Relations for extrema of the work of aggregation in micellar solutions

Exact relations for the dependence of the positions of extrema of the aggregation work on the aggregation number axis on the surfactant monomer concentration are found for a dilute micellar solution. Relations for the half-widths of the vicinities of the extrema of the aggregation work are determined also. These half-widths are determined by the condition that, within the boundaries of their values, the aggregation work deviates from its extreme values by a thermal unit. The relations derived are illustrated by calculations based on the droplet and quasi-droplet models of spherical aggregates of nonionic surfactant. Some important applications of the obtained results to the theory of relaxation in micellar solutions are considered.     

System of relaxation equations for materially isolated surfactant solution with spherical and cylindrical micelles

Analytical expressions for the direct and reverse fluxes of molecular aggregates over the first and second potential barriers of the aggregation work in the presence of spherical and cylindrical micelles in non-ionic surfactant solution were derived. Expressions for the sum (entering into kinetic equations of micellization) of direct and reverse fluxes of molecular aggregates over the first and second potential barriers of the aggregation work in the vicinity of the final equilibrium state of materially isolated surfactant solution were linearized. In the experimentally important range of the values of overall surfactant concentration in solution where the predominant contribution to the total surfactant amount is introduced by cylindrical micelles, we derived a closed system of two linearized relaxation equations determining the buildup (with time) of experimentally observed total concentrations of spherical and cylindrical micelles in the vicinity of the final equilibrium state of materially isolated surfactant solution. The case of the solutions of such surfactants, for which the spherical shape of a micelle appeared to be unrealizable due to the structure and packing conditions of molecules, was considered separately.Translated from Kolloidnyi Zhurnal, Vol. 67, No. 1, 2005, pp. 38–46. Original Russian Text Copyright © 2005 by Kuni, Shchekin, Rusanov, Grinin.     

Non-Steady-State Temperature Field of the Vapor-Gas Mixture in the Vicinity of a Growing Droplet: Balance of the Heat of Phase Transition

The non-steady-state temperature field of the vapor-gas medium in the vicinity of a droplet growing in supersaturated vapor is constructed. In the conduction problem, a time-dependent boundary condition is used which ensures the fulfillment of the balance condition of the heat of phase transition. The resultant temperature field is compared with the one obtained in the heat conduction problem with the equilibrium boundary condition on the surface of a droplet of a fixed radius. Although the solution with the equilibrium boundary condition does not ensure the balance between the heat released on the growing droplet and the heat distributed due to heat conduction in the vapor-gas medium, the difference between the two solutions is not very large. This difference is important for describing the homogeneous nucleation of supersaturated vapor in the vicinity of a growing droplet, as is indicated by comparison of the vapor supersaturation fields constructed with and without allowance for thermal effects, as well as with the use of solutions to the diffusion and heat conduction problems with various boundary conditions.__________Translated from Kolloidnyi Zhurnal, Vol. 67, No. 3, 2005, pp. 333–341.Original Russian Text Copyright © 2005 by Grinin, Zhuvikina, Gor.     

Statistico-probabilistic approach to taking account of the vapor depletion in the kinetics of homogeneous nucleation: a free-molecular regime of droplet growth

We propose a statistico-probabilistic approach to investigate the process of homogeneous formation of droplets in a vapor phase in the presence of an already formed and growing droplet under free-molecular regime of droplet growth after the instantaneous creation of initial vapor supersaturation. We find the probability density for the formation of a new, nearest (neighbor) droplet in the vicinity of an initially formed droplet. The mean distance between two neighboring droplets is also determined, as well as the average time lag for the formation of the nearest (neighbor) droplet; the latter quantity serves as an estimate for the duration of the nucleation stage. An estimate for the average number of droplets forming in unit volume by the end of the nucleation stage is also given. Our results are compared with the predictions of classical nucleation theory which assumes the density uniformity of a metastable phase. Where the proposed approach is applicable, there is observed qualitative agreement between the results. The underlying cause of this agreement is analyzed and the limits of applicability of the uniformity approximation are clarified.     

Statistical Regularities of Homogeneous Boiling-Up of Gas-Supersaturated Solutions

Statistical regularities of homogeneous boiling-up of liquid solutions supersaturated with gas are studied. The local gas concentration and local rate of homogeneous boiling-up of the solution are found in the vicinity of the bubble emerging in the solution at low gas solubility. The probability of the emergence of new bubbles in addition to the bubble originally emerged in the solution is calculated. The mean distance between two neighboring bubbles and the relative statistical scatter of this distance are found. The mean number of bubbles in a solution unit volume by the end of the stage of their emergence is estimated. The mean expectation time of the emergence of the neighboring bubble estimating the duration of the stage of bubble emergence is established. The proposed theory is extended to the case of homogeneous nucleation in the vapor–gas medium.     

Study of Relaxation in Micellar Solution by the Numerical Experiment

A numerical simulation of the relaxation process of surfactant micellar solution to a new equilibrium state is performed using model analytical representations for the main characteristics of micellar aggregates. Relaxation stages of molecular aggregate size distribution in the typical regions of aggregation number variations predicted by the analytical theory in two-flux approximation are revealed. Good agreement between the predicted values of the relaxation times of micellar solution and those obtained in numerical simulation is disclosed within the domain of applicability of two-flux approximation. Numerical algorithm proposed in this work makes it possible to study the relaxation process of micellar solution even in the case when two-flux approximation becomes inapplicable. The realization of numerical algorithm can be considered as a kind of experiment for studying the relaxation process of a model micellar solution.     

Nonstationary vapor concentration fields near the growing droplet of binary solution

Nonstationary vapor concentration fields near the droplet of binary solution growing in the vapor—gas mixture are revealed using the concepts of similarity. The revealed fields are determined with the exact account of the motion of droplet surface and refer to the times at which the droplet reaches sizes that provide for the diffusion regime of droplet growth. To obtain the self-similar solution of the problem of binary condensation, it is necessary to ensure a constant (in time) concentration of binary solution in the growing droplet. The velocities of an increase in the number of molecules and the radius of two-component droplet with time are found with allowance for the equation ensuring this solution. The conditions for the transformation of the self-similar solution of the problem of the condensation of two-component mixture into the solution, which was derived previously for the condensation of one component, are elucidated.