The article introduces the reader to the thermodynamic potential of classical thermodynamics, called the J-potential. It can be generally defined as the grand thermodynamic potential plus the volume of the system multiplied by some pressure (which may be the external pressure, the pressure in one of the phases in a multiphase system). Variants of the J-potential are considered. It is shown that the J-potential is perfectly suited for multiphase (heterogeneous) systems. In this work, it is applied to study the dependence of the chemical potential of a substance in a small vapor bubble, formed in the bulk liquid, on the bubble size.

Formation of a droplet around a spherical solid particle in supersaturated vapor is considered. The number and stability of equilibrium solutions in a closed small system are studied in the canonical ensemble in comparison to an open system in the grand canonical ensemble. Depending on the system's parameters, two modes exist in the canonical ensemble: the first one with only one solution and the second one with three solutions; the presence of the third solution is due to confinement. The analysis is conducted first on a macroscopic thermodynamic level of description, and then the results are supported by studies within two versions of classical density functional theory: the square-gradient approximation with the Carnahan-Starling equation of state for hard spheres on a completely wettable particle and the random-phase approximation with the fundamental measure theory on a poorly wettable particle. In the latter case, a solution breaking the spherical symmetry is observed at a small total number of molecules.

There exist many ways to calculate the contact angle of a sessile droplet on a planar surface on the basis of molecular modeling data, and the result may depend on the chosen approach. This paper presents molecular dynamics simulations of argon and water sessile droplets on simple model and more complex atomistic substrates. Several approaches for calculating contact angles are described. These approaches are based on an analysis of instantaneous droplet’s configurations and averaged density profiles. The latter are analyzed using density cutoffs and the Sobel filters. Parameters of the interaction between the planar surface and the fluid are being tuned in order to consider model lyophilic and lyophobic substrates. Dependences of the contact angle on the strength of interaction between the fluid and the substrate and on the droplet size are obtained. The apparent line tensions for the observed sessile droplets are evaluated.

A method is proposed for calculating low interfacial tension (IFT) based on molecular dynamics simulation of systems with superdense packing of surfactant molecules at the water–liquid hydrocarbon interface. The interfacial tension was calculated by the molecular dynamics method using the all-atom and coarse-grained models in water–alkane (decane, dodecane) two-phase systems in the presence of various individual surfactants. The following ionic and nonionic surfactants were considered: sodium dodecyl sulfate (SDS), cetyltrimethylammonium chloride (CTAC), sodium dodecylbenzenesulfonate (SDBS), sodium decet-6 sulfate C10E6SO4Na, hexaethylene glycol monodecyl ether (C10E6), triethylene glycol monononadecyl ether (C19E3), and octapropoxypentaethylene glycol monododecyl ether (C12P8E5). It was shown that the interfacial tension decreases to zero when surfactant adsorption increases to the limiting values.

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.

Disjoining pressures in thin liquid films around nanosized wettable spherical particles and in thin vapor layers around non-wettable particles were calculated as functions of the lyophility degree, film thickness, and particle size on the basis of the expression for the grand thermodynamic potential as a molecular density functional. A characteristic feature of this approach is the full consideration of hard-sphere molecular correlations using the fundamental measure theory in the density functional theory (DFT) and calculation of the complete dependence of the grand thermodynamic potential of the system on the stable droplet or bubble size. Although the newly calculated dependences of the disjoining pressure are in a qualitative agreement with those found using a simpler gradient version of the molecular density functional, the results of the two methods considerably differ quantitatively. It was confirmed that the disjoining pressure in a liquid film around a nanosized lyophilic particle increases with increasing particle size and lyophilicity.

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.

A new statistical thermodynamic model of inverse nonionic aggregates of surfactant molecules in nonpolar solvents is considered. This model admits the fluctuation coexistence of inverse spherical, globular, and spherocylindrical aggregates without activation barriers between them. The model is based on the assumption of a uniform bulk density of the number of molecular groups inside the core of an aggregate that can continuously transform into a sphere, a globule, and a spherocylinder. In this model, for any aggregation numbers, the work of aggregation depends not only on the aggregation numbers and the concentration of surfactant monomers, but also on two independent geometric parameters characterizing, at the same aggregation numbers, the deviation from the spherical form of the aggregate towards globular and spherocylindrical forms. Even in the range of small aggregation numbers, this fact leads to a significant difference between the equilibrium distribution function of aggregates, which depends on the aggregation number and two form parameters, and the one-dimensional distribution function in terms of aggregation numbers. It is shown that the optimal values of the form parameters, which minimize the work of aggregation, are in good agreement for spherocylindrical aggregates with the predictions of a purely geometric model of such aggregates under the additional assumption of a uniform surface density of molecular groups at the micelle core. The predictions of a new molecular thermodynamic model for the degrees of surfactant micellization in inverse aggregates of various forms at different surfactant concentrations are considered.

Theoretical results of studying thermodynamic and structural characteristics of free droplets and bubbles around nanosized solid heterogeneous spherical inclusions using phenomenological thermodynamic approach and various versions of the molecular density functional method are reviewed. In the case of a droplet in an undersaturated or supersaturated vapor, the central solid particle is assumed to be lyophilic and can be either electrically charged, or uncharged. In the case of a vapor bubble in a stretched liquid, the central particle is assumed to be lyophobic and, as in the previous case, can be electrically charged or uncharged. The structure of droplets and bubbles is described by equilibrium molecular density profiles. Thermodynamic characteristics imply the chemical potential of molecules in a droplet or bubble as a function of their size, the work of formation of equilibrium droplets or bubbles as a function of the chemical potential of molecules in the system, and the surface tension and disjoining pressure in droplets or bubbles as functions of its radius.

Equilibrium 3D density profiles in droplets around solid lyophilic spherical particles in under- and supersaturated vapor and in thin concentric vapor shells on lyophobic particles in stretched and stable liquid have been found within the molecular density functional theory with hard-sphere correlations taken into account via the fundamental measure theory. The existence of stable drops and bubbles is confirmed, their structure is described, and threshold values of the chemical potential of vapor and liquid for barrierless nucleation are found. The results show a qualitative agreement with previous ones obtained by us within other versions of molecular density functional theory.

The dependences of the pressure, the chemical potential and the free-energydensity on the number density of particles for a homogeneous system of hardspheres have been compared using the Carnahan–Starling equation of state, theRusanov equations of state and the 18-coefficient virial expansion. The full anda narrower ranges of the number density of hard spheres have been considered. Itis shown that in general the 6th-order Rusanov equations agree better with thevirial expansion over the full range of the number density, but this is due tothe high-density region. In the low- and medium-density regions, theCarnahan–Starling equation shows slightly better agreement with the18-coefficient virial expansion. Using the dependences of the chemical potentialand the free-energy density on the local number density of particles obtainedfrom the Carnahan–Starling equation of state, the truncated 6th-order Rusanovequation, and the 18-coefficient virial expansion within an integral densityfunctional theory, we have calculated the molecular density profiles in radiallynonuniform spherical small droplets and bubbles of an argon-like substance andplotted the surface tension of small droplets and bubbles vs the curvature oftheir equimolecular surface. It is shown that the choice of the equation ofstate affects the values of quantities characterizing the two-phase equilibrium,e.g., the value of the chemical potential or the surface tension at a flatinterface, and can shift the droplet or bubble size.

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 derivation of an expression for the minimal work of micellization (aggregation work) on the basis of the extended droplet model has been considered for the cases of direct and inverse spherical micelles. The contributions of the following factors to the aggregation work have been taken into account: the hydrophobic effect upon the formation of the core of a direct micelle; the electrostatic interaction upon the formation of the core of an inverse micelle; the influence of the conformations of hydrocarbon tails and polar head groups in the corona and core of the direct and inverse micelle, respectively; and the effect of the surface tension at the boundary between a micelle core and a solution. The equation of state of molecular groups on a micelle core surface has been shown to play an important role in describing the stabilization of both direct and inverse micelles. The account of the contributions to the aggregation work makes it possible to explain the mechanism of surfactant aggregation in a nonpolar solvent in the absence of water and ascertain the existence of a critical micelle concentration, as well as to determine the average aggregation numbers of dry inverse micelles at different total surfactant concentrations.

The classical density functional theory makes it possible to explicitly calculate the local density profiles, the components of the pressure tensor, and the thicknesses of thin interlayers between a lyophilic or lyophobic solid surface and, accordingly, gas or liquid phases at different values of the chemical potentials of the phases. Within the framework of a unified approach based on the gradient approximation of the classical density functional theory, it has been shown that, at certain values of parameters characterizing the wettability or nonwettability of a solid, equilibrium liquid films or vapor layers of a uniform thickness are formed around a spherical particle, if its surface is lyophilic or lyophobic, respectively. Mechanical and thermodynamic definitions have been given for the disjoining pressure in the spherical liquid or vapor interlayer around a solid particle, and the agreement between the definitions has been proven by calculations at different interlayer thicknesses and particle radii. It has been shown that the disjoining pressure in a vapor interlayer around a nanosized lyophobic particle decreases with an increase in particle radius, with this phenomenon being opposite to the situation with liquid films.

A classical square-gradient density functional theory is used for describing disjoining pressure in vapor layers between a lyophobic planar smooth solid substrate and stable bulk liquid, and between a lyophobic nanosized solid sphere and stretched bulk liquid. Profiles of local density, of normal and tangential components of the local pressure tensor in the interlayers are calculated and analyzed for planar substrate and spherical particle at several values of the chemical potential of the fluid and parameters controlling wettability of solid. The pressure tensor is found to be significantly anisotropic in the vapor layers near the smooth solid surfaces. Mechanical and thermodynamic definitions of the disjoining pressure in the spherical vapor layer were compared and shown to be consistent. In contrast to case of liquid films, the disjoining pressure in vapor layers near the nanosized lyophobic solid particle is found to be greater than near the planar wall. It has been directly shown that, depending on lyophobicity, the disjoining pressure isotherms change from nonmonotonic to monotonic functions of the vapor layer thickness.

All-atom molecular dynamics has been employed to study the processes of self-aggregation and solubilization in aqueous solutions that contain decane, ionic and nonionic surfactants, and additives of salts. In particular, micellization of an anionic surfactant (sodium dodecyl sulfate) in an aqueous solution has been simulated in the presence of a hydrocarbon (decane) at preset temperature and pressure and different initial surfactant and hydrocarbon concentrations in the solution. Moreover, self-aggregation has been simulated in systems containing water, decane, and a mixture of anionic (sodium dodecyl sulfate) and nonionic (hexaethylene glycol monodecyl ether, C 10 E 6 ) both in a salt-free solution and in the presence of sodium chloride, calcium chloride, or a mixture thereof. Diffusion coefficients have been calculated for aggregates consisting of hydrocarbon and surfactant aggregates, and the viscosities of corresponding aqueous solutions have been estimated. The viscosities have been calculated in simulation cells containing either one or several aggregates.

A general picture of relaxation in micellar solution of nonionic one-component surfactant on the basis of numerical solution of linearized set of the Becker–Döring equations for spherical and for cylindrical micelles has been analyzed in the form of a series in eigenvectors of the matrix of the kinetic coefficients. Two general characteristic cases have been considered as the initial conditions, the addition of monomers to the equilibrium system and the dilution of the equilibrium system. In both cases, the significant eigenvectors (relaxation modes) have been localized in the space of the aggregation numbers, that are responsible for the stages of ultrafast, fast, and slow relaxation, and corresponding eigenvalues (inverse relaxation times) have been selected from the huge number of all eigenvalues of the matrix of the kinetic coefficients. The analytical methods for finding the relaxation times of the ultrafast, fast and slow relaxation, recently developed and new ones proposed in this article, have been considered. The accuracy of the analytical calculations was controlled by comparison with much more resource-intensive computations using the matrix of the linearized equation. The analytical determination of the fast relaxation modes was based on the transition to the continual boundary-value problem with the potential for the distribution function of micelles over the aggregation numbers and using the perturbation theory. The spectrum was found numerically using the Runge–Kutta method. A new analytical solution for fast relaxation of the ensemble of cylindrical micelles with physically sound coefficients of monomer attachment to cylindrical micelles has been found with reducing the boundary value problem for the differential Becker–Döring–Frenkel equation to the equation for the Airy function.

The main result of the commented article is based on the use of an erroneous technique in the derivation of the equation for the contact angles of surface bubbles. A correct derivation gives the same Young equation as for sessile droplets, and therefore supplementary contact angles for bubbles and droplets. This cannot explain the presented results of simulations of nanosized droplets and bubbles where there are also several questions. We suggest a possible source for the difference in the contact angle size dependence for droplets and bubbles as related to the line tension and adsorptions at the interfaces.

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.