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.     

Dynamics of the droplet size distribution function during the gradual creation of a supersaturation state under different regimes of droplet growth

The method of the direct numerical integration of kinetic equations of droplet size distribution functions that was previously proposed by the authors is employed to solve the problem of condensation relaxation in a vapor-gas mixture during the creation of a supersaturation state at a finite rate. Two relaxation regimes are considered. In the static regime, the mixture is expanded at a constant rate until a preset supersaturation ratio is achieved; in the dynamic regime, the expansion is continued. Solutions are obtained for argon-cesium and argon-ethane mixtures, thus making it possible to study the dependence of the process character on the Knudsen number. The effects of the rate of the supersaturation creation and the relaxation regime on the droplet size distribution function are analyzed.     

Kinetics of droplet condensation through a double free-energy barrier

Results are presented for the kinetics of nucleation of liquid droplets from a one-component vapor phase on a planar lyophobic substrate patterned with a large number of easily wettable (lyophilic) circular domains. If the wettability of these lyophilic domains is characterized by a contact angle smaller than pi2, for intermediate values of the supersaturation, the condensation of a droplet on a lyophilic domain occurs through a free-energy barrier with two maxima, that is, through a double barrier. A simple model is proposed for the kinetics of droplet condensation through a double barrier that combines Kramers's [Physica (Utrecht) 7, 284 (1940)] transition rate theory with known results of nucleation theory. In the framework of this model, the solution is derived for the steady-state limit of the nucleation process. The number of lyophilic domains available for droplet condensation reduces with time as domains are occupied by droplets. The problem of droplet condensation through a double barrier is solved taking into account the effect of the time-dependent depletion in the number of available lyophilic domains.     

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.     

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.     

Sub-micrometer dropwise condensation under superheated and rarefied vapor condition

Phase change accompanying conversion of a saturated or superheated vapor in the presence of subcooled surfaces is one of the most common occurring phenomena in nature. The mode of phase change that follows such a transformation is dependent upon surface properties such as contact angle and thermodynamic conditions of the system. In present studies, an experimental approach is used to study the physics behind droplet growth on a partially wet surface. Superheated vapor at low pressures of 4-5 Torr was condensed on subcooled silicon surface with a static contact angle of 60° in the absence of noncondensable gases, and the condensation process was monitored using environmental scanning electron microscopy (ESEM) with sub-microscopic spatial resolution. The condensation process was analyzed in the form of size growth of isolated droplets before a coalescence event ended the regime of single droplet growth. Droplet growth obtained as a function of time reveals that the rate of growth decreases as the droplet increases in size. This behavior is indicative of an overall droplet growth law existing over larger time scales for which the current observations in their brief time intervals could be fitted. A theoretical model based on kinetic theory further support the experimental observations indicating a mechanism where growth occurs by interfacial mass transport directly on condensing droplet surface. Evidence was also found that establishes the presence of sub-microscopic droplets nucleating and growing between microscopic droplets for the partially wetting case.     

Postnucleation droplet growth in supersaturated gas with arbitrary vapor concentration

This work concerns the reexamination and extension of the current theory of phase transition dynamics for liquid droplets growing on soluble aerosols from a supersaturated gas mixture for the general case of arbitrary value of vapor concentration. We found that the inconsistency in the common treatment of the vapor diffusion, due to an implicit assumption of the constancy of gas density in the vicinity of a droplet by neglecting its dependency on temperature and vapor concentration, leads to the obvious discrepancy in the Maxwell expression for the growth rate regarding droplets of near critical size. Restoring the correct treatment of the vapor diffusion in terms of the mass concentration of water vapor and taking into the consideration variations of gas density in the vicinity of a droplet in compliance with the equation of state of moist air, we have obtained a new expression for the droplet growth rate valid for an arbitrary value of vapor concentration. The limitations imposed by the molecular kinetic fluxes to postnucleation diffusional growth of small droplets with a large Knudsen number are also reevaluated to include previously neglected physical effects. In particular, the essential contribution of the vapor molecular energy flux into the total kinetic molecular heat flux as well as the temperature variations of mean thermal velocities of air and vapor molecules in the vicinity of the droplet interface have been taken into consideration. Surprisingly significant differences have been found in new expressions derived for the droplet growth rate and droplet temperature, even in the limit of small vapor concentration, if comparing with commonly used results. These findings could help with better interpretation of experimental measurements to infer more reliable data for the mass and thermal accommodations coefficients.     

Self-similar solution to the problem of vapor diffusion toward the droplet nucleated and growing in a vapor-gas medium

The problem of vapor diffusion toward a droplet nucleated and growing in the diffusion regime is exactly solved using the similarity theory. The surface motion of droplets is taken into account in the solution. The constructed nonstationary concentration field of vapor satisfies the diffusion equation, the boundary condition of equilibrium on the surface of growing droplet, and the initial homogeneous condition. According to the found solution, the radius of a droplet is proportional to the square root of the time of its growth. Far from the critical point, at a low ratio between the densities of excess vapor and a liquid droplet, the proportionality coefficient coincides with that resulting from an approximate solution. The balance between the numbers of molecules removed from vapor and those composing a growing droplet exactly corresponds to the obtained solution.     

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.     

Experimental studies of the vapor phase nucleation of refractory compounds. VI. The condensation of sodium

In this paper we discuss the condensation of sodium vapor and the formation of a sodium aerosol as it occurs in a gas evaporation condensation chamber. A one-dimensional model describing the vapor transport to the vapor/aerosol interface was employed to determine the onset supersaturation, in which we assume the observed location of the interface is coincident with a nucleation rate maximum. We then present and discuss the resulting nucleation onset supersaturation data within the context of nucleation theory based on the liquid droplet model. Nucleation results appear to be consistent with a cesium vapor-to-liquid nucleation study performed in a thermal diffusion cloud chamber.     

Stochastic self-assembly of incommensurate clusters

Nucleation and molecular aggregation are important processes in numerous physical and biological systems. In many applications, these processes often take place in confined spaces, involving a finite number of particles. Analogous to treatments of stochastic chemical reactions, we examine the classic problem of homogeneous nucleation and self-assembly by deriving and analyzing a fully discrete stochastic master equation. We enumerate the highest probability steady states, and derive exact analytical formulae for quenched and equilibrium mean cluster size distributions. Upon comparison with results obtained from the associated mass-action Becker-D?ring equations, we find striking differences between the two corresponding equilibrium mean cluster concentrations. These differences depend primarily on the divisibility of the total available mass by the maximum allowed cluster size, and the remainder. When such mass "incommensurability" arises, a single remainder particle can "emulsify" the system by significantly broadening the equilibrium mean cluster size distribution. This discreteness-induced broadening effect is periodic in the total mass of the system but arises even when the system size is asymptotically large, provided the ratio of the total mass to the maximum cluster size is finite. Ironically, classic mass-action equations are fairly accurate in the coarsening regime, before equilibrium is reached, despite the presence of large stochastic fluctuations found via kinetic Monte-Carlo simulations. Our findings define a new scaling regime in which results from classic mass-action theories are qualitatively inaccurate, even in the limit of large total system size.     

Condensation of a vapor bubble in submicrometer container

Condensation of a spherically symmetric submicrometer size vapor bubble is studied using diffuse interface hydrodynamic model supplemented by the van der Waals equation of state with parameters characteristic for argon. The bubble, surrounded by liquid, is held in a container of constant volume with temperature of the wall kept fixed. The condensation is triggered by a sudden rise of the wall temperature. We find that in the same container and subjected to a similar increase of the wall temperature the condensation process is totally different from the opposite process of droplet evaporation. In particular, the rapid change of the wall temperature excites the wave, which hits the interface and compresses the bubble, leading to a considerable increase of the temperature inside. The condensation of the submicrometer size bubble takes tens of nanoseconds, whereas evaporation of the same size droplet lasts roughly 50 times longer. In contrast to evaporation the condensation process is hardly quasistationary.     

Key thermodynamic characteristics of nucleation on charged and neutral cores of molecular sizes in terms of the gradient density functional theory

The gradient density functional theory and the Carnahan–Starling model formulated for describing the contribution of hard spheres have been used to calculate the profiles of condensate density in small critical droplets formed via homogeneous nucleation, as well as in stable and critical droplets formed via heterogeneous nucleation on solid charged and neutral condensation cores of molecular sizes. The calculations performed for water and argon at different values of condensate chemical potential have yielded the heights of the activation barriers for homoand heterogeneous nucleation as functions of vapor supersaturation at preset system temperatures. The interaction of condensate molecules with a solid core has been described by the resultant potential of molecular attractive forces. In the case of a charged core, the long-range Coulomb potential of electric forces has additionally been taken into account. Dielectric permittivities have been calculated as known functions of the local density of the fluid and temperature. The radius of the equimolecular droplet surface has been chosen as a variable describing the droplet size. Dependences of the chemical potential of condensate molecules in a droplet on its size have been plotted for water and argon with allowance for the action of capillary, electrostatic, and molecular forces. It has been shown that the role of the molecular force potential in heterogeneous nucleation increases with the size of condensation cores.     

Kinetics of the droplet formation at the early and intermediate stages of the spinodal decomposition in homopolymer blends

The kinetics of the droplet formation during the spinodal decomposition (SD) of the homopolymer blends has been studied by numerical integration of the Cahn‐Hilliard‐Cook equation. We have found that the droplet formation and growth occurs when the minority phase volume fraction, f_m , approaches the percolation threshold value, f_thr = 0.3 ± 0.01. The time for the formation of the disperse droplet morphology (coarsening time) depends only on the equilibrium minority phase volume fraction, f_m . f_m approaches its equilibrium value logarithmically at the late SD stages, and, therefore, the coarsening time decreases exponentially as the average volume fraction or the quench depth decrease. Since the temporal evolution of the total interfacial area does not depend on the quench conditions and blend morphology, the average droplet size and the droplet number density is determined by the coarsening time. Within the time scale studied, the droplet number density decreases with time as t ^{–0.63±0.03}; the average mean curvature decreases as t ^{–0.35±0.05}; the average Gaussian curvature decreases as t ^{–0.42±0.03}, and the average droplet compactness ˜V/S^3/2 where S is the surface area and V is the volume) approaches a spherical limit logarithmically with time. The droplets with larger area have lower compactness and in the low compactness limit their area is a parabolic function of compactness. The size and shape distribution functions have been also investigated.     

Formation and Release of Cobalt(II) Sorption and Precipitation Products in Aging Kaolinite-Water Slurries

A comprehensive simulation of the coarsening mechanism coalescence-induced coalescence (CIC) is developed to predictthe growth rate of inviscid droplets in a viscous matrix fluid. In CIC, the shape relaxations of coalescing droplets establish flow fields that drive other droplets into contact, thus creating a cascade of coalescence events. It is believed that CIC is responsible for droplet growth in some demixed polymer solutions, such as isotactic polypropylene (iPP) and diphenyl ether (DPE). A cascade of coalescence events is simulated using a three-dimensional molecular dynamics-like simulation of a dispersed two-phase isopycnic fluid system. The coalescence-induced flow is driven mostly by the strong gradients in curvature at the neck of a coalescing pair of droplets, and the flow is modeled analytically by approximating it as due to a ring of point forces. The resultant velocity of each droplet in the suspension is calculated by superimposing all of the coalescence-induced flow fields and applying Faxen's Law. The mean droplet size grows like t(xi), where t is the coarsening time and xi a growth exponent that increases with increasing minority phase volume fraction varphi. Good agreement with experimental values of xi (0.22/=0.23. It is also shown that the droplet size distribution broadens for semidilute suspensions (varphi/=0.54). A phenomenological kinetic theory of coalescence is proposed. It is believed that in nondilute emulsions, CIC can account for coarsening that has been attributed previously to more traditional coalescence mechanisms. Copyright 2000 Academic Press.     

A kinetic approach to the theory of heterogeneous nucleation on soluble particles during the deliquescence stage

Deliquescence is the dissolution of a solid nucleus in a liquid film formed on the nucleus due to vapor condensation. Previously, the kinetics of deliquescence was examined in the framework of the capillarity approximation which involves the thermodynamic interfacial tensions for a thin film and the approximation of uniform density therein. In the present paper we propose a kinetic approach to the theory of deliquescence which avoids the use of the above macroscopic quantities for thin films. The rates of emission of molecules from the liquid film into the vapor and from the solid core into the liquid film are determined through a first passage time analysis whereas the respective rates of absorption are calculated through the gas kinetic theory. The first passage time is obtained by solving the single-molecule master equation for the probability distribution of a "surface" molecule moving in a potential field created by the cluster. Furthermore, the time evolution of the liquid film around the solid core is described by means of two mass balance equations which involve the rates of absorption and emission of molecules by the film at its two interfaces. When the deliquescence of an ensemble of solid particles occurs by means of large fluctuations, the time evolution of the distribution of composite droplets (liquid film+solid core) with respect to the independent variables of state is governed by a Fokker-Planck kinetic equation. When both the vapor and the solid soluble particles are single component, this equation has the form of the kinetic equation of binary nucleation. A steady-state solution for this equation is obtained by the method of separation of variables. The theory is illustrated with numerical calculation regarding the deliquescence of spherical particles in a water vapor with intermolecular interactions of the Lennard-Jones kind. The new approach allows one to qualitatively explain an important feature of experimental data on deliquescence, namely the occurrence of nonsharp deliquescence, a feature that the previous deliquescence theory based on classical thermodynamics could not account for.     

New approach to the kinetics of heterogeneous unary nucleation on liquid aerosols of a binary solution

The formation of a droplet on a hygroscopic center may occur either in a barrierless way via Kohler activation or via nucleation by overcoming a free energy barrier. Unlike the former, the latter mechanism of this process has been studied very little and only in the framework of the classical nucleation theory based on the capillarity approximation whereby a nucleating droplet behaves like a bulk liquid. In this paper the authors apply another approach to the kinetics of heterogeneous nucleation on liquid binary aerosols, based on a first passage time analysis which avoids the concept of surface tension for tiny droplets involved in nucleation. Liquid aerosols of a binary solution containing a nonvolatile solute are considered. In addition to modeling aerosols formed through the deliquescence of solid soluble particles, the considered aerosols constitute a rough model of "processed" marine aerosols. The theoretical results are illustrated by numerical calculations for the condensation of water vapor on binary aqueous aerosols with nonvolatile nondissociating solute molecules using Lennard-Jones potentials for the molecular interactions.     

Homogeneous nucleation of droplets from supersaturated vapor in a closed system

Kinetic equations describing homogeneous nucleation kinetics within standard model are solved numerically under the condition of a constant number of molecules in the considered system. It has consequences to decrease the supersaturation of the supersaturated vapor during the process of the formation of small droplets of a new phase. The decrease of supersaturation occurs in a short time and reaches some value which remains unchanged for a relatively long time (quasistationary regime), especially at lower initial supersaturations. This time interval decreases with increasing value of the initial supersaturation. In the quasistationary regime the nucleation rate reaches its stationary value. At higher initial supersaturation, the rate of formation of nuclei goes to some maximum value corresponding to the stationary nucleation rate and then decreases with time due to the decrease of supersaturation.     

仿生超疏水表面的抗冷凝失效研究进展

在存在一定过冷度或蒸汽过饱和度的条件下,水蒸汽可在固体表面凝结成核.随着过冷度增大,液滴成核半径将随之减小,冷凝液滴的生长融合将无法避免地发生在超疏水表面不可或缺的微/纳米结构内.若液滴不能及时排出,则会滞留在表面结构内并挤出空气,形成局部浸润,导致材料表面的超疏水性能下降或失效,甚至引起泛洪.本文首先总结了表面因冷凝...