Coarsening Of Droplets Research Articles

Formation and coarsening of Ga droplets on focused-ion-beam irradiated GaAs surfaces

We have investigated the formation and coarsening of Ga droplets on focused-ion-beam (FIB) irradiated GaAs surfaces. To separately examine formation and coarsening, Ga droplets were fabricated by Ga+ FIB irradiation of GaAs substrates with and without pre-patterned holes. We determined the droplet growth rate and size distribution as a function of FIB energy following irradiation. The data suggest a droplet formation mechanism that involves Ga precipitation from a Ga-rich layer, followed by droplet coarsening via a combination of diffusion and Ostwald ripening or coalescence via droplet migration (dynamic coalescence).

Coarsening through directed droplet coalescence in fluid-fluid phase separation

Phase-separation dynamics of an asymmetric mixture of an isotropic dopant in a nematogenic fluid is presented. We show that, on steady cooling, the nucleating nematic drops move down the dopant concentration gradient, with a velocity that is dependent on the cooling rate and concentration gradient. This propulsion of the drops leads to a mechanism of droplet coarsening, where radius of a drop scales with time as R(t) approximately t. Various mechanisms for droplet propulsion are discussed.

Reduced droplet coarsening in electromagnetically levitated and phase-separated Cu–Co alloys by imposition of a static magnetic field

The paper reports a novel method for investigating the kinetics of liquid-phase separation, which combined conventional electromagnetic levitation with a superconducting magnet. The experiments on Cu50Co50 alloys revealed a singular size distribution for the minor Cu-rich phase in a Co-rich matrix under a magnetic field of 2 T, which differed from a bimodal distribution under a zero field. Such an effect of the magnetic field is similar to that of reduced gravity during parabolic flights.

MANY DROPLET PATTERN IN THE CYLINDRICAL PHASE OF DIBLOCK COPOLYMER MORPHOLOGY

The Ohta–Kawasaki density functional theory of diblock copolymers gives rise to a nonlocal free boundary problem. In a proper range of the block composition parameter and the nonlocal interaction parameter, an equilibrium pattern of many droplets exists in a general planar domain. A sub-range of the parameters is identified where the multiple droplet pattern is stable. This stable droplet pattern models the cylindrical phase in the diblock copolymer morphology. Each droplet is close to a round disc. The boundaries of the droplets satisfy an equation that involves the curvature of the boundaries and a quantity that depends nonlocally on the whole pattern. The locations of the droplets are determined via a Green's function of the domain. In constructing the droplet pattern we overcome three obstacles: interface oscillation, droplet coarsening, and droplet translation.

Deterministic model for matrix solidification in liquid–liquid thermally induced phase separation

A deterministic approach was taken to predict membrane morphologies resulting from the matrix solidification step that occurs in the formation of microporous membranes via liquid–liquid thermally induced phase separation (L–L TIPS). Many studies have examined the growth rate of droplets in the isothermal coarsening stage of membrane formation, but few have attempted to extend this information into the subsequent processing steps of matrix solidification, diluent extraction/exchange, and drying. This research is the first to take into account the compositional change of the matrix phase during non-isothermal quenching of an L–L TIPS system that has gone through droplet coarsening. In the research presented here on matrix solidification, the approach taken utilized Monte-Carlo routines to provide quantitative information on cell size and cell size distribution for a representative L–L TIPS system, isotactic poly(propylene)–diphenyl ether. The model accounts for diluent expulsion from the polymer-rich phase, continued droplet growth during the solidification process, and shrinkage due to crystallization of the polymer-rich phase. The predicted structures were compared to experimentally formed membranes.

In situ annealing of GaN dot structures grown by droplet epitaxy on (1 1 1) Si substrates

The effect of in situ annealing was investigated on GaN dots grown on Si(1 1 1) substrates by droplet epitaxy. In particular, the effect of irradiation by nitrogen plasma during annealing was examined. The density of GaN dots on the sample annealed with nitrogen plasma was approximately 1.0×10 11 cm −2, which was much higher than the density of dots on the sample without nitrogen plasma (1.2×10 8 cm −2). X-ray photoelectron spectroscopy showed that the Ga 2p peaks of the samples annealed both with and without irradiation by nitrogen plasma consisted mainly of Ga–N peaks, while that of the unannealed sample contained an extremely large Ga–O component, which is observed at the higher binding energy side of the Ga–N peak. These results indicate that the degree of nitridation of the dots was much improved by annealing. The high density of GaN dots on the sample annealed with nitrogen plasma, in which case the dot diameters were as small as 20 nm, was presumably enabled through the suppression of both the re-evaporation of Ga atoms and the coarsening of Ga droplets by irradiation by the surface by nitrogen plasma.

Late-stage coarsening of oil droplets of excess oil in microemulsions following a temperature quench

An oil-in-water microemulsion of the system C 12 E 5 /water/ n-decane with a fixed surfactant-to-oil volume fraction ratio φ s /φ o of 0.85 is studied. This clear microemulsion is temperature-quenched from a clear one-phase region (26 °C) with spherical microemulsion droplets to an unstable region (13 °C) where the excess oil phase is in equilibrium with the microemulsion droplets. Using a high-resolution version of a laboratory small-angle X-ray scattering (SAXS) instrument, the evolution of the excess oil and its growth as a function of time is studied. Unlike coarsening in alloys and polymer blends, a maximum in the intensity at a finite scattering vector is not observed. This clearly indicates that the droplet of the excess oil is formed homogeneously and that there is no depletion zone around the formed droplets. The scattering curves are quantitatively analysed as a function of time. The microemulsion droplets are modeled as polydisperse core-shell ellipsoidal particles, using molecular constraints and the oil droplets are modeled as polydisperse spheres. The average radius of the oil droplets increases with a scaling exponent in agreement with that of coarsening of precipitates in alloys and phase separations in polymer blends, suggesting some degree of universality of coarsening processes.

Stochastic diffusion interactions and coarsening in a system of droplets growing from a supersaturated gas mixture

In this work we study diffusion interactions among liquid droplets growing in stochastic population by condensation from supersaturated binary gas mixture. During the postnucleation transient regime collective growth of liquid droplets competing for the available water vapor decreases local supersaturation leading to the increase of critical radius and the onset of coarsening process. In coarsening regime the growth of larger droplets is prevailing noticeably broadening the droplet size-distribution function when the condensation process becomes more intensive than the supersaturation yield. Modifications in the kinetic equation are discussed and formulated for a stochastic population of liquid droplets when diffusional interactions among droplets become noteworthy. The kinetic equation for the droplet size-distribution function is solved together with field equations for the mass fraction of disperse liquid phase, mass fraction of water vapor component of moist air, and temperature during diffusion-dominated regime of droplet coarsening. The droplet size and mass distributions are found as functions of the liquid volume fraction, showing considerable broadening of droplet spectra. It is demonstrated that the effect of latent heat of condensation considerably changes coarsening process. The coarsening rate constant, the droplet density (number of droplets per unit volume), the screening length, the mean droplet size, and mass are determined as functions of the temperature, pressure, and liquid volume fraction.

Coarsening mechanism of phase separation caused by a double temperature quench in an off-symmetric binary mixture

We study phase-separation behavior of an off-symmetric fluid mixture induced by a "double temperature quench." We first quench a system into the unstable region. After a large phase-separated structure is formed, we again quench the system more deeply and follow the pattern-evolution process. The second quench makes the domains formed by the first quench unstable and leads to double phase separation; that is, small droplets are formed inside the large domains created by the first quench. The complex coarsening behavior of this hierarchic structure having two characteristic length scales is studied in detail by using the digital image analysis. We find three distinct time regimes in the time evolution of the structure factor of the system. In the first regime, small droplets coarsen with time inside large domains. There a large domain containing small droplets in it can be regarded as an isolated system. Later, however, the coarsening of small droplets stops when they start to interact via diffusion with the large domain containing them. Finally, small droplets disappear due to the Lifshitz-Slyozov mechanism. Thus the observed behavior can be explained by the crossover of the nature of a large domain from the isolated to the open system; this is a direct consequence of the existence of the two characteristic length scales.

The role of interfacial rheological properties on Ostwald ripening in emulsions

The coarsening of emulsion droplets by Ostwald ripening is studied by means of numerical simulations in which time-dependent (elastic) interfacial behaviour is taken into account. Theoretical calculations on the dissolution of a single emulsion droplet in an infinite medium at saturated conditions show that the dissolution process can be stopped only when the interfacial tension goes to zero. When interfacial stress relaxation is included, which prevents a continuous zero interfacial tension, no stabilisation of the dissolution process is observed and the droplet dissolves completely. In the case of an ensemble of droplets, numerical calculations on the coarsening of emulsion droplets with finite interfacial elasticity show that a stable situation occurs at finite interfacial tensions of the droplets. This applies for a closed system with the same assumptions as those made in the Lifshitz–Slyozov–Wagner (LSW) theory. The coarsening behaviour strongly depends on the saturation of the dispersed phase in the continuous phase. If the system is in contact with atmosphere, saturation will finally go to unity and stabilisation will only occur for zero interfacial tension of the droplets. For an ensemble of droplets in a closed system, the calculations show that stress–relaxation of the interface causes the Ostwald-ripening process to continue, so no stable situation is reached. Stabilisation can only be accomplished by adding insoluble species to the dispersed phase, by using particles as stabilisers or by micro-encapsulation of the emulsion droplets by thick insoluble interfacial layers, which have a thickness that is in the order of the radius of the droplet.

The role of interfacial rheological properties on Ostwald ripening in emulsions

The coarsening of emulsion droplets by Ostwald ripening is studied by means of numerical simulations in which time-dependent (elastic) interfacial behaviour is taken into account. Theoretical calculations on the dissolution of a single emulsion droplet in an infinite medium at saturated conditions show that the dissolution process can be stopped only when the interfacial tension goes to zero. When interfacial stress relaxation is included, which prevents a continuous zero interfacial tension, no stabilisation of the dissolution process is observed and the droplet dissolves completely. In the case of an ensemble of droplets, numerical calculations on the coarsening of emulsion droplets with finite interfacial elasticity show that a stable situation occurs at finite interfacial tensions of the droplets. This applies for a closed system with the same assumptions as those made in the Lifshitz–Slyozov–Wagner (LSW) theory. The coarsening behaviour strongly depends on the saturation of the dispersed phase in the continuous phase. If the system is in contact with atmosphere, saturation will finally go to unity and stabilisation will only occur for zero interfacial tension of the droplets. For an ensemble of droplets in a closed system, the calculations show that stress–relaxation of the interface causes the Ostwald-ripening process to continue, so no stable situation is reached. Stabilisation can only be accomplished by adding insoluble species to the dispersed phase, by using particles as stabilisers or by micro-encapsulation of the emulsion droplets by thick insoluble interfacial layers, which have a thickness that is in the order of the radius of the droplet.

Isothermal aggregation of Bi atoms embedded in a soda borate glass: Coarsening of liquid nanodroplets and atomic diffusion

The process of nucleation and growth of liquid Bi nanodroplets embedded in a soda borate glass submitted to isothermal annealing at different temperatures was studied by small-angle x-ray scattering (SAXS) and transmission-electron microscopy. The experimental results indicate that the formation and growth of Bi droplets occur in two successive stages after a short incubation period. The first is characterized by the nucleation and growth of spherical droplets promoted by atomic diffusion and aggregation of isolated Bi atoms and the second one by a subsequent droplet coarsening. The experimental functions describing the time variation of the droplet average radius and density number at advanced stages of the growth process agree with the classical Lifshitz-Slyozov-Wagner (LSW) theory. However, the radius distribution was demonstrated to be well described by a log-normal function thus differing from the prediction of the LSW model. The atomic diffusion coefficient of Bi was determined from SAXS results for several annealing temperatures and, from it, the activation energy for the diffusion process was inferred.

Effect of Convection on Domain Growth During Demixing Transitions in Fluids

Current analytical work on the effect of convection on the late stages of spinodal decomposition in liquids is briefly described. The morphology formed during the spinodal decomposition process depends on the relative composition of the two species. Droplet spinodal decomposition occurs when the concentration of one of the species is small. Convective transport has a significant effect on the scaling laws in the late-stage coarsening of droplets in translational or shear flows. In addition, convective transport could result in an attractive interaction between non-Brownian droplets which could lead to coalescence. The effect of convective transport for the growth of random interfaces in a near-symmetric quench was analysed using an area distribution function, which gives the distribution of surface area of the interface in curvature space. It was found that the curvature of the interface decreases proportional to time t in the late stages of spinodal decomposition, and the surface area also decreases proportional to t .

Effect of Surface Elasticity on Ostwald Ripening in Emulsions

The coarsening of droplets in an emulsion with a size distribution that initially is given by the Lifshitz−Slyozov−Wagner (LSW) distribution is studied by means of numerical calculations taking into account elastic interfacial behavior. Droplets smaller than the critical radius will shrink while droplets larger than the critical radius will grow. For a zero interfacial elasticity the stationary LSW distribution is obtained and its coarsening rate matches theoretical values. The critical droplet radius, number-averaged droplet radius, and volume-surface-averaged droplet radius increase with time. Low interfacial elasticity with respect to initial interfacial tension causes the initial LSW distribution to become bimodal. The size distribution of the coarsening peak can still be described by the LSW distribution. The smaller peak that accumulates in time has an average radius that depends on the ratio between the interfacial elastic modulus and the interfacial tension. For large ratios (E/σ > 1), the system goes within a short time to a stable situation without changes in particle size with time.

Convection-Induced Patterns in Phase-Separating Polymeric Fluids

The process of pattern formation and droplet coarsening has been studied for ternary polymeric fluids in which phase separation was induced by solvent evaporation. For a particular range of solvent evaporation rates and thicknesses of liquid layers, ordered hexagonal patterns were formed at the liquid film-air interface. We ascribe this effect to B\'enard-Marangoni convection induced by solvent evaporation and estimate conditions generating periodic two-phase structures in polymeric films. Droplet coarsening rate featured a crossover from $R\ensuremath{\sim}{t}^{0.89}$ to $R\ensuremath{\sim}{t}^{0.67}$ coinciding with the onset of convection and was explained by convection-induced stabilization of droplets against coalescence.

Temporal behavior of the number density of particles during Ostwald ripening

The kinetic behavior of the number of particles per unit volume or area, N, is examined theoretically for different dimensionalities of the dispersed phase ( d) and the diffusion field ( D). For particles that are either perfectly spherical ( d=3) or circular ( d=2) a kinetic equation for N is presented which is valid for D = 3, 2 and 1. The temporal exponents are also presented in terms of d and D. It is shown that the conventionally accepted kinetic law for N, i.e. N ∝ t−, where m = d/(3 + d − D), be replaced by an equation containing two terms. This equation has the form N = At− +Btu−p, where A and B are constants related to the physical parameters of the system, and p = (d+1)/(3 + d − D). The second term is not negligible with respect to the first, and arises because the volume fraction is not constant during Ostwald ripening. This equation is evaluated for self-consistency by the analyses of data on four systems: 1. Coarsening of γ′ (Ni 3Al) precipitates in an Ni-Al alloy ( d = D = 3); 2. The results of a computer modeling experiment ( d = D = 2); 3. Coarsening of ‘islands’ in a diblock copolymer film ( d = D = 2); 4. Coarsening of droplets of succinonitrile on a quartz substrate ( d = 3; D = 2). Self-consistency is tested by comparing values of A predicted using data on the kinetics of growth of the average particle with those obtained from the analysis of data on the temporal behavior of N. Self-consistency also demands that B be non-zero, and that different estimates of B obtained from different methods of data analysis be nearly equal. All the data sets examined fulfill these self-consistency requirements. Moreover, the values of B for the γ′/Ni-Al alloy are in excellent agreement with that calculated using reasonably well-known physical constants. The importance of the second term in the equation for N can also account for the discrepancy that is often observed between values of m obtained from plots of log N versus log t and the values expected on the basis of conventional theory.

New mechanisms of droplet coarsening in phase-separating fluid mixtures

We propose here a new mechanism of droplet coarsening in phase-separating fluid mixtures. In contrast to the conventional understanding that there are no interactions between droplets in the late stage of spinodal decomposition, we demonstrate the existence of interactions between droplets that is caused by the coupling between diffuse concentration change around droplets. We show the possibility that this mechanism plays an important role in droplet phase separation together with Brownian-coagulation mechanism. We also discuss the coupling between hydrodynamic and diffusion modes, namely, “collision-induced collision” phenomena.

A new coarsening mechanism of droplet spinodal decomposition

We propose here a new coarsening mechanism (‘‘diffusion-coupling mechanism’’) of droplet phase separation. For a system of high droplet density, the inhomogeneous distribution of droplets and the resulting anisotropic, selective coupling between the diffusion fields around neighboring droplets lead to the formation of the closed iso-concentration lines including more than two droplets. The excess energy associated with the concentration gradient in the matrix phase around neighboring droplets tries to reduce the length of the iso-concentration lines and causes the attractive interaction between the droplets. This mechanism could accelerate the droplet coarsening in spinodal decomposition.

Coarsening of three-dimensional droplets by two-dimensional diffusion: Part I. Experiment

An experimental study of diffusional coarsening, or Ostwald ripening, in a liquid-liquid two-phase system is described. An experiment performed at its isopycnic point, 42°C, allowed observations for the long times required to investigate coarsening. A holographic technique was instrumental in this work. Holograms takenin situ permit investigation of details regarding both the influence of local environmental conditions on individual droplet size histories and measurement of global averages. This study utilized a 100 μm pathlength test cell. The discrete phase was nucleated on one wall of the cell. This configuration resembles island formation in thin film growth. Observation of Ostwald ripening over a period of 1×107 s (∼4 mo.) reveals that droplet number decays as t−0.733 and the average radius increases as t−0.247, in the asymptotic limit. This shows good agreement with theoretical predictions for diffusional growth of spherical caps on a two-dimensional substrate which is a valid approximation for the geometry of this experiment. Part I of this paper describes the experimental results. Part II discusses a numerical model for droplet growth in a comparison with the experimental results.

Coarsening of three-dimensional droplets by two-dimensional diffusion: Part II. Theory

Theoretical modeling of coarsening among a finite cluster of precipitates is implemented, using the multipole expansion method. This method requires the diffusion field to behave quasi-statically. Two approximate solutions were developed, one to monopolar order, and other to the dipolar order. The conventional Gibbs-Thomson equilibrium relationship was used as the boundary condition at the precipitate-matrix interface. Part I of this paper considers a liquid-liquid system in a mixed-dimensional geometrical configuration, wherein three-dimensional precipitates interact via a diffusion field constrained in two dimensions. This kind of geometric configuration is often encountered in island evolution dynamics and phase segregation in thin films. The initial experimental configuration of droplets provides the initial condition for the simulation. Both monopole and dipole approximations closely follow the experimentally observed scaling laws, characteristic for the mixed-dimensional coarsening (N−4/3 and\(\bar R\) 4, varied linearly with time, where N is the number of droplets in the experimental field of view, and\(\bar R\) is the average droplet radius). Good agreement is observed for time evolution of radii of some individual precipitates. Certain deviations appearing among the two approximate solutions and the experimental data are discussed.