ R . The dashed line in Fig. 2D shows that the equiv-alent prediction for the droplet radius explains the mean growth dynamicsof cavitated droplets. In fact, this analysis is also valid for the Neo-Hookeanmodel shown in Fig. 2C since droplets also start growing around the sametime and absorb all excess material in this case. Taken together, this anal-11sis indicates that the large droplets are mono-disperse because they startgrowing at the same time and grow with the same rate. However, whilethese conditions are met artificially by our setup of the Neo-Hookean model,they are self-organized in the Breakage model by controlling which dropletscavitate.The final droplet size can be estimated by evaluating Eq. [10] at thetime t final = ∆ c/α when the final temperature is reached. For simplicity, weconsider the case where droplets are large compared to the cavitation thresh-old R cav , which also implies t cav (cid:28) t final and leads to V final ≈ ∆ c/ ( nc in ) . Thisapproximation correctly predicts that the final droplet volume is independentof the quench rate α in the Neo-Hookean model where the droplet density n is set by the initial condition; see Fig. 3A. Conversely, in the Breakage model,the density of cavitated droplets might depend on the quench rate α , whichcould explain the observed size-dependence shown in Fig. 3B. Number and size of cavitated droplets dependon quench rate and cavitation thresholds
To understand why faster cooling leads to more and smaller droplets, wenext focus on the cavitation process in the Breakage model. Since cavitateddroplets suppress further cavitation in their vicinity, we hypothesize that thissuppression is less efficient when the system is cooled faster, implying thatmore droplets can cavitate overall.To estimate the final density n of cavitated droplets, we analyze a sim-plified theoretical model. The main idea is to study a fixed density n ofcavitated droplets and test whether additional droplets could cavitate in thissituation. The best estimate is then the lowest value of n where no moredroplets cavitate. For simplicity, we consider a homogeneous distribution ofcavitated droplets, allowing us to focus on a single droplet of radius R = R cav in a spherically symmetric domain of volume n − . We then obtain the con-centration field c ( r ) around the droplet by solving the diffusion equation withthe boundary condition c ( R cav ) = c eq ( P ∞ , T ) ; see SI. Additional cavitationtakes place in the dilute phase if there is a droplet whose critical concentra-tion c cav = c eq ( P cav , T ) is lower than the actual concentration c at its posi-tion. Note that the cavitation pressures P cav are randomly distributed sincethe elastic matrix is heterogeneous. However, since cavitation only happens12 r o p l e t d e n s i t y -4 -2 Rate [s -1 ] A B Mesh heterogeneity
NumericalTheory NumericalTheory
Figure 4:
Suppression of cavitation by large droplets explains nu-merical data. (A) Density n of cavitated droplets from numerical simula-tions (black symbols) compared to the analytical prediction (red line) as afunction of the mesh heterogeneity η . (B) n as a function of the rate α withwhich the saturation concentration decreases. (A,B) Model parameters aregiven in Fig. 2, except η/m = 3 · E µm in panel B.for low P cav , it is sufficient to specify the associated cumulative distributionfunction F ( P cav ) to linear order around the lower bound P mincav , F ( P cav ) = P cav − P mincav η Θ( P cav − P mincav ) , (11)where P cav − P mincav (cid:28) η . Here, Θ( x ) is Heaviside’s function and η describeshow widely the small cavitation pressures are distributed. η thus quantifiesthe heterogeneity of the mesh. Considering a homogeneous density m ofnucleated droplets, we can then calculate the expected value of droplets thatcavitate in the volume n − . This theory is self-consistent if exactly onedroplet cavitates in this volume, which provides an implicit condition for thesought density n of cavitated droplets; see SI.The theory does not have any adjustable parameters and we thus compareit directly to our numerical simulations. Fig. 4A shows that the density n ofcavitated droplets decreases when fewer droplets nucleate (smaller m ) or cav-itation thresholds P cav are wider distributed (more heterogeneous network,higher η ). This is because these two parameters define how many nucleateddroplet possess a low enough threshold to cavitate. Conversely, Fig. 4B showsthat more droplets cavitate when the system is cooled faster. Since the total13mount of material taken up by droplets is conserved, this implies smallerdroplets for faster cooling, consistent with Fig. 3B. While our theory showsthe same trends as the numerical simulations, it consistently overestimates n by roughly a factor of in most cases. This is likely because we assumed ahomogeneous distribution of the droplets with the lowest cavitation thresh-old, while in reality two droplets with low threshold might out-compete eachother, effectively leading to a higher cavitation threshold than we anticipate.However, our theory indicates that the cavitated droplets deplete the dilutephase, thus suppressing further cavitation. Since this depletion is diffusion-limited, decreasing temperature slowly implies stronger suppression, leadingto fewer and larger droplets. Increasing cooling rates cause bidisperse emul-sions
We showed that the number and size of the cavitated droplets depends onthe depletion of the dilute phase and thus the cooling rate. This impliesthat additional droplets could cavitate when the cooling rate is increased,while lowering the cooling rate should merely slow down droplet growth.Indeed, experiments by Rosowski et al. showed a bimodal droplet size dis-tribution when the cooling rate was rapidly increased in the middle of theexperiment [11]. To explain this observation, we perform a numerical simula-tion where we rapidly increase the cooling rate well after the first generationof droplets has cavitated. This results in a second generation of cavitateddroplets, which then grow together with the previously cavitated ones; seeFig. 5. We show in the SI that other size distributions are possible whenthe rate is changed multiple times. Taken together, this demonstrates thatdifferent droplet size distributions can be engineered by adjusting the coolingprotocol.
Heterogeneous nucleation might explain more cav-itated droplets in stiffer systems
So far, we have investigated how the density and sizes of the observed dropletsdepend on the cooling rate α . Another important observation of Style et14 B F r e q u e n c y Cavitated droplet radius 1500100050000510 Time t [s] D r o p l e t r a d i u s Figure 5:
Increasing cooling rate yields bidisperse emulsion.
Shownare the droplet radii as a function of time (A) and the droplet size distribution(B) from a numerical simulation with η/m = · E µm , E =
80 kPa, and α = · -6 s -1 ν -1 for t <
960 s , then α = · -5 s -1 ν -1 . Other parametersas in Figure 2.al. [10] is that the droplet density n increases linearly with the Young’smodulus E of the elastic matrix. This implies that stiffer matrices lead tosmaller droplets. Unfortunately, it is difficult to connect E , which measuresthe macroscopic response of the matrix to small strains, to the microscopicdetails required by our model. We thus next consider several possibilities toelucidate which microscopic picture could explain the experimental data.In the simplest case, the bulk modulus E is connected to the pressurecurve P ( R ) . For example, the Neo-Hookean model implies P NH ( R → ∞ ) = E ; see Eq. [8]. The relation is more complicated for the breakage model,but recent experiments [21] indicate that both the cavitation pressure P cav aswell as the pressure P ∞ exhibited by large droplets scale with E . Using thisscaling in our model, we obtain slightly smaller droplet densities for stiffermatrices, opposite to what we expect from the experiments; see Fig. 6A.Consequently, the scaling of the pressure with E cannot explain the observeddata.Our model would yield more (and smaller) cavitated droplets when thedensity m of nucleated droplets was increased. We thus speculate that stiffersystems nucleate more droplets. Indeed, we can explain the observed linearincrease of the density n of cavitated droplets with E by postulating that m strongly increases with E ; see Fig. 6B. So far, it is not clear how dropletsactually nucleate in the elastic network, but it is likely that heterogeneous15 B Young modulus [kPa]0 400 800 1000600200051015 0 10000.51.01.5 [kPa]400 800 1000600200 Young modulus [kPa]12 D r o p l e t d e n s i t y ExperimentalNumericalTheory
Figure 6:
Increasing nucleation density could explain stiffness de-pendence (A) Droplet density n as a function of Young’s modulus E . Ournumerical (black symbols) and analytical (red line) theory, based on a lin-ear scaling of pressures with E , cannot explain the experimental data (bluedashed line, [10]). (B) Predicted nucleation site density m as a function of E to match the measured n ( E ) shown in the inset. Inset: n ( E ) from exper-iments (blue dashed line, [10]) and numerical simulations (black symbols).(A,B) Model Parameters are α = · -5 s -1 ν -1 , and given in Fig. 2nucleation plays a role. For instance, the cross-linking molecules that areused to create the PDMS matrix could act as nucleation sites. In this case,stiffer gels would have more nucleated droplets simply because they containmore cross-linkers [10, 11, 21]. Moreover, stiffer networks might be morehomogeneous [25], which would be capture by a smaller mesh heterogene-ity η . Taken together, these two effects might explain our prediction that theparameter m/η increases strongly with E . Conclusions
We identified a novel mechanism to create monodisperse emulsions, wheresome growing droplets break the surrounding elastic matrix in a cavitationevent. While these droplets become macroscopic, most droplets stay con-strained by the matrix and do not grow significantly beyond mesh size. Thecavitation barrier imposed by the elastic matrix thus separates the stochasticnucleation phase from a deterministic growth phase. The resulting cavitateddroplets have correlated positions and similar sizes, which can be controlled16y the cooling rate. Our model agrees quantitatively with experiments [10,11] and it suggests how this mechanism can be used to create microscopicpatterns in technological applications.Monodisperse emulsion also emerge in other situations of driven phaseseparation. For instance, supplying more droplet material externally [26],internally using solubility gradients [27], or by chemical reactions [28] alllead to narrower droplet size distributions than expected from the standardLifshitz-Slyozov argument [29]. In all these cases, the diffusive flux betweendroplets that normally drives Ostwald ripening is dominated by the flux ofthe supplied droplet material. In our system, all droplets additionally startgrowing at similar times, because they cross the cavitation barrier at similarsaturation concentration. Taken together, this ensures that droplets reachsimilar sizes, despite multiple opposing processes: Beside the heterogeneitiesin the elastic properties that causes the dispersion in our model, thermalfluctuations might also contribute. Moreover, both Ostwald ripening, drivenby surface tension, and Elastic ripening, driven by stiffness gradients overlong length scales [11, 13], will affect the droplet size distribution in realisticsystems. It will be interesting to study all these interactions in the future.We expect similar behaviors for biomolecular condensates, which oftenform as a response to changes in temperature, pH, salt concentration, orprotein concentration in cells [30–33]. Moreover, chemical modifications,like post-translational modifications, allow cells to actively regulate conden-sates [34, 35]. All these changes could in principle drive droplet formation,similar to the cooling in our example. Biomolecular condensates are also typ-ically constrained by elastic matrices [3, 8, 9], which can limit their growth.Moreover, biopolymer gels often rearrange dynamically, implying that themechanical stress exerted by droplets can relax and they can grow furtherakin to the cavitation event in our model. Beyond our current description, therearrangement implies visco-elastic behavior [36–38] and biopolymer gels alsooften display strain-stiffening [39]. There is also the possibility of dropletswetting the mesh instead of excluding it completely [16]. Taken togetherwith the fact that the size of typical condensates is comparable to the gel’smesh size, we thus expect a rich phenomenology. Our theory provides arobust starting point for such future investigations.17 aterials and Methods
The numerical simulations where performed using the py-pde python package[40] using an explicit Euler stepping with a second order discretization of thespatial derivative.
Acknowledgments
We thank Eric Dufresne, Pierre Ronceray, and Robert W. Style for a criticalreview of the manuscript and helpful discussions. For further discussions,we also thank Tal Cohen, Stefanie Heyden, and Noah Ziethen. Funding wasprovided by the Max Planck Society.
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Pollak, D. A. Weitz, andF. C. MacKintosh, “Cross-link-governed dynamics of biopolymer networks”,Phys. Rev. Lett. , 238101 (2010). Q. Wen and P. A. Janmey, “Polymer physics of the cytoskeleton”, CurrentOpinion in Solid State and Materials Science , 177–182 (2011). C. Storm, J. J. Pastore, F. C. MacKintosh, T. C. Lubensky, and P. A. Jan-mey, “Nonlinear elasticity in biological gels”, Nature , 191–194 (2005). D. Zwicker, “Py-pde: a python package for solving partial differential equa-tions”, Journal of Open Source Software , 2158 (2020). C. A. Weber, D. Zwicker, F. Jülicher, and C. F. Lee, “Physics of activeemulsions”, Reports on Progress in Physics , 064601 (2019).21 upplementary Material1 Elastic energy of a growing cavity We consider a droplet nucleated inside a cavity of an elastic material. Thiscavity has original radius A and we look for the elastic energy of the systemonce the droplet has grown and expanded the cavity to a radius a . Theelastic energy F E of the system is F E = (cid:90) ω d r , (S.1)where ω is the energy density of the system and depends on the elastic prop-erties of the media. Assuming a spherical cavity and a perfectly homogenoussystem, the pressure P exerted on the droplet is P = − (cid:90) σr d r , (S.2)where σ is the biaxial stress and r the radial coordinate of the deformed solid.We parametrize the system using the radial stretch λ = r/R , where R is theradial coordinate in the original (non-deformed) solid. The two coordinatessystems are related through volume conservation, R − A = r − a . (S.3)Using the biaxial stress definition σ ( λ ) = λ ω d λ , (S.4)we obtain the pressure P in terms of the energy density, P = (cid:90) a/A λ − ω d λ d λ , (S.5)where we have used λ → at the system’s boundary. Integrating by partswe obtain P = ωλ − (cid:12)(cid:12)(cid:12)(cid:12) a/A + (cid:90) a/A λ ω ( λ − d λ . (S.6)22efining P = A ω ( a/A ) a − A , (S.7)and using the definition of Elastic Energy F E , we get an expression for theelastic energy in terms of the pressure exerted on the droplet F E = 4 π ( a − A )3 ( P − P ) . (S.8)We next check whether this mechanical definition is consistent with the ther-modynamic definition ∂F E ∂V = P, (S.9)where F E is differentiated with respect to the expanded cavity radius V =4 πa / . Differentiating [S.8], ∂F E ∂V = P − P + ( a − A )3 a (cid:18) ∂P∂a − ∂P ∂a (cid:19) , (S.10)using the definition of P , see [S.7], we find ∂F E ∂V = P + 13 a (cid:18) ( a − A ) ∂P∂a − A ∂ω∂a (cid:19) . (S.11)Finally, using [S.5], we obtain ∂P∂a = A a − A ∂ω∂a , (S.12)which combined with [S.11] recovers the thermodynamic pressure given by[S.9]. We now look for a simple expression for the free energy of a growing droplet ina phase separating system. Analysing how this energy changes with volumewill give us an approximation of the effect of an external pressure in thissystem. 23iven a free energy density f and a systems’s size V sys , the total freeenergy is F = V f ( φ in ) + ( V sys − V ) f ( φ out ) + γA + F E ( V ) , (S.13)where φ in is the volume fraction inside the growing droplet, φ out is the volumefraction outside, F E is, as before, the elastic energy, V is the droplet’s volume,and A is the droplet’s surface area.We assume material conservation in the system during the droplet’s ex-pansion, V sys ¯ φ = ( V sys − V ) φ out + V φ in , (S.14)where ¯ φ is the average concentration in the system. Assuming small changesin the dilute phase concentration, we can expand the free energy density, f ( φ out ) ≈ f ( φ ) + f (cid:48) ( φ ) (cid:18) V sys ¯ φ − V φ in V sys − V − φ (cid:19) . (S.15)We can then reorder the free energy as F ≈ F − V g + γA + F E , (S.16)with g = f ( φ ) − f ( φ in ) + f (cid:48) ( φ )( φ in − φ ) (S.17)and F = V sys (cid:2) f ( φ ) + f (cid:48) ( φ )( ¯ φ − φ ) (cid:3) . (S.18)Using the definition of osmotic pressure, Π = − f ( φ ) + f (cid:48) ( φ ) φ , (S.19)we express g using the difference in osmotic pressures, g = Π in − Π − (cid:2) f (cid:48) ( φ in ) − f (cid:48) ( φ ) (cid:3) φ in . (S.20)Finally, using the definition of chemical potential µ = νf (cid:48) ( φ ) , with ν themolecular volume, we obtain g = Π in − Π − (cid:2) µ in − µ (cid:3) c in , (S.21)where c in = φ in /ν is the number concentration inside the droplet. In the caseof an incompressible dense phase, i.e. where φ in is constant, we find that thedriving strength g is independent of droplet volume.24 Stability analysis
We showed in the previous section that the driving strength g is typicallyindependent of the droplet volume V . Phase separation is favorable (in theabsence of surface tension and elastic effects), when g > . To see how elasticeffects affect the phase separation, we differentiate [S.16] with respect to thedroplet volume, ∂F∂V = − g + 2 γR + P E , (S.22)where we have used [S.9] to derive the elastic pressure. Therefore, dropletgrowth is favourable if g > γR + P E . (S.23)A droplet will thus grow as long as the driving strength g is bigger than thetotal pressure difference between the inside and outside of the droplet.We next study the stability of a steady state, which exists when g = P ( R ∗ ) . This state is stable if ∂ F∂V (cid:12)(cid:12)(cid:12)(cid:12) R = R ∗ = ∂∂V (cid:18) γR + P E ( R ) (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) R = R ∗ > . (S.24)Therefore, a droplet will be stable if the pressure increases with increasingradius.The same stability condition can be obtained from the dynamical equa-tions presented in the main manuscript, d R i d t = DR i c in (cid:2) c ( (cid:126)x i ) − c eq (cid:0) P ( R i ) , T (cid:1)(cid:3) . (S.25)A linear stability analysis shows that the droplet radius is stable if c eq in-creases with droplet radius. Since c eq is a monotonically increasing functionof P ( R ) , the stability condition reduces to ∂∂R (cid:18) γR + P E ( R ) (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) R = R ∗ > , (S.26)which is equivalent to Eq. [S.24]. 25 Droplet nucleation
We here study the droplet nucleation behavior by considering the energynecessary to cross the nucleation barrier in the absence of elastic effects. Weconsider a supersaturated homogeneous solution with concentration c anduse the driving strength g , Eq. S.21, to estimate the energy change due toa droplet nucleating. We assume that the differences in pressure relax muchmore quickly than the differences in chemical potential, implying that we canapproximate the driving strength as g ≈ [ µ out − µ in ] c in . (S.27)To estimate its value, we thus need to determine the chemical potentials µ out and µ in outside and inside the droplet, respectively.Assuming that the nucleated droplet is small, the chemical potential out-side is close to that of the homogeneous phase, which can be estimated usingideal solution theory, µ out ≈ k B T log c out . To obtain an upper bound on thenucleation rate, we seek the strongest possible driving strength and thus use c out ≈ c .We estimate the chemical potential µ in inside the droplet by consideringthe phase separated system, which reached an equilibrium between a dilutephase with concentration c eqout and a droplet phase with concentration c eqin . Inequilibrium, the chemical potential of both phases are identical, and we thushave µ eqin = µ eqout ≈ k B T log( c − ∆ c ) , (S.28)where c − ∆ c is a lower bound for the concentration in the dilute phase.Finally, we assume that the chemical potential inside the droplet is con-stant during equilibration, µ in ≈ µ eqin , to obtain the estimate g ≈ k B T c in log (cid:18) c c − ∆ c (cid:19) . (S.29)Consequently, the change in free energy ∆ F due to one droplet of radius R is ∆ F ≈ πR γ − π R g , (S.30)where γ = 4 . / m is the surface tension measured in the experiments [10].The location of the maximum of this curve corresponds to the nucleationradius R nuc = 2 γg ≈ .
09 nm , (S.31)26here we have used the measured value c in k B T = 11 MPa [11]. The associ-ated energy barrier is ∆ F nuc = 16 πγ g ≈ . · − J ≈ . k B T . (S.32)The probability P nuc of droplet nucleation can then be estimated using clas-sical nucleation theory, P nuc = ke − ∆ F nuc /k B T ≈ k · − . (S.33)The pre-factor k in this theory is difficult to estimate, but this expressionshows that the nucleation rate is suppressed by − and homogeneous nu-cleation is thus very unlikely in this system.The fact that droplets appear in the experiments suggests that they arenucleated by alternative paths. We thus propose that heterogeneous nucle-ation is crucial. In heterogeneous nucleation droplets are nucleated aroundsurfaces or imperfections, which effectively lower the nucleation barrier. Theinteraction of the droplet with these nucleation sites scales with the contactarea, which scales with ∝ R if the size of the nucleation site is about R nuc or larger. Consequently, the primary effect of heterogeneous nucleation is tolower the effective surface tension. We thus assume that our system for smalldroplets has an effective surface tension that is smaller than the measuredsurface tension for large cavitated droplets. We thus do not discuss surfacetension effects in the main text and rather assume that droplet nucleationhappens quickly. In the Breakage model, we use a simplified pressure curve, which is pa-rameterized by the mesh size (cid:96) , the cavitation radius R cav , the cavitationpressure P cav , the heterogeneity parameter η , and the final pressure P ∞ forlarge radii; see Fig. S.1. The mesh size (cid:96) only determines when the pressurestarts to increase, and therefore the slope of the pressure curve. Consideringthe cavitated droplet density n , introducing variation in the mesh size (cid:96) isequivalent to variations in P cav for different droplets. Beside this, varyingthe mesh size (cid:96) changes the size of the small droplets, but since they do not27 roplet radius R P r e ss u r e P R cav ‘ P ∞ P cav η Figure S.1:
Stress-strain relation of the Breakage model.
Pressure P asa function of the droplet radius R , which is imposed in numerical simulations.The plot indicates the mesh size (cid:96) , the radius R cav at which the dropletcavitates, the maximal pressure P cav the mesh can exert, and the pressure P ∞ after breakage. The shaded area shows the possible values for P cav giventhe distribution parameter η .affect the number of cavitated droplets, we do not study this effect further.We thus for simplicity only vary P cav and keep (cid:96) the same for all droplets.Moreover, numerical simulations indicate that the cavitation radius R cav hasbasically no influence on the cavitated droplet density n ; see Fig. S.2. Giventhese results we choose not to vary R cav between droplets or simulations, andkeep η/m as our only free parameter to fit experimental data. In this section we estimate the relevant timescale for Ostwald ripening in oursystem to asses its relevance for the cavitated droplets. Ostwald ripening isdriven by surface tension and the relevant scale is the capillary length scale28 avitation radius D r o p l e t d e n s i t y Figure S.2:
Droplet density is independent of cavitation radius
Den-sity n of cavitated droplets as a function of the cavitation radius R cav . Modelparameters as in Fig. 2 of the main manuscript, except η/m = 3 · E µm . (cid:96) γ of the system [41], (cid:96) γ = 2 γc in k B T ≈ . , (S.34)where we have used the surface tension γ = 4 . / m of macroscopic droplets [10].Linear stability analysis shows that the fastest growing mode λ is given by [28] λ = D(cid:96) γ R c eq c in . (S.35)Considering a typical radius R = 10 µm of a cavitated droplet, togetherwith D = 50 µm /s and c eq /c in ≈ c sat (300 K) /c in = 0 . , we find λ ≈ · -6 s -1 . Consequently, the timescale τ = λ − of Ostwald ripening betweenthe cavitated droplets is τ ≈
130 hr . (S.36)Interestingly, the same expression is obtained when considering the criticalradius R c of the Lifshitz–Slyozov scaling law [41] R c ∝ (cid:18) D(cid:96) γ c eq c in t (cid:19) / . (S.37)We thus conclude that Ostwald ripening is a slow process in our system andwe can neglect it. 29 r o p l e t r a d i u s Neo-Hookean Model Breakage Model
A BC D time: 200s time: 800s time: 200s time: 800s
Time t [s] Time t [s] NumericalTheoryFinal temperature 5.0100.0
Figure S.3:
Breakage stops newly nucleated droplets from growing. (A,B) Snapshots of a typical simulation with a constant nucleation rate forNeo-Hookean (left) and Breakage pressure curves (right). Disks indicatedroplets, while the heat map indicates the concentration c in the dilute phase,gray scale indicates depth in the z-axis. (C,D) Droplet radii over time forall nucleated droplets. p nuc = 0.5 s -1 , other parameters as in Figure 2 of themain text. To make a more realistic comparison of the Neo-Hookean model and theBreakage model, we performed numerical simulations with a simple dropletnucleation protocol. The simulations were initialized without any dropletsand at each time step a droplet might nucleate with a probability p nuc . Thenew droplet appears with R < (cid:96) . The typical simulations presented in FigureS.3 show that in the Neo-Hookean model the new droplets can grow freely,leading to a wide range of final radii. In contrast, in the Breakage model,most new droplets get stuck at mesh size and do not grow further, thusproducing a monodispersed emulsion of cavitated droplets.30 o l i d i s p e r s i t y -4 -2 Rate [s -1 ] A B Mesh heterogeneity
ExperimentalNumerical
Figure S.4:
Polydispersity increases at higher droplet densities.
Poly-dispersity, defined as the standard deviation of the droplet radius divded byits mean, for (A) different mesh heterogeneities η/mE and (B) different rates α . Experimental data from [10]. To quantify the position correlations of droplets in our system we defined thepair correlation function g ( r ) which gives the probability of finding a dropletat distance r from the reference droplet. It is formally defined as g ( r ) = V sys N (cid:42)(cid:88) i δ ( (cid:126)r − (cid:126)r i ) (cid:43) , (S.38)where the sum is over all cavitated droplets, the average is over differentensembles, N is the number of cavitated droplets, and V sys the system’svolume. In practice we calculate it as g ( r ) = h ( r ) V sys πN r ∆ r , (S.39)where h ( r ) is the histogram of the distances between droplets (a total of N ( N − elements) and ∆ r is the histogram’s bin size. Finally, to collapse thedifferent curves we normalized by the mean droplet distance / (4 πn )] / .31
100 200 0 100 200 0 100 2000100200 time: 200s time: 800s time: 1300s
Droplet radius
Time
Figure S.5:
Double quench experiment produces a bimodal distribu-tion.
Left panel: Droplet radii as a function of time displaying a bimodaldistribution of the cavitated droplets and several small ones kept at meshsize. Subsequent panels: 2-D projection of a typical time evolution in thissystem, showing droplets starting to cavitate, grown first group of cavitateddroplets, and final state of the system. Smaller droplets are shown as blackdots for didactic purposes and might appear as inside cavitated dropletsdue to 3-D projection. Parameters are η/m = · E µm , E =
80 kPa,first cooling rate α = · -6 s -1 ν -1 , second cooling rate α = · -5 s -1 ν -1 .Other parameters as in Figure 2. n We here provide details on the analytical theory to predict the density n of cavitated droplets. This is based on the simplified picture that a singlelarge droplet depletes a surrounding volume of radius L in a diffusion limitedprocess. To obtain the concentration field around this droplet, we solveEq. (6) in a spherical domain of radius L = [3 / (4 πn )] / with boundaryconditions ∂ r c | r = L = 0 and c ( r = R cav ) = c sat ( t ) exp ( P ∞ / ( c in k B T )) . Defining ¯ c = c + αt exp ( P ∞ / ( c in k B T )) turns Eq. (6) into ∂ t ¯ c = D ∇ ¯ c + α exp (cid:18) P ∞ c in k B T (cid:19) − c in (cid:88) i d V i d t δ ( (cid:126)x i − (cid:126)x ) , (S.40)with the simpler boundary conditions ∂ r ¯ c | r = L = 0 and ¯ c ( r = R cav ) = c exp ( P ∞ / ( c in k B T )) .Assuming radial symmetry and steady state, the field around a cavitated32roplet of radius R cav reads x‘ ¯ c ( r ) = (cid:104) c + α D (cid:0) R cav − r (cid:1) + αL D (cid:18) R cav − r (cid:19)(cid:21) exp (cid:18) P ∞ c in k B T (cid:19) . (S.41)We thus simplified the cavitation scenario by assuming that once a dropletcavitates it absorbs material in a sphere of radius L around them.To account for heterogeneity in the cavitation pressures P cav , we describethem through their cumulative distribution function F ( P cav ) , which gives thefraction of droplets whose cavitation pressure is lower than P cav . Since onlydroplets with the lowest cavitation thresholds will cavitate, we only needto describe the lower end of F ( P cav ) . Since there must be a lowest, positivecavitation pressure P mincav , we assume a linear expansion around this minimum, F ( P cav ) = P cav − P mincav η Θ( P cav − P mincav ) , (S.42)where η describes how widely distributed the lower thresholds are. This ex-pression can be rewritten using Eq.(5) to obtain the cumulative distributionin terms of the equilibrium concentrations. F ( c, t ) = (cid:20) c in k B Tη log (cid:18) cc − αt (cid:19) − P mincav η (cid:21) Θ (cid:18) c in k B T log (cid:18) cc − αt (cid:19) − P mincav (cid:19) . (S.43)Therefore, given a density m of nucleated droplets, the density of dropletswith cavitation threshold below c is m F ( c, t ) . For this theory to be self-consistent, the aforementioned volume V = n − must have only one dropletwith a cavitation threshold below the concentration field, i.e. π (cid:90) L m F ( c ( r ) , ¯ t ) r dr. (S.44)Here, we take the time ¯ t such that the equilibrium concentration of the n -thdroplet matches the total amount of material in the system c , ¯ t = c α (cid:20) − exp (cid:18) − P mincav − ηn/mc in k B T (cid:19)(cid:21) . (S.45)33 B F r e q u e n c y Cavitated droplet radius 100050000510 Time t [s] D r o p l e t r a d i u s Figure S.6:
Increasing rate stepwise leads to increasingly moredroplet cavitating.
Simulation increasing the material rate α twice duringthe cooling process. A) Droplet radii as a function of time displaying a tri-modal distribution of the cavitated droplets and several small ones kept atmesh size. B) Radii probability distribution. Parameters are η/m = · E µm , first cooling rate α = α = · -6 s -1 ν -1 , second cooling rate α = 4 α ,and third cooling rate α = 16 α . Other parameters as in Figure 2.We can now combine [S.41]-[S.45] to obtain an implicit relation for the cavi-tated droplet density n , which we solve numerically to obtain the lines shownin the main text.
10 Temperature protocol changes
Numerical simulations, where the cooling rate is increased after droplet cav-itation, show that a new group of droplets can cavitate if the new rate ishigh enough. Examples of this type of simulations are shown in Fig. S.5 andFig. S.6. We show in [S.41] that the concentration profile around a cavitateddroplet depends strongly on the material rate α . If this rate is suddenlyincreased, the new concentration profile might be higher than the cavitationconcentration c cav of some of the small droplets, thus causing their cavita-tion. This process can then be repeated to cavitate additional droplets, asdepicted in Fig. S.6. Therefore, controlling the rate αα