Stationary concentration of a binary solution in a growing drop and the time of its establishment

Abstract

In the general case, the problem of describing the growth of a drop of a binary solution in a mixture of vapors of its constituent substances and a passive gas is quite difficult to solve. To solve this problem, even in the relatively simple case of diffusion-controlled drop growth, it is usually assumed [1, 2] that the solution concentration in the drop rapidly reaches a stationary value. To confirm this assumption, the time dependence of the concentration of an ideal binary solution in drops at various initial compositions was numerically calculated [2]. However, these numerical calculation results [2] show that the stationary solution concentration in a drop may be reached over quite a long time. It has recently been studied [3] how the stationary concentration of an ideal binary solution in a drop growing under diffusion control is reached, but neither has the value of this stationary solution concentration been found nor has the law under which the stationary concentration is reached with time been explicitly determined. In this work, we will study the case where the number density of molecules of each component in a twocomponent mixture of vapors that have condensed to form a drop noticeably exceeds the number density of molecules of each component in a two-component mixture of vapors being in equilibrium with this drop. In this interesting case, each of the two components of the vapor mixture intensely condenses to the drop. For diffusion-controlled drop growth, we will obtain expressions for the time dependence of the drop radius and the numbers of molecules of the substances that have condensed to the drop from the vapor mixture. We will find the time in which these expressions and the diffusioncontrolled mode of drop growth become valid after drop nucleation. We will find the stationary solution concentration in the drop growing under diffusion control and also will derive the power law under which the stationary solution concentration is reached with time. We will show that the stationary solution concentration in the growing drop is reached with time rather rapidly in comparison with the relatively slow drop growth. Unlike recent works [1‐3], we will need no assumption of ideality of the solution in the drop. Let us consider a drop of a binary solution that has nucleated in an initially uniform vapor‐gas medium of a passive gas and a two-component mixture of vapors of the same substances as in the drop. After nucleation, the drop grows because of the condensation of each of the two components of the vapor mixture. The amount of the passive gas in the vapor‐gas mixture is assumed to be much larger than the amount of the vapors. The drop grows under diffusion control when the drop radius R noticeably exceeds the free path length λ of molecules of the vapors in the passive gas, i.e., when R ≥ R 0 , where (1) is the drop radius above which the drop grows under diffusion control. The large relative amount of the passive gas in the vapor‐gas medium ensures isothermal conditions of condensation and also allows one to ignore the Stefan flow of the vapor‐gas medium and the effect of the diffusion flows of molecules of the vapors on each other. Let x i , i = 1, 2, be the number of molecules of component i in the drop at a current moment of time and ν i , i = 1, 2, be the partial volume occupied by a molecule of component i in the solution within the drop. As Grinin and Lezova [3], for simplicity, we assume the volumes ν 1 and ν 2 to be constant. It was shown [3] that the solution within the drop is uniform. Obviously,