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.

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.

Abstract—The dependence of the thermodynamic surface tension of a small droplet formed on a molecular-size condensation nucleus on the droplet size, nucleus size, molecular field parameters, and nuclear charge in the case of an ion is investigated. Calculations have been performed for molecules of supersaturated argon vapor at different values of the chemical potential of the molecules in the framework of the gradient density functional theory (DFT) and the Carnahan–Starling model to take into account the contribution of hard spheres. The interaction of argon molecules with an uncharged condensation nucleus has been described by the Lennard–Jones potential. In the case of an ion, the long-range Coulomb potential of electric forces is additionally taken into account. The dielectric constant is defined as a function of the local density of the number of argon molecules. As a variable describing the size of the droplet, the radius of the equimolecular surface of the droplet is chosen. The obtained dependences of the surface tension of the droplets have been compared with the dependence of the surface tension on the size of droplet without a condensation nucleus. When the effect of the solvation layer is discarded, the dependence of the surface tension on the radius of the equimolecular surface of a small droplet with a condensation nucleus exhibits similar behavior as in the absence of the nucleus with almost the same negative Tolman correction. The effect of the rigidity constant, however, is clearly influenced by the existence of a condensation nucleus.  It is shown that when the first solvation layer is divided around the condensation nucleus, the dependence of surface tension on the radius of the equimolecular surface of a small droplet with a condensation nucleus exhibits similar behavior with almost the same negative Tolman correction as in the absence of a nucleus, but with a different correction, namely, with effective rigidity constant for the surface layer.

Existence of stable and unstable spherical shell-like vapor interlayers (or concentric nanobubbles) near lyophobic surfaces has been proved and their thermodynamic properties have been studied within the square gradient density functional theory. The equilibrium density profiles around lyophobic nanoscopic particles (without and with electric charge) in a stretched argon-like liquid have been computed. The combination of the Carnahan–Starling and the mean-field models for fluid–fluid interaction and the total Lennard-Jones potential of interaction between the particle and fluid molecule have been used. The lyophobicity of the particle has been controlled by the energy parameter for attraction of molecules of the particle and the fluid molecules. This parameter has been taken considerably smaller than the energy parameter for fluid–fluid molecular attraction. As a result, two equilibrium radial density profiles corresponding to two concentric vapor shells around particle were found at a fixed value of the condensate chemical potential below its value for the flat equilibrium. It was shown that the smaller shell is related to the minimum of the work of the vapor shell formation and represents a stable nanobubble, while the larger shell corresponds to the maximum of this work and refers to the unstable critical nanobubble. The equimolecular radii of the stable and unstable concentric nanobubbles increase with the radius of the core particle. The curve of the dependence of the chemical potential of fluid molecules in the bubble with the lyophobic core particle on the bubble radius has a minimum, below which heterogeneous nucleation of bubbles becomes thermodynamically barrierless. The appearance of the electric charge on the particle shifts the minimum of the condensate chemical potential deeper and inhibits bubble nucleation. The dependence of the bubble surface tension on the radius of the equimolecular dividing surface and the charge of the particle has been found for bubble at heterogeneous nucleation and compared with that for bubble and droplet at homogeneous nucleation and droplet at heterogeneous nucleation on lyophilic particle.

The square-gradient density functional theory has been used to study the bulk pressure and normal/tangential components of the local pressure tensor at several chemical potential values in a small droplet without and with a completely wettable solid core and a flat liquid thin film on a solid substrate. The Lennard-Jones fluid with the Carnahan-Starling model for the hard-sphere contribution to intermolecular interactions and mean-field attraction contribution have been used to describe the condensate. The interaction between the solid spherical core (and flat substrate) molecules and the condensate molecules has been taken to be stronger than the condensate-condensate interaction. The inhomogeinity within the curved liquid film on a solid core (and within a liquid film on flat substrate) has been found to be an effect of the interaction (or overlapping) of the interfacial layers of the solid-liquid and liquid-vapor interfaces. The disjoining pressure for a flat liquid thin film on a solid substrate in the undersaturated vapor has been computed for various film thicknesses as the difference between bulk values of the pressures for vapor and liquid phases. This disjoining pressure has been compared with that of small droplets condensed on solid cores found through the thermodynamic and mechanical routes for a range of values of the droplet and core sizes. The disjoining pressure was smaller for a curved film and was dependent on the solid core size.

The results of molecular modeling of ionic surfactant self-aggregation in sodium dodecyl sulphate (SDS) aqueous solutions, both salt-free and with addition of NaCl, are reported. The modeling has been based on all-atom molecular dynamics simulations at several SDS concentrations. Formation of one or three ionic aggregates of different size has been observed in the molecular dynamics runs. The computed trajectories of molecules and ions have been used to study the effects of aggregation on local densities of water molecules and counterions, and on the diffusivities of the aggregates themselves. With finding the mean force potential, the degree of counterion binding for aggregates with different aggregation numbers has been estimated. The results have been compared with the classical square-gradient density functional computations for a single ionic micelle in polar solvent. In addition, the influence of size of the simulation box on the transport properties of SDS aggregates has been investigated. In particular, we have found how the diffisivity of an ionic aggregate with a given aggregation number depends on the total surfactant concentration. The diffusivities of aggregates with aggregation numbers 16, 32, 48, and 64 at fixed total surfactant concentration have been computed. With the help of the Stokes-Einstein relation, the viscosity of the micellar solution at different total surfactant concentrations has been calculated. Time dependencies of the number of aggregates in the simulation box and of the average aggregation number during the self-assembly of surface active ions have been studied.

A numerical study of size-dependent effects in the thermodynamics of a small droplet formed around a solid nanoparticle has been performed within the square-gradient density functional theory. The Lennard-Jones fluid with the Carnahan-Starling model for the hard-sphere contribution to intermolecular interaction in liquid and vapor phases and interfaces has been used for description of the condensate. The intermolecular forces between the solid core and condensate molecules have been taken into account with the help of the Lennard-Jones part of the total molecular potential of the core. The influence of the electric charge of the particle has been considered under assumption of the central Coulomb potential in the medium with dielectric permittivity depending on local condensate density. The condensate density profiles and equimolecular radii for equilibrium droplets at different values of the condensate chemical potential have been computed in the cases of an uncharged solid core with the molecular potential, a charged core without molecular potential, and a core with joint action of the Coulomb and molecular potentials. The appearance of stable equilibrium droplets even in the absence of the electric charge has been commented. As a next step, the capillary, disjoining pressure, and electrostatic contributions to the condensate chemical potential have been considered and compared with the predictions of classical thermodynamics in a wide range of values of the droplet and the particle equimolecular radii. With the help of the found dependence of the condensate chemical potential in droplet on the droplet size, the activation barrier for nucleation on uncharged and charged particles has been computed as a function of the vapor supersaturation. Finally, the work of droplet formation and the work of wetting the particle have been found as functions of the droplet size.

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.

A scheme of computation of the condensate chemical potential per molecule as a function of the droplet equimolecular radius for stable and critical droplets on uncharged or charged spherical particle of molecular size at heterogeneous nucleation has been considered. The scheme is based of the gradient density functional theory (DFT) with the van der Waals (vdW) and Carnahan–Starling (СS) models for the hard-sphere contribution to intermolecular interaction in liquid and vapor phases and interfaces. The particle serving as a condensation center in the case of heterogeneous nucleation has been characterized by an attractive short-range molecular potential and the long-range electric Coulomb potential. The dielectric permittivities of the droplet–vapor systems have been taken as known functions of the local condensate density and temperature for polar and nonpolar fluids. Detailed numerical calculations of the density profiles in critical and stable equilibrium droplets at water or argon nucleation in presence of the capillary, electrostatic and molecular forces have been performed. Dependence of the condensate chemical potential in the droplet on the droplet equimolecular radius has been analyzed in the case of homogeneous and heterogeneous nucleation and compared for water and argon within the vdW and СS models for the hard-sphere part of the equation of state.