THESE DE DOCTORAT Spécialité: Chimie Physique

Populations of droplets or particles dispersed in aliquid may evolve through Brownian collisions, aggregation, andcoalescence. We have found a set of conditions under which thesepopulations evolve spontaneously toward a narrow sizedistribution. The experimental system consists of poly(methylmethacrylate) (PMMA) nanodroplets dispersed in a solvent(acetone) + nonsolvent (water) mixture. These droplets carryelectrical charges, located on the ionic end groups of themacromolecules. We used time-resolved small angle X-rayscattering to determine their size distribution. We find that thedroplets grow through coalescence events: the average radius ⟨R⟩increases logarithmically with elapsed time while the relativewidth σR/⟨R⟩ of the distribution decreases as the inverse square root of ⟨R⟩. We interpret this evolution as resulting fromcoalescence events that are hindered by ionic repulsions between droplets. We generalize this evolution through a simulation ofthe Smoluchowski kinetic equation, with a kernel that takes into account the interactions between droplets. In the case ofvanishing or attractive interactions, all droplet encounters lead to coalescence. The corresponding kernel leads to the well-known“self-preserving” particle distribution of the coalescence process, where σR/⟨R⟩ increases to a plateau value. However, for dropletsthat interact through long-range ionic repulsions, “large + small” droplet encounters are more successful at coalescence than“large + large” encounters. We show that the corresponding kernel leads to a particular scaling of the droplet-size distributionknown as the “second-scaling law” in the theory of critical phenomena, where σR/⟨R⟩ decreases as 1/√⟨R⟩ and becomesindependent of the initial distribution. We argue that this scaling explains the narrow size distributions of colloidal dispersionsthat have been synthesized through aggregation processes. ■ INTRODUCTIONThe fate of unstable systems in the colloidal domain isdetermined by the outcome of encounters between colloidalobjects, e.g., particles or droplets. This is the case of emulsions,made of droplets of one liquid dispersed into anothernonmiscible liquid. Since the two liquid phases are nonmiscible,there is an interfacial free energy cost proportional to the areaof interface. Droplets encounter each other through Browniancollisions and may then aggregate and coalesce, since this willreduce the area of interface and thus the total interfacial freeenergy. How will the population of droplets evolve throughthese coalescence events? In surfactant stabilized emulsions, itis usually found that encounters of two large droplets are moresuccessful than those of a large and a small droplet. Indeed ifthe limiting step for coalescence is the rupture of the surfactantfilm that separates droplets, then the larger area of the filmseparating two large droplets will lead to a higher coalescenceprobability. Consequently, large droplets will grow faster thanthe rest of the population, and the distribution of droplet sizeswill become more and more polydisperse as film rupture eventsoccur. Thus, the effect of coalescence in emulsions is usually toyield a few very large drops that grow at the expense of the restof the population. The same trend is predicted in dry foams.A similar fate is observed in dispersions of colloidal particles,where interfacial free energy or interfacial reactions may causethe individual particles to aggregate. Experimental observationsand numerical observations show that in most cases theseaggregates turn out to have disordered self-similar structures,and size distributions that are quite polydisperse. Indeed,during collisions interfacial forces maintain the particles in theconfiguration in which they stuck to each other. The repetitionof such aggregation events yields clusters with branched, bushystructures where the branches are more likely to react than theinner surfaces of the aggregates. In the reaction limited clusteraggregation (RLCA) processes, where only a fraction of allencounters are successful, the larger clusters are more successfulat reacting, again because they have a larger surface areaavailable for reactions, and the population of clusters becomes Received: February 7, 2013Revised: March 22, 2013Published: April 9, 2013Article pubs.acs.org/Langmuir © 2013 American Chemical Society5689dx.doi.org/10.1021/la400498j | Langmuir 2013, 29, 5689−5700pastel-00921639,version1-20Dec2013 more and more polydisperse as the reactions proceed. In thediffusion limited cluster aggregation (DLCA), where all theencounters are successful, the population of clusters reaches aconstant polydispersity, lower than for RLCA but often higherthan what is desired in practical purposes.The view that aggregation/coalescence processes always leadto polydisperse populations of disordered aggregates is socommonly held that it has often been taken as a universal trend.Consequently, colloid chemists who attempt to synthesizemonodisperse collections of colloidal particles take greatprecautions to avoid aggregation/coalescence: a classical reviewby Sugimoto states that: “Thus, as a rule, inhibition ofcoagulation is necessary for the preparation of monodisperseparticles”. If possible, colloid chemists try to grow the particlesaccording to the “nucleation and growth” scheme. In thisscheme, the supersaturation of the solution is kept at a lowvalue, so that the number of successful nuclei is small, and thesenuclei are kept apart form each other, so that no aggregationtakes place, while the nuclei grow through monomer addition.However, a few authors such as Zukoski and co-workers andMatijevik and co-workers have reported that monodispersepopulations of colloidal particles can be obtained at very highsupersaturation, through processes in which a large number ofvery small nuclei are generated, which subsequently aggregatewith each other to form the final particles. This view hasrecently received additional experimental support from thework of Polte and co-workers. Such reports contradict theidea that aggregation processes always lead to polydispersepopulations. Furthermore, the possibility of using highsupersaturations to produce the desired particles is attractive,since it opens the way to higher productivity and simplerprocesses.In order for an aggregation/coalescence process to lead to a“monodisperse” population, it is necessary that the reactions ofsmall objects with larger ones be more frequent than thereactions of large with large objects. This requires a particulartype of interaction between these objects. In the present work,we show that colloidal droplets of a concentrated polymersolution, dispersed in a nonsolvent, may interact in this way iftheir surfaces carry electrical charges bound to the macro-molecules, and the nonsolvent is water or a solution of waterwith a polar solvent. These droplets repel each other throughionic long-range interactions, and these repulsions are strongerfor larger droplets, because they carry more electrical charges.We use a model system comprising macromolecules ofpoly(methyl methacrylate) (PMMA) in a solvent (acetone)that is mixed quickly with water, using a stopped-flow device.We follow the kinetics of the droplet coalescence processthrough ultrafast small angle X-ray scattering (SAXS). Whereasprevious works have used stopped-flow SAXS to studymetal or oxide particle formation, involving reactive species, ourwork monitors for the first time the formation of organicparticles without any chemical reaction (Ouzo effect). TheSAXS spectra show that the droplet population evolves throughcoalescence events, and yet this population propagates torelatively narrow size distributions. We show that these resultscan be explained within the frame of coalescence eventshindered by a size-dependent repulsive interdroplet potential.The observation of coalescence events that decrease thepolydispersity of a distribution of droplets is also importantbecause it represents a new scaling of the solutions of theSmoluchowski equation. By scaling, we mean that thepopulations that result from processes that run according tothe Smoluchowski equation have size distributions that are allsimilar to each other, and can be deduced from each other by asimple transformation such as the application of a scale factor.The well-known first scaling is obtained for Brownian collisionsand corresponds to a size distribution that keeps a constantrelative width σR/⟨R⟩, while the radius grows as the power 1/3of time. Here we identify the second scaling through kineticMonte Carlo simulations and comparison with the kineticSAXS experiments. Ionic repulsions between droplets lead tothis new scaling law where the relative width σR/⟨R⟩ decreasesas⟨R⟩−1/2, while the radius grows logarithmically with elapsedtime. This brings a new perspective on the concept of colloidalmetastability.This paper is organized in the following way. At first, wepresent the phase diagram of the PMMA−acetone−watersystem, and we describe the solvent-shifting method that weused to produce droplets of concentrated PMMA solutiondispersed in the nonsolvent mixture (water−acetone). Next wedescribe how the growth of these droplets was monitoredthrough ultrafast SAXS, and how we extracted the sizedistributions of droplets through an inversion of the scatteringcurves. Then we write the kernel of the Smoluchowski equationaccording to the interactions between droplets, and comparethe predictions from this equation between the case of nointeractions and the case of repulsive interactions with theexperimental evolution of the droplet populations observedthrough SAXS. We show that coalescence without repulsionsleads to a high polydispersity with the first scaling law, whereascoalescence of repelling droplets leads to a low polydispersitywith the second scaling law. Finally we show how the outcomeof any process involving the coalescence of droplets or particles,limited by ionic repulsions, may be predicted through theSmoluchowski approach. ■ EXPERIMENTAL SECTIONMaterials. Poly(methyl methacrylate) (PMMA) was purchasedfrom Aldrich. Size exclusion chromatography in THF yielded a weight-average molar mass of 14700 g/mol with a polydispersity of 1.54.Acetone was purchased from Aldrich with an assessed purity of 99.9%and milli-Q water was adjusted to pH = 10 using a sodium hydroxideaqueous solution; the ionic strength of water was set with sodiumchloride to 1.5 × 10−2 mol.L−1.Methods. Controlled and Rapid Solvent-Shifting with aStopped-Flow Apparatus. We used a Bio-Logic SFM-400 stopped-flow instrument, consisting of four motorized syringes and threemixers. The last mixer was coupled to a thin-walled flow-throughquartz capillary cell (1.3 mm in diameter with a wall thickness ofapproximately 10 μm). The total dead time was estimated to below 5ms. The first syringe was filled with a solution of PMMA in acetone ata mass fraction of 10−2 (volume fraction 0.66 × 10−2) . The lastsyringe was filled with water at a pH of 10 at an ionic strength of 1.5 ×10−2 mol.L−1. The two middle syringes were filled with pure acetone.One experiment consists of one injection; however, severalexperiments are necessary to perform a time-resolved sequence. It isthus essential to achieve an excellent reproducibility between thesuccessive injections. The mixing reproducibility is ensured by the useof the stopped-flow apparatus; however, it is also important to preventany mixing contamination of the syringes between two successiveexperiments. The main problem arises from Marangoni effects insidethe device, which may cause contamination in the syringes, since theyare opened to the whole device during an experiment.We designed a strict protocol to avoid these difficulties. ThePMMA/acetone solution was injected with the water in the finalmixing cell and a set of spectra was acquired. Immediately after thesemeasurements, pure acetone from the intermediate syringes wasinjected in order to create an acetone buffer between the PMMA/LangmuirArticle dx.doi.org/10.1021/la400498j | Langmuir 2013, 29, 5689−57005690pastel-00921639,version1-20Dec2013 acetone solution and the water syringes. Then the syringes of thestopped flow were closed manually until the next experiment wasperformed. We used a video camera to follow the macroscopic eventstaking place in the capillary cell and we checked that the cell was filledhomogeneously with no turbid domains, within the experimental timeframe.Ultrafast Small Angle X-ray Scattering (SAXS) with a Stopped-Flow Apparatus. SAXS experiments were performed on samplesprepared through rapid mixing of the PMMA/acetone solution withwater, injected in the capillary flow-cell. SAXS patterns were collectedat the European Synchrotron Radiation Facility, at the high-brilliancebeamline ID02 (wavelength = 0.1 nm, detector distance 8 m, cross-section of the beam at the sample holder = 0.3 mm). The X-rayscattered intensity was recorded using a FReLoN (fast-readout low-noise) Kodak CCD detector in the q-range of 0.01−0.6 nm−1 with ahigh sensitivity and count rate capability. This unique instrument witha monochromatic, highly collimated, and intense beam makes itpossible to perform millisecond time-resolved stroboscopic experi-ments of SAXS.The duration of each measurement was 5 ms and the minimal timeinterval between two measurements was 200 ms. To obtain data in therange 5−205 ms, we thus had to perform several experiments withvarious initial measurement times. This requires good reproducibilityof all the processes, which is possible when using a stopped-flow devicewith the protocol detailed above.The SAXS data was then reduced, i.e., normalized, grouped intoone-dimensional spectra, and background-subtracted. The backgroundwas the mixture of acetone and water made through a stopped-flowexperiment with the same parameters as for the PMMA/acetonesolutions. The intensities were obtained on an absolute scale.Experimental Coalescence Laws through Inversion of the SAXSData. We used a numerical optimized inversion procedure, based onthe Titchmarsh formula, to obtain the size distribution of the dropletsfrom the SAXS data. This method computes the size distributionsfrom the deviations of the scattering signal to the Porod law, usingShannon’s theorem for the available amount of information. Theprocedure does not require extrapolation of the data beyond theexperimental q-range, and it is known to provide a reliable and stableway to assess possible broad distributions. The quality of the inversionprocedure was assessed by calculating the scattering curves from theinverted size distributions and comparing them with the experimentalspectra.Theoretical Coalescence Laws through Kinetic Monte CarloSimulations of Charged Droplets. We realized a number of kineticMonte Carlo simulations to solve the coalescence Smoluchowskiequations including van der Waals and electrostatic interactions.Systems containing up to N = 2 × 10 monomers, with monodisperseinitial conditions, have been studied that way. The general algorithmfor the simulations is as follows.At each time step: • All the probabilities per unit of time of possible coalescenceevents are listed. Probabilities are noted:Kij, with i and j thelabels of the colliding droplets.• The average time for the next coalescence event to occur is τ =1/∑ i,jKij . Then, the normalized event probabilities are τKij.• A time increment, δt, is chosen randomly with the exponentialdistribution of parameter τ (i.e., f(δt) = 1/τ exp(−δt/τ)).• An event is chosen randomly in the list of all the possiblecoalescence events, according to the probabilities τKij. Thisevent is realized, and the current time is incremented by δt. The sequence of coalescence events goes on this way until theexperimental times are reached. That way, we have access to thetheoretical evolution of the physical observables, such as the averagevalue and the standard deviation of the droplet radius. ■ RESULTSSolvent-Shifting Produces a Collection of Dropletswith Ionized Interfaces. Solvent-shifting is a mixing methodthat throws a solution into a state that is far from equilibrium,and uses the subsequent evolution to produce a fine dispersionof the solute. Typically, a polymer/good solvent solution israpidly mixed with a nonsolvent to cross the equilibrium phaseseparation boundary (binodal line). The equilibrium state isthen the coexistence of two macroscopic phases, which isquantitatively described by the phase diagram of the ternarysystem. However, interactions between the separating domainsmay restrict the phase separation to the colloidal scale. Thesystem is then conveniently described as a homogeneouscolloidal system, i.e., a dispersion of droplets in a liquid.Nonetheless, since the phase diagram indicates the composi-tions of the dispersed phase and of the dispersing liquid, itsstudy is valuable to understand the nonequilibrium pathways.We therefore first locate the solvent-shifting pathways in thephase diagram, describe the phase separation process, and thenshow that the system, trapped in a nonequilibrium state, can bedescribed as a colloidal one-phase system.We used the system PMMA/acetone (good solvent) + water(nonsolvent). The phase diagram of this ternary system hasbeen determined experimentally by Aubry and co-workersand analytically through the Flory−Huggins−Stockmayertheory by Cohen-Addad. Figure 1 presents this phase diagram in coordinates that are the ratio,φPMMA/φa, ofPMMA volume fraction to acetone volume fraction, and thewater volume fraction, φw. At low water volume fractions thehomogeneous solution of PMMA in the acetone-rich solventmixture is stable, whereas at high water volume fractions thestable state is the coexistence of a polymer-rich phase, whichwill constitute the droplets, with a water−acetone solutioncontaining a small amount of polymer. These regions areseparated by the equilibrium phase separation line, or binodalline. Most of this binodal line is located at very low volumefractions, and for this reason a log scale was used for thehorizontal axis.Figure 1. Phase diagram of the PMMA/acetone/water system.Horizontal axis: ratio of the PMMA volume fraction to the acetonevolume fraction,φPMMA/φa, logarithmic scale. Vertical axis: watervolume fraction,φw. The solvent-shifting path is the ascendant verticalline. When the system crosses the binodal line (red), phase separationtakes place. High values of the supersaturation are quickly reached.The composition used in the present experiments is indicated by afilled diamond.LangmuirArticle dx.doi.org/10.1021/la400498j | Langmuir 2013, 29, 5689−57005691pastel-00921639,version1-20Dec2013 In a solvent-shifting experiment, the solution starts at φw = 0(no water) and follows a vertical path during solvent shifting(addition of water). With moderate water additions (φw = 0.2−0.5), the system reaches the two-phases region where it issupersaturated with respect to the polymer. The super-saturation S is the ratio of the actual polymer concentrationto its equilibrium concentration, which can be read at theintersection of the tie line with the binodal line. Since thebinodal line is quite flat, a small water addition above thebinodal line produces a huge supersaturation. The compositionused in the present study is displayed on the phase diagram andcorresponds to a PMMA/acetone volume ratio φPMMA/φa = 6.6× 10−3 and water volume fraction φw = 0.33. The super-saturation is above 3000. The phase diagram allows thecalculation of the equilibrium swelling of a macromolecule inthe mixed solvent water/acetone. Knowing the radius of acollapsed macromolecule from its molar mass and density,Rcollapsed = 1.7 nm, we can deduce the radius of themacromolecule in the mixed solvent at equilibrium to R1 =2.2 nm.In dilute conditions, metastable dispersions of PMMAdroplets with sizes around 100 nm and low polydispersity areobtained. Metastability requires repulsive barriers between thedroplets to prevent relaxation toward the equilibrium macro-scopic phase separation. We found through electrophoreticmobility measurements that these droplets are negativelycharged with a pH-dependent variation that is typical ofcarboxylic acid moieties (see Supporting Information). We havedemonstrated in previous work that even macromolecules thathave no ionizable monomers can carry a charge at their endgroups, where the initiator started the polymerization duringthe synthesis of the macromolecules. If the initiator does notbear any ionizable group, the macromolecules cannot provideany charge, and the surface charge of the polymer waterinterface vanishes. In that case, the dispersion does not showmetastability, in contrast to the case of oil droplets that alwayscarry a surface charge due to fatty acid impurity adsorption andreaction with hydroxide ions. We can obtain the surfacecharge density from the variation with pH of the electro-phoretic mobility of the final polymer droplets σ = 0.025 ±0.005 e.nm−2. Assuming a simple geometrical model in whichthe macromolecules at the interface of the polymer dropletcontribute to the surface charge, we can estimate the surfacecharge density σ = 0.0236 e.nm−2.Solvent-shifting is also sensitive to other parameters such asthe PMMA/acetone ratio and the mixing conditions. In thiswork we kept these parameters constant, by choosing a setPMMA/acetone ratio and operating under fast mixingconditions. This was important since bad mixing leads tomixtures of regions of different compositions, which evolve atdifferent rates, leading to larger sizes and broader distributionsof the droplet population. The most favorable hydrodynamicsituation is thus the fast, turbulent mixing of the two liquids, asachieved in the stopped flow device.We investigate the mechanism through which monodispersedroplets are obtained in conditions where homogeneoussolvent-shifting is achieved through fast mixing and theresulting colloidal droplets repel through ionic interactions.Ultrafast SAXS Monitors the Droplet Population inReal Time. The PMMA/acetone solution was mixed withwater and injected in the SAXS capillary cell in 5 ms. This steptriggered the first measurement time, which took 5 ms.Measurements were performed using several initial times inorder to obtain time-resolved scattering patterns. In Figure 2,we plot I(q).q as a function of q, for six measurement times.Such a plot displays efficiently the temporal evolution of thespectra. Droplets Grow by Coalescence. The average size of thescatterers can be determined through model-independentanalysis of the SAXS data. Indeed, whenlimq→∞q I(q) is finite,the surface area of the scattering phase is given by the ratio oftwo invariants, namely ∫π=→∞∞AVq I q q I q qlim ( )/ ( ) dq4