Heat exchange between phases and kinetics of condensation relaxation for different regimes of droplet growth

The mathematical model and numerical solution of the problem relevant to the heat exchange between a droplet and a vapor-gas medium are described for different regimes of the condensation droplet growth. The effect of the heat exchange between phases on the kinetics of the condensation relaxation after supersaturation is instantaneously generated in a vapor-gas mixture is studied based on the results obtained. Cesium-argon and ethane-helium mixtures are considered, with the droplet growth regime being free-molecular in the former case and varying from transient to continual in the latter case. The data obtained in terms of the general formulation of the problem and those resulting from the assumptions simplifying the behavior of droplet temperature are compared to determine the applicability domain of the aforementioned assumptions.     

Dynamics of variations in size and composition of supercritical droplet at binary condensation

A general analytical solution is found in quadratures for the radius and concentration of a solution droplet, which isothermally grows or evaporates in a diffusion or free-molecular regime in a binary mixture of vapors. The obtained solution describes the dynamics of variations in the size and composition of a super-critical droplet during the binary condensation in mixed vapors at an arbitrary initial droplet composition. It is shown that, at small (linear) deviations of the growth regime and droplet composition from the stationarity, these quadratures lead to the results that were recently obtained for the composition relaxation in a growing droplet. Moreover, it is demonstrated that, in terms of the nonlinear theory, when the deviation of solution concentration in a droplet from its stationary value is not small, it is invalid to use the law of stationary variations in the size of a droplet with time to describe the relaxation process for its chemical composition.     

Kinetics of binary condensation on the droplet growing in diffusion regime

A relaxation equation determining the regular tendency of the concentration of binary solution in the growing droplet to the stationary value, at which there is a self-similar solution to the problem of the condensation in a binary mixture, is derived. An analytical solution of the relaxation equation is obtained and it is demonstrated that the stationary value of concentration is achieved via the power law. The time interval that elapses from the emergence of the droplet until the diffusion regime of droplet growth and derived relaxation equation become effective is revealed. The stationary value of concentration is found for the model of ideal solution.     

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.     

Study of the volume condensation process in supersaturated vapor by the direct numerical solution of the kinetic equation for the droplet size distribution function

The volume condensation of supersaturated vapor is investigated by the direct numerical solution of the basic kinetic equation for the droplet size distribution function by analogy with the corresponding solution of the Boltzmann kinetic equation. The proposed consideration of the condensation growth of droplets is applicable at any Knudsen number. The method is tested by the example of vapor condensation under the conditions of the rapid development of supersaturation in a vapor-gas mixture as a result of its adiabatic expansion. In a wide range of Knudsen numbers, the results of the modeling are compared with those obtained by the moment method.     

Statistical Characteristics of the Process of Homogeneous Nucleation in the Vapor–Gas Medium during Free Molecular Regime of the Growth of Forming Droplets

Statistical approach to the study of the process of homogeneous nucleation of droplets in the vapor–gas medium in the presence of originally generated growing droplet at free molecular regime of droplet growth after the instantaneous creation of initial vapor supersaturation is proposed. The probability density of the creation of a new droplet in the vicinity of originally generated droplet is found. The mean distance between two neighboring droplets and the relative scatter of this distance are determined. The mean expectation time for the appearance of neighboring droplet estimating the duration of the droplet nucleation stage is found. The average number of droplets in a unit volume of the vapor–gas medium by the end of the droplet nucleation stage is estimated. The results obtained are compared with the predictions of the theory based on the assumption of the homogeneity of metastable phase.     

Statistico-probabilistic approach to taking account of the vapor depletion in the kinetics of homogeneous nucleation: a free-molecular regime of droplet growth

We propose a statistico-probabilistic approach to investigate the process of homogeneous formation of droplets in a vapor phase in the presence of an already formed and growing droplet under free-molecular regime of droplet growth after the instantaneous creation of initial vapor supersaturation. We find the probability density for the formation of a new, nearest (neighbor) droplet in the vicinity of an initially formed droplet. The mean distance between two neighboring droplets is also determined, as well as the average time lag for the formation of the nearest (neighbor) droplet; the latter quantity serves as an estimate for the duration of the nucleation stage. An estimate for the average number of droplets forming in unit volume by the end of the nucleation stage is also given. Our results are compared with the predictions of classical nucleation theory which assumes the density uniformity of a metastable phase. Where the proposed approach is applicable, there is observed qualitative agreement between the results. The underlying cause of this agreement is analyzed and the limits of applicability of the uniformity approximation are clarified.     

Nonstationary vapor concentration fields near the growing droplet of binary solution

Nonstationary vapor concentration fields near the droplet of binary solution growing in the vapor—gas mixture are revealed using the concepts of similarity. The revealed fields are determined with the exact account of the motion of droplet surface and refer to the times at which the droplet reaches sizes that provide for the diffusion regime of droplet growth. To obtain the self-similar solution of the problem of binary condensation, it is necessary to ensure a constant (in time) concentration of binary solution in the growing droplet. The velocities of an increase in the number of molecules and the radius of two-component droplet with time are found with allowance for the equation ensuring this solution. The conditions for the transformation of the self-similar solution of the problem of the condensation of two-component mixture into the solution, which was derived previously for the condensation of one component, are elucidated.     

Dynamics of variations in size and composition of a binary droplet in a mixture of two condensing vapors and a passive gas under arbitrary initial conditions

The dynamics of variations in the size and composition of a droplet in a mixture of two vapors and a noncondensable carrier gas has been studied at an arbitrary initial droplet size and two limiting initial concentrations of the solution in the droplet corresponding to a pure first or pure second component. The conditions for nonmonotonic variations in the droplet radius with time have been analyzed. The physical situation has been investigated for the course of binary condensation, in which, at an initial stage, before the droplet begins to grow and the stationary concentration of the solution in it is established, the droplet size markedly decreases. The opposite situation is also considered, in which the droplet grows at the initial stage and then passes to the regime of monotonic evaporation.     

On the evolution of a multicomponent droplet during nonisothermal diffusion growth or evaporation

A set of equations has been derived for the size, composition, and temperature of a multicomponent droplet of a nonideal solution during its diffusion nonisothermal condensation growth or evaporation in a multicomponent mixture of vapors with an incondensable carrier gas. In addition to complete equations for material and heat transfer in the vapor-gas medium surrounding the droplet, the derived set, in the general case, describes the nonstationary growth or evaporation of the droplet under arbitrary initial conditions (initial size and temperature of the droplet and the concentrations of the nonideal multicomponent solution in it) and the establishment of the stationary values of the composition, temperature, and the rate of variations in the size of the droplet with allowance for heat effects and diffusion and thermodiffusion material transfer, Stefan flux, motion of the droplet surface, and the nonideality of the solution in the droplet. A simplified set of equations obtained without taking into account the contributions from the flow, cross effects, and thermal expansion in the equations of the material and heat transfer in the vapor-gas medium has been considered. Equations describing growth/evaporation in the stationary regime have been analyzed for droplets of ideal multicomponent solutions.     

Times of metastable droplet relaxation to equilibrium states

Times of metastable droplet relaxation to their equilibrium state are calculated at saturated vapor pressures, depending on the droplet size. It is shown that for small droplets with radius R = 6 molecular diameters (or ~2 nm) the relaxation times are ~1 ns (which is comparable to the characteristic flight times of rarefied gas molecules). For large droplets with radius R ~ 800 molecular diameters, the relaxation times are as long as 10 μs. At a fixed droplet radius (6 ≤ R ≤ 800), the range of variation in relaxation time from the melting point to the critical temperature does not exceed one order of magnitude: the lower the temperature, the slower the relaxation process.     

Layered droplet microstructures in sheared emulsions: finite-size effects

We investigate the influence of confinement on the steady state microstructure of emulsions sheared between parallel plates, in a regime where the average droplet dimension is comparable to the gap width between the confining walls. Utilizing droplet velocimetry, we find that the droplets can organize into discrete layers under the influence of shear. The number of layers decreases from two (at relatively higher shear rates) to one (at lower shear rates), as the drops grow slightly larger due to coalescence. We argue that the layering and overall composition profile may be controlled by the interplay of droplet collisions (which can cause separation of droplet centers in the velocity gradient direction), droplet migration toward the centerline (due to wall effects), and droplet packing constraints. We also study the effects of mixture composition on droplet microstructure, and summarize these results in the form of a morphology diagram in the parameter space of mass fraction and shear rate. We find that formation of strings of the suspended phase (reported earlier by our group in flow-visualization studies on confined emulsions) is observed over a broad composition window. We also find a stable (nontransient) morphology wherein the droplets are arranged in highly ordered pearl-necklace chain structures.     

Analysis of the Results of Numerical Simulation of the Supersaturated Vapor Condensation Relaxation

The process of bulk vapor condensation from the vapor-gas mixture after the fast creation of the supersaturated state as a result of vapor expansion is discussed. Scaling relations are derived for the time of condensation relaxation and droplet number density. The principal possibility of the experimental determination of nucleation rate is demonstrated based on the results obtained. The effect of initial phase, frequency, and pulsation amplitude on the condensation relaxation is analyzed for the case, when thermodynamic parameters of a mixture in the process of condensation are subjected to perturbations in the form of low-amplitude harmonic pulsations. The domain of applicability of results obtained on the scale of pulsation frequencies is determined.     

Thermal effects accompanying stationary binary condensation of vapors into overcritical droplet

A set of equations is derived to calculate the stationary temperature and concentration of a solution in a overcritical droplet with regard to the heat release accompanying the condensation of a binary mixture of vapors in a diffusion or free-molecular regime. In the approximation of an ideal solution, relations are found for the stationary temperature of droplet growing under the conditions of strong and weak thermal effects. For the general case and the cases of strong and weak thermal effects, the temperature and concentration of the droplet and the coefficient of the thermal deceleration of the droplet growth are calculated as functions of the density of a passive gas. The influence of the condensation heat values of the first and second components of the mixture on the stationary temperature and concentration of the solution in the growing droplet is investigated separately.     

Self-similar theory of nonisothermal vapor condensation on a droplet growing in a vapor-gas medium

Rigorous self-similar solutions to the joint problems of vapor diffusion toward a droplet growing in a vapor-gas medium and the removal of heat released during vapor condensation are found. An equation for the temperature of a droplet ensuring the existence of a self-similar solution is derived. This equation sets the constancy of the temperature of a droplet throughout the time of its growth and unambiguously determines this temperature. In the case of the strong heat effects, when the rate of droplet growth decreases substantially, the analytical solution to this equation is obtained. This temperature coincides precisely with the temperature, which is established in the droplet at the diffusion regime of its growth. At the found droplet temperature, interconnected fields of vapor concentration and temperature of vapor-gas medium around the droplet are expressed through the initial (prior to the droplet nucleation) parameters of a vapor-gas medium. These parameters are used to express the dependence of the radius of a droplet on the time at the diffusion regime of its growth and the time required to establish the diffusion regime of droplet growth. The case of weak heat effects is also studied.     

Kinetics of the Establishment of Quasi-Steady-State Regime of Overcoming Activation Barrier of Nucleation on the Macroscopic Wettable Nuclei

The solution of the kinetic equation of nucleation on macroscopic wettable condensation nuclei was constructed for the initial (incubation) stage. The solution thus constructed determines the times of relaxation to quasi-steady-state distribution of droplets generating on droplet nuclei in the vicinity of maximum of the work of droplet heterogeneous formation as well as the relaxation to quasi-equilibrium droplet distribution throughout the entire region located to the left of this vicinity at the droplet size axis. The dependence of relaxation times on the height of activation barrier of nucleation, size of nuclei, their nature, and characteristics of matter comprising condensate was elucidated. It was shown when the non-steady-state rate of nucleation becomes actually equal to the quasi-steady-state rate of nucleation.     

Influence of dynamic interfacial tension on droplet formation during membrane emulsification

Membrane emulsification is a promising and relatively new technique for producing emulsions. The purpose of this study was to better understand the influence of interfacial tension on droplet formation during membrane emulsification. Droplet formation experiments were carried out with a microengineered membrane; the droplet diameter and droplet formation time were studied as a function of the surfactant concentration in the continuous phase. These experiments confirm that the interfacial tension influences the process of droplet formation; higher surfactant concentrations lead to smaller droplets and shorter droplet formation times (until 10 ms). From drop volume tensiometer experiments we can predict the interfacial tension during droplet formation. However, the strong influence of the rate of flow of the to-be-dispersed phase on the droplet size cannot be explained by the predicted values. This large influence of the oil rate of flow is clarified by the hypothesis that snap-off is rather slow in the studied regime of very fast droplet formation.     

Study of nonsteady diffusional growth of a droplet in a supersaturated vapor: treatment of the moving boundary and material balance

A new mathematical treatment of the problem of droplet growth via diffusion of molecules from a supersaturated vapor is presented. The theory is based on a semiquantitative analysis with good physical arguments and is justified by its reasonable predictions. For example it recovers the time honored growth law in which, to a high degree of approximation, the droplet radius increases with the square root of time. Also, to a high degree of approximation, it preserves material balance such that, at any time, the number of molecules lost from the vapor equals the number in the droplet. Estimates of the remaining approximational error are provided. On another issue, we show that, in contrast, the conventional treatment of droplet growth does not maintain material balance. This issue could be especially important for the nucleation of another droplet in the vicinity of the growing droplet where the rate of nucleation depends exponentially on supersaturation. Suggestions for further improvement of rigor are discussed.