An analytical and numerical study of diffusion growth of overcritical gas bubbles at degassing of the supersaturated-by-gas solution with the explicit full-scale influence of viscous and capillary forces on internal pressure in the bubbles has been presented. This study is based on our recent semi-analytical approach to the same problem (Kuchma and Shchekin, 2021). Generally, this approach allows one to find how the growth rate of overcritical bubbles depends on gas supersaturation, its diffusivity and solubility in solution, solution viscosity, surface tension at bubble surface and how it changes from zero for critical bubbles at unstable equilibrium with solution to sufficiently large values for large overcritical bubbles. A special question concerns approaching the widely used in kinetics of bubble nucleation stationary diffusion regime of growth and not so widely used self-similar non-stationary diffusion regime. As a first step, we have found analytical formulas for the bubble growth rate and the correction function (which takes into account the balance in the number of gas molecules that have left the liquid solution and came into the growing bubble) of small overcritical gas bubbles at strong viscosity of the solution and full account of capillary pressure in the bubbles as a function of the bubble radius. As a second step, we derived asymptotic formulas for the case of low viscosity and small radii of bubbles. As a third step, we obtained the formulas for the bubble growth rate and the correction function at large overcritical radii. Finally, we numerically evaluated the joint effects of viscous and capillary forces on the rate of gas bubble growth at any radius of the overcritical bubble within the wide range of viscosities of the supersaturated-by-gas solution and confirmed all the asymptotic analytical results.

A detailed statistical description of the evolution of supersaturated-by-gas solution at degassing has been presented on the basis of finding the time-dependent distribution in radii of overcritical gas bubbles. The influence of solution viscosity and capillarity via internal pressure in the bubbles on this distribution has been considered until the moment when the gas supersaturation drops due to depletion and stops nucleation of new overcritical gas bubbles. This study is based on our previous results for the nonstationary growth rates of overcritical bubbles depending on gas supersaturation, diffusivity and solubility in solution, solution viscosity, and surface tension on bubble surface. Other important factors are linked with the initial rate of homogeneous gas bubble nucleation and coupling between diffusivity and viscosity in the solution. Here, we numerically studied how all these factors affect the time-dependent distribution function of overcritical bubbles in their radii, maximal and mean bubble radii, and the time-dependent swelling ratio of a supersaturated-by-gas solution in a wide range of solution viscosities.

The traditional approach of the mean-field supersaturation to kinetic description of nucleation is based on assumptions that homogeneous nucleation of overcritical particles of a new phase in a closed system occurs uniformly over the volume of the system and is synchronous with a decrease in the mean supersaturation of the metastable phase. The approximation of the mean supersaturation field also implies that the transport of molecules of the metastable phase into the growing particles of the new phase is slow and stationary. We have found in this work that, in the diffusion regime of the particle growth, the approach of the mean-field of supersaturation at the end of the first stage of formation of overcritical droplets in a supersaturated vapor requires low volatility of the condensing liquid, and realizes in the case of the stage of nucleation of overcritical gas bubbles in a solution supersaturated with gas only at extremely low solubility of the gas in the solution. In particular, for condensation of water vapor and degassing ethanol supersaturated by gas with moderate or high solubility at atmospheric pressure, the approximation of the mean-field supersaturation cannot be strictly justified. We demonstrated here that there are no such restrictions when using the excluded volume approach in the kinetic description of the phase transition. Along with that we have shown, that the excluded volume approach describes fast self-similar diffusion growth of particles of a new phase at the nucleation stage, leading to the formation of a cellular structure at the next stage of the phase transformation, the stage of intense decrease in the supersaturation of the metastable system.

The article describes the effect of phase transition heat on the temperature of a closed multicomponent vapor–gas metastable phase and on growing supercritical droplets and their size distribution at the stage of homogeneous formation and growth (nucleation stage) of the supercritical droplets. It is assumed that, between essentially supercritical droplets and the vapor–gas medium, a stationary diffusion transfer of condensing vapor molecules and heat is established, and, then, both the composition and temperature remain unchanged and the same for all supercritical droplets. A set of equations is derived to determine the composition, temperature, and growth rate of the essentially supercritical droplets via the initial temperature and supersaturation of vapors. Expressions are obtained for the deviation of vapor–gas medium temperature from its initial value and for the droplet size distribution function as depending on time.

The regularities of non-stationary diffusion growth of overcritical gas bubbles and kinetics of their distribution in sizes in a supersaturated-by-gas liquid solution on the nucleation stage have been analytically described by taking into account the full-scale influence of viscous and capillary forces on pressure in the overcritical bubbles. The results are general and not limited by values of gas supersaturation and gas solubility in the surrounding liquid solution. It is shown how the nonuniform concentration profile of the dissolved gas in supersaturated solution around the growing bubble changes with time and distance from the center of the overcritical bubble and gradually transforms into a stationary (at low solubility and moderate supersaturation of the dissolved gas) or self-similar profile (at large solubility and supersaturation of the dissolved gas). The kinetic theory of the nucleation stage with the excluded volume has been extended to the case of non-stationary gas concentration profiles due to viscous and capillary forces. The general approach has been illustrated in the limiting case of negligible viscous but significant capillary contributions to the vapor pressure in the bubble and in the case when the approximation of the mean field of gas supersaturation can be applied.

The regularities of changing chemical composition and size of a ultra-small multicomponent gas bubble growing in a viscous solution have been analyzed. The full-scale effects of solution viscosity and bubble curvature at non-stationary diffusion of arbitrary number of dissolved gases with any value of gas supersaturations and solubilities in the surrounding liquid solution have been taken into account. The non-uniform concentration profiles of gas species in supersaturated solution around the growing bubble with changing composition have been found as a function of time and distance from the bubble center. Equations describing transition to stationary concentrations of gases in the bubble with increasing radius have been obtained. Analytic asymptotic solutions of these equations for a ternary system have been presented.

A review of the theoretical data accumulated for the last decade on the diffusion kinetics of the stage of homogeneous nucleation of liquid droplets and gas bubbles in multicomponent systems has been presented. In addition to the previously known results, the review contains new relations and discussions that represent the further development of own studies. Thermodynamic expressions that relate the composition of critical droplets and bubbles occurring at unstable equilibrium with metastable multicomponent systems to the sizes of new-phase particles and degrees of supersaturation in the systems have been discussed. The dynamics of the growth of individual multicomponent supercritical droplets and bubbles at the stage of nucleation has been described at arbitrary values of vapor supersaturation for droplets and gas solubility in solutions for bubbles. The kinetics of the nucleation stage has been considered for ensembles of droplets and bubbles within the framework of the mean-field description of supersaturations and the excluded-volume approach. A relation has been shown between the excluded-volume approach to the description of the nucleation stage and the Kolmogorov crystallization theory.

A new analysis of evolution of an ensemble of supercritical (in size) droplets in the atmosphere of several condensing vapors has been presented. The analysis has been performed for the nucleation stage of formation and growth of the supercritical droplets in a closed system with a fixed amount of condensing species. The nucleation stage starts with appearance of supercritical droplets and finishes when nucleation rate of new critical droplets in the closed system ceases due to vapor depletion by the growing supercritical droplets. Here, we extend the mean-field theory for the nucleation stage of gas bubbles formation at degassing of a solution of several dissolved gases, which was published recently [A. E. Kuchma et al., J. Chem. Phys. 148, 234103 (2018)], to the nucleation stage of multicomponent nucleation and growth of supercritical droplets at isothermal conditions. An approach, which allows one to find all vapor supersaturations and the distribution of supercritical droplets in sizes as functions of time on the nucleation stage, has been proposed here for a real multicomponent solution and illustrated in the case of ideal multicomponent solution in supercritical droplets.

A set of equations describing the evolution of the volume and composition of evaporating sessile binary droplets consisting of infinitely miscible liquids has been employed to consider the evaporation dynamics of sessile droplets of aqueous solutions of sulfuric acid, 1-propanol, and ethanol in the atmosphere of humid air. It has been shown that experimentally observed features of droplet evolution may be explained with allowance for the nonideality of a solution in a droplet and the influence of thermal effects. In particular, the numerically obtained values of parameters at which the realization of this or that scenario is observed for the evolution of droplets of aqueous 1-propanol and ethanol solutions agree with the corresponding experimental data.

Experimental data have been obtained on the complete diffusion evaporation of a sessile microdroplet of an aqueous 1-propanol solution on a hydrophobized polished quartz substrate in air at atmospheric pressure and room temperature. During the evaporation of the sessile droplet, time dependences have been determined for its key thermodynamic and geometric parameters, i.e., the contact angle, base surface area, and volume. It has been revealed that the character of time variations in the contact angle depends on the initial alcohol concentration in the droplet and air humidity. At a high alcohol concentration and a low air humidity, the droplet contact angle monotonically decreases throughout the evaporation process. The contact angle of a solution droplet with a prevailing content of water varies in several stages. In this case, the monotonic reduction in the contact angle is, at a certain moment, replaced by a stage of its growth. The comparison of the maximum contact angle of an evaporating droplet with the contact angle of a sessile droplet of pure water enables one to determine the amount of alcohol in a studied droplet by the end of this stage. The residual alcohol amount governs the subsequent evolution of the droplet up to its complete evaporation.

A new kinetic analysis of degassing and swelling of a decompressed liquid solution with several dissolved gases has been presented. The analysis has been performed for the nucleation stage of formation and growth of supercritical gas bubbles in a closed system under conditions of a limited availability of the dissolved species. The nucleation stage is an important stage of degassing, on which a certain size distribution of gas bubbles is formed, being the starting point for further growth. This stage starts with the appearance of supercritical gas bubbles and is widely completed when the nucleation rate of supercritical gas bubbles diminishes by a decimal order. Neglecting the role of the Laplace pressure in large supercritical bubbles, we were able to introduce the concept of total gas supersaturation and to develop a theory of this stage for liquid solutions with arbitrary number and any values of supersaturations and solubilities of the dissolved gases. First, we have considered slowly growing bubbles within the mean-field approach assuming a stationary diffusion of gases to bubbles at moderate total gas supersaturation. In the case of large total gas supersaturation, we have built a description of fast growing bubbles on the basis of the extended excluded volume approach with nonstationary nonuniform diffusion shells around the bubbles and mean-field mixing of the concentration of gases at the external boundaries of the shells. A main novel feature of the developed theory is its ability to predict the kinetic behavior of the whole ensemble of bubbles with different sizes under changes in the initial gas composition in the liquid solution at its fast decompression. It has been shown that the effects of nonstationary diffusion may be very significant in the growth of multicomponent bubbles and, in particular, are responsible for a significant swelling of a decompressed liquid solution. Distribution of supercritical bubbles in sizes as a function of concentrations of solute gases at any moment of the nucleation stage, the duration of the nucleation stage, and the swelling ratio at the end of the nucleation stage have been determined.

A new comprehensive analysis of growth dynamics of an ensemble of gas bubbles on the nucleation stage at degassing in a gas-supersaturated liquid solution is presented. The Laplace pressure in supercritical bubbles is strictly taken into account in the analysis based on the mean-field approach to nucleation kinetics. The whole distribution of supercritical bubbles in sizes, gas supersaturation in the liquid solution at any time at the nucleation stage as well as the total duration of the nucleation stage have been described analytically. The scaled universal forms independent of specific parameters of the gas and solvent components have been found for the maximal bubble radius and the reduction of mean gas supersaturation as functions of time. It has been shown that the Laplace pressure in the supercritical bubbles considerably affects the bubble distribution at the beginning of nucleation stage but becomes less important to the end of the nucleation stage.

Diffusion evaporation of a sessile binary droplet in an atmosphere of a noncondensable carrier gas has been considered. For a droplet consisting of two infinitely miscible liquids, a relation between the current values of solution concentration and volume of the droplet has been derived in an explicit form under the ideal solution approximation. It has been shown that the volume of a sessile binary droplet may, as well as the volume of a free binary droplet, vary nonmonotonically with time. The evaporation of a droplet of an aqueous sulfuric-acid solution has been considered in detail taking into account the nonideality of the solution. Time variations in the volume, base area, and contact angle have been experimentally measured for the sessile droplet of an aqueous sulfuric-acid solution on a hydrophobized substrate. The experimental data obtained at different initial humidities of water-vapor and droplet-solution concentrations have been analyzed within the theory of the stationary isothermal diffusion evaporation of a sessile binary droplet.