A. E. Kuchma

Dynamics of gas bubble growth in a supersaturated solution with Sievert's solubility law

This paper presents a theoretical description of diffusion growth of a gas bubble after its nucleation in supersaturated liquid solution. We study systems where gas molecules completely dissociate in the solvent into two parts, thus making Sieverts solubility law valid. We show that the difference between Henrys and Sieverts laws for chemical equilibrium conditions causes the difference in bubble growth dynamics. Assuming that diffusion flux is steady we obtain a differential equation on bubble radius. Bubble dynamics equation is solved analytically for the case of homogeneous nucleation of a bubble, which takes place at a significant pressure drop. We also obtain conditions of diffusion flux steadiness. The fulfillment of these conditions is studied for the case of nucleation of water vapor bubbles in magmatic melts.

A self-similar regime of droplet growth with allowance for the Stefan flux and dependence of diffusion coefficient on vapor-gas medium composition

An analytical self-similar description is formulated for the problem of nonstationary diffusion droplet growth in a vapor-gas medium with allowance for the Stefan flux of the medium, themotion of the surface of a growing droplet, and an arbitrary concentration dependence of the diffusion coefficient of a condensing vapor in the medium. Analytical expressions are found for the diffusion profile of condensing vapor concentration and for the droplet growth rate at a constant diffusion coefficient of the vapor and at a coefficient linearly dependent on vapor concentration. The combined allowance for the Stefan flux of the medium, the motion of the surface of the growing droplet, and the dependence of the vapor diffusion coefficient in the medium on the vapor concentration is shown to result in renormalization of the rate of droplet growth related to the stationary diffusion regime. At small deviations from the stationarity, the Stefan flux, nonstationary diffusion, and dependence of the diffusion coefficient on the vapor-gas mixture composition lead to corrections of the same order of magnitude.

Nucleation stage in supersaturated vapor with inhomogeneities due to nonstationary diffusion onto growing droplets

An analytical description of the nucleation stage in a supersaturated vapor with instantly created supersaturation is given with taking into account the vapor concentration inhomogeneities arising as a result of depletion due to nonstationary diffusion onto growing droplets. This description is based on the fact, that the intensity of the nucleation of new droplets is suppressed in spherical diffusion regions of a certain size surrounding previously nucleated droplets, and remains at the initial level in the remaining volume of the vapor–gas medium. The value of the excluded volume (excluded from nucleation) depends on the explicit form of the vapor concentration profile in the space around the growing droplet, and we use for that the unsteady self-similar solution of the time-dependent diffusion equation with a convective term describing the flow of the gas–vapor mixture caused by the moving surface of the single growing droplet. The main characteristics of the phase transition at the end of the nucleation stage are found and compared with those in the theory of nucleation with homogeneous vapor consumption (the theory of mean-field vapor supersaturation). It is shown that applicability of the mean-field approach depends on smallness of the square root of the ratio of the densities of metastable and stable phases. With increasing the temperature of the supersaturated vapor or for liquid or solid solutions, this smallness weakens, and then it would be more correct to use the excluded volume approach.

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.

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.

The stage of nonisothermal nucleation of supercritical particles of a new phase under nonstationary conditions of particle diffusion growth and heat transfer to a medium

An analytical theory has been formulated for the stage of nonisothermal nucleation of supercritical particles in a metastable medium with instantaneously generated initial supersaturation. The theory takes into account the nonuniformities of metastable substance concentration and temperature, which result from the nonstationary diffusion of the substance to growing particles and the nonstationary transfer of the heat of the phase transition from the particles to the medium. The formulated theory extends the approach based on the concept of excluded volume that has recently been used in the theory of the stage of nucleation under isothermal conditions. This approach implies that the nucleation intensity of new particles is suppressed in spherical diffusion regions with certain sizes that surround previously nucleated supercritical particles and remaining unchanged in the rest of the medium. It has been shown that, when self-similar solutions are used for nonstationary equations of substance diffusion to particles and heat transfer from the particles, the ratio between the excluded volume and the particle volume is independent of particle size, thereby enabling one to analytically solve the integral equation for the excluded volume throughout a system as a time function at the stage of nucleation. The main characteristics of the phase transition have been found for the end of the stage of nucleation. Comparison has been carried out with the characteristics obtained in terms of the isothermal and nonisothermal nucleation theory upon uniform vapor consumption and heat dissipation (the mean-field approximation of vapor supersaturation and temperature).

Steady-state composition of a two-component gas bubble growing in a liquid solution: Self-similar approach

The paper presents an analytical description of the growth of a two-component bubble in a binary liquid-gas solution. We obtain asymptotic self-similar time dependence of the bubble radius and analytical expressions for the nonsteady profiles of dissolved gases around the bubble. We show that the necessary condition for the self-similar regime of bubble growth is the constant, steady-state composition of the bubble. The equation for the steady-state composition is obtained. We reveal the dependence of the steady-state composition on the solubility laws of the bubble components. Besides, the universal, independent from the solubility laws, expressions for the steady-state composition are obtained for the case of strong supersaturations, which are typical for the homogeneous nucleation of a bubble.

Equations for the evolution of a growing or evaporating free microdroplet under nonstationary conditions of diffusion and heat transfer in a multicomponent vapor–gas medium

A set of equations has been derived for the nonstationary composition, size, and temperature of a growing or evaporating multicomponent microdroplet of a nonideal solution under arbitrary initial conditions. Equations for local nonstationary diffusion molecular and heat fluxes in a mixture of a multicomponent vapor with a noncondensable carrier gas have been obtained within the framework of nonequilibrium thermodynamics with allowance for hydrodynamic flow of the medium. The derived closed set of equations takes into account the nonstationarity of the diffusion and heat transfer, effect of thermodiffusion and other cross effects in the multicomponent vapor–gas medium, the Stefan flow, and droplet boundary motion, as well as the nonideality of the solution in the droplet. The general approach has been illustrated by the consideration of the multicomponent medium at low concentrations of vapors taking into account its thermal expansion due to the Stefan flow in the case of a nonstationary diffusion regime of the nonisothermal condensation growth of a one-component droplet.

Three stages of water microdroplet evaporation on hydrophobized surface: Comparison between steady-state theory and experiment

Three consecutive stages of evaporation of a sessile water microdroplet are studied both theoretically and experimentally under the conditions of contact angle hysteresis. The influence of thermal effects on the dynamics of droplet evaporation is quantitatively investigated on the basis of the obtained experimental results. The features of droplet evolution are analyzed at the final stage, when both the contact angle and the radius of the droplet base decrease with time. It is shown that evaporation at this stage also occurs in a steady-state regime, but the average droplet temperature approaches the ambient temperature.

The Stefan outflow in a multicomponent vapor–gas atmosphere around a droplet and its role for cloud expansion

Abstract A new comprehensive analysis of Stefan׳s flow caused by a free growing droplet in the vapor–gas atmosphere with several condensing components is presented. This analysis, based on the nonstationary heat and material balance and diffusion transport equations, shows the appearance of the Stefan inflow in the vicinity of the growing droplet and the Stefan outflow at large distances from the droplet as a consequence of nonisothermal condensation. For an ensemble of droplets in the atmospheric cloud, this outflow provides an increase of the total volume of the cloud, which can be treated as cloud thermal expansion and leads to the rise of the cloud as a whole due to increasing its buoyancy. We have formulated the self-similar solutions of the nonstationary diffusion and heat conduction equations for a growing multicomponent droplet and have derived analytical expressions for the nonstationary velocity profile of Stefan׳s flow and the expansion volume of the vapor–gas mixture around the growing droplet. To illustrate the approach, we computed these quantities in the case of droplet of stationary composition in air with several specific vapors ( C 2 H 5 OH / H 2 O ; H 2 SO 4 / H 2 O ; H 2 O ).