Surface tension of water droplets upon homogeneous droplet nucleation in water vapor

A method has been proposed for determining interfacial free energy from the data of molecular dynamics simulation. The method is based on the thermodynamic integration procedure and is distinguished by applicability to both planar interfaces and those characterized by a high curvature. The workability of the method has been demonstrated by the example of determining the surface tension for critical nuclei of water droplets upon condensation of water vapor. The calculation has been performed at temperatures of 273–373 K and a pressure of 1 atm, thus making it possible to determine the temperature dependence of the surface tension for water droplets and compare the results obtained with experimental data and the simulation results for a “planar” vapor–liquid interface.     

Role of nearest-neighbor drops in the kinetics of homogeneous nucleation in a supersaturated vapor

A theory of simultaneous nucleation and drop growth in a supersaturated vapor is developed. The theory makes use of the concept of "nearest-neighbor" drops. The effect of vapor heterogeneity caused by vapor diffusion to a growing drop, formed previously, is accounted for by considering the nucleation of the nearest-neighbor drop. The diffusional boundary value problem is solved through the application of a recent theory that maintains material balance between the vapor and the drop, even though the drop boundary is a moving one. This is fundamental to the use of the proper time and space dependent vapor supersaturation in the application of nucleation theory. The conditions are formulated under which the mean distance to the nearest-neighbor drop and the mean time to its appearance can be determined reliably. Under these conditions, the mean time provides an estimate of the duration of the nucleation stage, while the mean distance provides an estimate of the number of drops formed per unit volume during the nucleation stage. It turns out, surprisingly, that these estimates agree fairly well with the predictions of the simpler and more standard approach based on the approximation that the density of the vapor phase remains uniform during the nucleation stage. Thus, as a practical matter, in many situations, the use of the simpler and less rigorous method is justified by the predictions of the more rigorous, but more complicated theory.     

Evaluation of surface tension and Tolman length as a function of droplet radius from experimental nucleation rate and supersaturation ratio: metal vapor homogeneous nucleation

Zinc and silver vapor homogeneous nucleations are studied experimentally at the temperature from 600 to 725 and 870 K, respectively, in a laminar flow diffusion chamber with Ar as a carrier gas at atmospheric pressure. The size, shape, and concentration of aerosol particles outcoming the diffusion chamber are analyzed by a transmission electron microscope and an automatic diffusion battery. The wall deposit is studied by a scanning electron microscope (SEM). Using SEM data the nucleation rate for both Zn and Ag is estimated as 10(10) cm(-3) s(-1). The dependence of critical supersaturation on temperature for Zn and Ag measured in this paper as well as Li, Na, Cs, Ag, Mg, and Hg measured elsewhere is analyzed. To this aim the classical nucleation theory is extended by the dependence of surface tension on the nucleus radius. The preexponent in the formula for the vapor nucleation rate is derived using the formula for the work of formation of noncritical embryo [obtained by Nishioka and Kusaka [J. Chem. Phys. 96, 5370 (1992)] and later by Debenedetti and Reiss [J. Chem. Phys. 108, 5498 (1998)]] and Reiss replacement factor. Using this preexponent and the Gibbs formula for the work of formation of critical nucleus the dependence of surface tension on the radius R(S) of the surface of tension is evaluated from the nucleation data for above-mentioned metals. For the alkali metals and Ag the surface tension was determined to be a strong function of R(S). For the bivalent metals (Zn, Hg, and Mg) the surface tension was independent of radius in the experimental range. A new formula for the Tolman length delta as a function of surface tension and radius R(S) is derived by integration of Gibbs-Tolman-Koenig equation assuming that delta is a monotonic function of radius. The formula derived is more correct than the Tolman formula and convenient for the elaboration of experimental data. Using this formula the values of delta are determined as a function of R(S) from the experimental nucleation data. It is determined that all the metals considered are characterized by strong dependence of delta on radius; for the bivalent metals delta changes sign.     

On the reduced moment in the transient regime of homogeneous nucleation

One of the parameters characterizing the evolution of nucleation in the transient regime is the so-called reduced moment, a dimensionless quantity. This parameter describes the steepness with which the nucleation rate approaches its steady state. Until recently, very little had been known about this parameter in real systems, although a widely quoted 1969 theory [D. Kashchiev, Surf. Sci. 14, 209 (1969)] existed that formally described nucleation in the transient regime. This theory has been shown to be incorrect in its implication about the reduced moment. Molecular dynamics simulations have recently greatly clarified what happens in the transient regime. It turns out that the reduced moment depends strongly on the size of the nucleus under consideration, and, for a rapidly quenched liquid, it substantially exceeds unity for small nuclei but approaches unity as nuclei increases in size. The objective of this paper is to illustrate the behavior of the reduced moment and to show how this behavior is a natural consequence of the kinetics of the nucleation process.     

The kinetics of homogeneous and two-step nucleation during protein crystal growth from solution

Many experimental reports for the kinetics of crystal nucleation and growth, from an isothermal solution, point to a sigmoidal-like behavior for the process. Here we consider three different nucleation models from the literature and show that all lead to sigmoidal or sigmoidal-like behavior for the kinetics of nucleation. A two-step nucleation process is known to occur within certain supersaturated protein solutions, and it is demonstrated in this report how the sigmoidal law yields kinetic information for the two-step and homogeneous nucleation modes. We propose here that two-step solute-rich associates form in the solution around seed nuclei that are already present at or near the point in time when the solution is prepared. Using this hypothesis, we are able to model the time-dependent volume of the two-step phase per unit volume of solution and show that this compares well with reported experimental data. A kinetic model is given for the proposed process, which leads to a sigmoidal rate law. Additionally, a relation between the initial and final nuclei densities and the induction time is derived. As a result of this study, the conclusion is that two-step activity increases with increasing initial supersaturation or increasing salt concentration.     

On the steady-state solutions of the kinetics of homogeneous nucleation

A general solution to the steady-state equations for the kinetics of homogeneous nucleation is determined, in agreement with irreversible thermodynamics. Most of the steady-state distributions of clusters present a minimum. Our theory provides a conjecture for understanding the apparent contradictions existing in the present experimental data.     

Influence of initial conditions on homogeneous nucleation kinetics in a closed system

The formation of nuclei of a new phase from the supersaturated mother phase in a closed system is studied. The depletion of the mother phase due to phase transition is taken into account. Basic kinetic equations describing such process are solved numerically to determine the number density of nuclei of newly forming phase and nucleation rate. It is shown that in contrary to the standard nucleation model, when the depletion of the mother phase is not taken into account, the initial size distribution of the clusters affects considerably the nucleation process at higher supersaturations. Our model starts with the equilibrium size distribution of clusters up to various cluster sizes in the undercritical region. At lower supersaturation the formation of nuclei is similar to the standard model because of the low depletion of the mother phase. At higher supersaturation, the depletion of the mother phase plays an important role and some extremal value appears at the size distribution of nuclei, which is not observed in the standard model. The extremum in the size distribution is not a consequence of the coalescence process itself, but it is caused rather by the depletion of the mother phase during the phase transformation.     

Effect of the surface-stimulated mode on the kinetics of homogeneous crystal nucleation in droplets

The thermodynamics of surface-stimulated crystal nucleation demonstrates that if at least one of the facets of the crystal is only partially wettable by its melt, then it is thermodynamically more favorable for the nucleus to form with that facet at the droplet surface rather than within the droplet. So far, however, the kinetic aspects of this phenomenon had not been studied at all. In the present paper, a kinetic theory of homogenous crystal nucleation in unary droplets is proposed by taking into account that a crystal nucleus can form not only in the volume-based mode (with all its facets within the droplet) but also in the surface-stimulated one (with one of its facets at the droplet surface). The theory advocates that even in the surface-stimulated mode crystal nuclei initially emerge (as subcritical clusters) homogeneously in the subsurface layer, not "pseudo-heterogeneously" at the surface. A homogeneously emerged subcritical crystal can become a surface-stimulated nucleus due to density and structure fluctuations. This effect contributes to the total rate of crystal nucleation (as the volume-based mode does). An explicit expression for the total per-particle rate of crystal nucleation is derived. Numerical evaluations for water droplets suggest that the surface-stimulated mode can significantly enhance the per-particle rate of crystal nucleation in droplets as large as 10 microm in radius. Possible experimental verification of the proposed theory is discussed.     

New approach to the kinetics of heterogeneous unary nucleation on liquid aerosols of a binary solution

The formation of a droplet on a hygroscopic center may occur either in a barrierless way via Kohler activation or via nucleation by overcoming a free energy barrier. Unlike the former, the latter mechanism of this process has been studied very little and only in the framework of the classical nucleation theory based on the capillarity approximation whereby a nucleating droplet behaves like a bulk liquid. In this paper the authors apply another approach to the kinetics of heterogeneous nucleation on liquid binary aerosols, based on a first passage time analysis which avoids the concept of surface tension for tiny droplets involved in nucleation. Liquid aerosols of a binary solution containing a nonvolatile solute are considered. In addition to modeling aerosols formed through the deliquescence of solid soluble particles, the considered aerosols constitute a rough model of "processed" marine aerosols. The theoretical results are illustrated by numerical calculations for the condensation of water vapor on binary aqueous aerosols with nonvolatile nondissociating solute molecules using Lennard-Jones potentials for the molecular interactions.     

Experimental study of homogeneous nucleation from the bismuth supersaturated vapor: evaluation of the surface tension of critical nucleus

The homogeneous nucleation of bismuth supersaturated vapor is studied in a laminar flow quartz tube nucleation chamber. The concentration, size, and morphology of outcoming aerosol particles are analyzed by a transmission electron microscope (TEM) and an automatic diffusion battery (ADB). The wall deposit morphology is studied by scanning electron microscopy. The rate of wall deposition is measured by the light absorption technique and direct weighting of the wall deposits. The confines of the nucleation region are determined in the "supersaturation cut-off" measurements inserting a metal grid into the nucleation zone and monitoring the outlet aerosol concentration response. Using the above experimental techniques, the nucleation rate, supersaturation, and nucleation temperature are measured. The surface tension of the critical nucleus and the radius of the surface of tension are determined from the measured nucleation parameters. To this aim an analytical formula for the nucleation rate is used, derived from author's previous papers based on the Gibbs formula for the work of formation of critical nucleus and the translation-rotation correction. A more accurate approach is also applied to determine the surface tension of critical drop from the experimentally measured bismuth mass flow, temperature profiles, ADB, and TEM data solving an inverse problem by numerical simulation. The simulation of the vapor to particles conversion is carried out in the framework of the explicit finite difference scheme accounting the nucleation, vapor to particles and vapor to wall deposition, and particle to wall deposition, coagulation. The nucleation rate is determined from simulations to be in the range of 10(9)-10(11) cm(-3) s(-1) for the supersaturation of Bi(2) dimers being 10(17)-10(7) and the nucleation temperature 330-570 K, respectively. The surface tension σ(S) of the bismuth critical nucleus is found to be in the range of 455-487 mN/m for the radius of the surface of tension from 0.36 to 0.48 nm. The function σ(S) changes weakly with the radius of critical nucleus. The value of σ(S) is from 14% to 24% higher than the surface tension of a flat surface.     

Simple correction to the classical theory of homogeneous nucleation

Formation of the new disperse phase via homogeneous nucleation plays a fundamental role wherever the first-order phase transitions occur. Inconsistent temperature dependence of the nucleation rates and poor agreement of theoretical critical supersaturations with experimental data for a number of substances are fundamental problems of the classical nucleation theory (CNT). Here we show that these problems can be solved with a simple empirical correction to CNT. Despite its simplicity, the corrected CNT (CCNT) accurately predicts temperature dependences and absolute values of the critical supersaturations for both organic and inorganic substances with widely varying properties at different ambient conditions and it works surprisingly well in a wide size range down to few molecules. The difference in predictions of CCNT and other versions of the classical nucleation theory commonly used in analyzing experimental data is discussed. It has been found that CCNT consistently gives better agreement with experimental data than other versions of classical nucleation theory.     

Study of nonsteady diffusional growth of a droplet in a supersaturated vapor: treatment of the moving boundary and material balance

A new mathematical treatment of the problem of droplet growth via diffusion of molecules from a supersaturated vapor is presented. The theory is based on a semiquantitative analysis with good physical arguments and is justified by its reasonable predictions. For example it recovers the time honored growth law in which, to a high degree of approximation, the droplet radius increases with the square root of time. Also, to a high degree of approximation, it preserves material balance such that, at any time, the number of molecules lost from the vapor equals the number in the droplet. Estimates of the remaining approximational error are provided. On another issue, we show that, in contrast, the conventional treatment of droplet growth does not maintain material balance. This issue could be especially important for the nucleation of another droplet in the vicinity of the growing droplet where the rate of nucleation depends exponentially on supersaturation. Suggestions for further improvement of rigor are discussed.     

To the theory of homogeneous nucleation: Cluster energy

An attempt is made to critically analyze the modern state of the theory of homogeneous nucleation as concerns its ability to describe experiments with high accuracy. An analysis of the experimental data led us to conclude that the dependence of the nucleation rate on supersaturation and temperature T was not described by the theory, which underestimates the critical cluster size compared with the Gibbs-Thomson equation. The possibility of applying density functional theory (one of the latest achievements in the theory of homogeneous nucleation) was questioned. Within this theory, the Gibbs-Thomson equation remains valid even outside the classic capillary approximation. It is suggested that, to bring theory in consistency with experiment, certain fundamental propositions of the theory of nucleation should be revised. The inclusion of an additional contribution to the Gibbs energy of a cluster caused by the size dependence of the specific heat capacity of the cluster decreases the critical cluster size compared with the value calculated by the Gibbs-Thomson equation. The calculated dependence of nucleation rate on supersaturation was in agreement with the experimental results.     

Importance of heat capacity determination in homogeneous nucleation: application to progesterone

Progesterone is known to exist under different crystallographic forms in the solid state. The thermodynamic stable form (I), melts at 129.2 °C (402.35 K) under atmospheric pressure. After melting and cooling a metastable form (II) can be obtained which melts at 122 °C (395.15 K). This uncommon behaviour can be explained with the theory of nucleation, only if heat capacity of the different forms are known.