A. K. Shchekin

VALIDITY OF TOLMAN'S EQUATION : HOW LARGE SHOULD A DROPLET BE?

Surface tension and the length δ (distance between the Gibbs surface of tension Rs and the equimolar surface Re) of simple liquid droplet (Lennard-Jones and Yukawa) are computed over a wide range of droplet sizes up to about 4×106 molecules. The study is based on the Gibbs theory of capillarity combined with the density–functional approach to gas–liquid nucleation. Since this method provides behavior of the surface tension fully consistent with the tension of the planner surface, the constant in Tolman’s equation δ∞ can be determined unequivocally from the asymptotic behavior of σs. Comparison of the tension given by Tolman’s equation against the result of exact thermodynamic relations reveals that Tolman’s equation is valid only when the droplet holds more than 106 molecules for the simple fluid systems near their triple points, in contrast to the conventional wisdom that Tolman’s equation may be applicable down to droplets holding a few hundreds of molecules.

Role of surface forces in heterogeneous nucleation on wettable nuclei

Abstract The condition has been formulated and the role of the surface forces at a solid-liquid interface has been established for the barrierless heterogeneous formation of a droplet on a macroscopic wettable insoluble solid nucleus in supersaturated vapour. The threshold value of the vapour supersaturation starting from which heterogeneous nucleation occurs as a barrierless process and all thermodynamic characteristics of heterogeneous formation of a droplet which are necessary for kinetic description of phase transition below the threshold have been found as a function of parameters of the exponential and the power-law approximations to the work of wetting of the nucleus associated with the surface forces and size of condensation nuclei: the coordinates and the half-widths of the potential hump and the potential well in the curve of the work of heterogeneous formation of a droplet; the activation barrier which droplets have to overcome by fluctuations in the process of formation of the stable phase.

Generalization of the Gibbs–Kelvin–Köhler and Ostwald–Freundlich equations for a liquid film on a soluble nanoparticle

A derivation of chemical equilibrium equations for a spherical thin film of solution around a soluble solid nanoparticle is presented. The equations obtained generalize the Gibbs-Kelvin-Kohler and Ostwald-Freundlich equations for a soluble particle immersed in the bulk phase. The generalized equations describe the dependence of the chemical potentials of a condensate and dissolved nanoparticle matter in the thin solution film, the condensate saturated pressure, and the solubility of the nanoparticle matter on the film thickness, and the nanoparticle size with account of the disjoining pressure of the liquid film.

Excess equimolar radius of liquid drops.

The curvature dependence of the surface tension is related to the excess equimolar radius of liquid drops, i.e., the deviation of the equimolar radius from the radius defined by the macroscopic capillarity approximation. Based on the Tolman [J. Chem. Phys. 17, 333 (1949)] approach and its interpretation by Nijmeijer et al. [J. Chem. Phys. 96, 565 (1991)], the surface tension of spherical interfaces is analyzed in terms of the pressure difference due to curvature. In the present study, the excess equimolar radius, which can be obtained directly from the density profile, is used instead of the Tolman length. Liquid drops of the truncated and shifted Lennard-Jones fluid are investigated by molecular dynamics simulation in the canonical ensemble, with equimolar radii ranging from 4 to 33 times the Lennard-Jones size parameter σ. In these simulations, the magnitude of the excess equimolar radius is shown to be smaller than σ/2. This suggests that the surface tension of liquid drops at the nanometer length scale is much closer to that of the planar vapor-liquid interface than reported in studies based on the mechanical route.

Dynamic Light Scattering Study of Cetyltrimethylammonium Bromide Aqueous Solutions

Cetyltrimethylammonium bromide (CTAB) aqueous solutions are studied by dynamic light scattering method in a wide concentration range covering the first and second critical micelle concentrations (CMC1 and CMC2, respectively). Nonmonotonic and ambiguous behavior of diffusion coefficients D with an increase in concentrations above CMC1 is revealed. An increase in the D values in the first decade of CTAB concentration above CMC1 agrees with known published data for aqueous solutions of ionic surfactants. It is shown that an increase in the ionic strength of solution with the addition of KBr leads to a decrease in the positive slope of the dependence of diffusion coefficients on CTAB concentration up to zero at 0.05 M KBr. Two relaxation processes corresponding to large and small D values are simultaneously observed in micellar solutions, beginning with 0.03 M CTAB concentration. The data obtained are compared with published data, as well as with the results of viscosity measurements. The performed analysis indicates that the observed relaxation processes are explained by the coexistence of spherical and nonspherical micelles. It is established that micelles acquire a cylindrical shape at CTAB concentrations ranging from 0.2 to 0.25 M. Hydrodynamic radii and lengths of micelles are calculated.

Thermodynamics of droplet formation around a soluble condensation nucleus in the atmosphere of a solvent vapor.

An expression for the work of formation of a spherical droplet condensing on a soluble condensation nucleus out of a solvent vapor is derived. The dependence of the formation work on the solvent vapor chemical potential and the droplet and the nucleus residue sizes is analyzed. The balance of the solute matter between the liquid film and the nucleus residue and the effect of overlapping the surface layers of the thin film have been taken into account. It is shown that the equations of the chemical equilibrium of a solute and a solvent in the droplet, resulting from the generating properties of the formation work, coincide with the generalized Gibbs-Kelvin-Kohler and Ostwald-Freundlich equations. The numerical solution of these equations at a fixed number of molecules of the nucleus matter (at an initial size of the nucleus specified) has been performed in the case of the solvent vapor undersaturated over the bulk liquid solvent phase. The solution links the equilibrium sizes of the droplet and the soluble nucleus residue with the chemical potential or the pressure of the solvent vapor saturated over the droplet. It also determines the limiting sizes of the droplet with small nucleus residue above which the chemical equilibrium of the residue surface and the solution film does not exist. The existence of the limiting sizes is responsible for the specific behavior of the droplet thermodynamic characteristics and the work of droplet formation at deliquescence transition from the droplet state with a partly dissolved nucleus to the state of complete dissolution of the nucleus.

The microphysical effects in nonisothermal nucleation

The role of microphysical effects in the ability of a small new phase nucleus to emit molecules into the vapor is revealed on a thermodynamic basis. It is shown that dependence of the surface tension of a nucleus on its temperature and the dependence of the vaporisation heat on the nuclear surface curvature are insignificant in nonisothermal nucleation. The meaning of the quantities used in the one-dimensional theory of isothermal nucleation has been clarified from the point of the two-dimensional nonisothermal theory. Statistical and kinetic treatment of the microphysical effects in the emitting rate of a nucleus has been performed. An association of the microphysical effects with deviation of the kinetic equation of nonisothermal nucleation from the Fokker–Planck form has been examined.

Micellization and relaxation in solution with spherical micelles via the discrete Becker-Döring equations at different total surfactant concentrations

A numerical description of micellization and relaxation to an aggregate equilibrium in surfactant solution with nonionic spherical micelles has been developed on the basis of a discrete form of the Becker-Döring kinetic equations. Two different models for the monomer-aggregate attachment-detachment rates have been used, and it has been shown that the results are qualitatively the same. The full discrete spectrum of characteristic times of micellar relaxation and first relaxation modes in their dependence on equilibrium monomer concentration have been found with using the linearized form of the Becker-Döring kinetic equations. Overall time behavior of surfactant monomer and aggregate concentrations in micellization and relaxation at large initial deviations from final equilibrium has been studied with the help of nonlinearized discrete Becker-Döring kinetic equations. Comparison of the computed results with the analytical ones known in the limiting cases from solutions of the linearized and nonlinearized continuous Becker-Döring kinetic equation demonstrates general agreement.

Thermodynamic Characteristics of Micellization in the Droplet Model of Surfactant Spherical Molecular Aggregate

The dependence of the work of the molecular aggregate formation on the aggregation number and surfactant monomer concentration in solution that has the key role for the theory of micellization was studied on the basis of a simple realistic droplet model of spherical aggregate composed of surfactant molecules (the o/w micelle type). Analytical formulas were derived for the coordinates of maximum and minimum of aggregate formation work on the aggregation number axis arising with an increase in the concentration of micellar solution. Model calculations of the thermodynamic characteristics of the kinetics of micellization were performed for premicellar and micellar regions of aggregate sizes within a wide range of solution concentration including the critical micellization concentration.

The effects of external electric field in thermodynamics of formation of dielectric droplet

Abstract The methods of deriving the relations describing the influence of the external electric field on thermodynamic characteristics and shape of a dielectric liquid droplet are considered. It is shown that the uniform external electric field enhances nucleation for a fixed vapor chemical potential due to decreasing the chemical potential per molecule in droplet and the work of droplet formation. The results are obtained by taking into account the axisymmetric prolongation of the droplet in the direction of the external electric field.