Generalized Kelvin equation and pseudospinodal in nucleation theory

The classical Kelvin equation, relating the size of the critical cluster to the supersaturation, is inadequate for very small, molecular-sized clusters emerging at deep quenches observed in recent nucleation experiments. Using statistical mechanical considerations, we propose a generalization of the Kelvin equation applicable up to the vicinity of the pseudospinodal, where the nucleation barrier is approximately k(B)T. The supersaturation at the pseudospinodal is expressed in terms of the second virial coefficient. It is shown that near the pseudospinodal the critical cluster size is close to the coordination number in the liquid phase. Comparisons with computer simulations are presented.     

A thermodynamically consistent kinetic framework for binary nucleation

The traditional theory for binary homogeneous nucleation follows the classical derivation of the nucleation rate in the supposition of a hypothetical constrained-equilibrium distribution in the calculation of the cluster evaporation rate. This model enables calculation of the nucleation rate, but requires evaluation of the cluster distribution and cluster properties for an unstable equilibrium with supersaturated vapor. An alternate derivation of the classical homomolecular nucleation rate eliminated the need for this nonphysical approximation by calculating the evaporative flux at full thermodynamic equilibrium. The present paper develops that approach for binary nucleation; the framework is readily extended to ternary nucleation. In this analysis, the evaporative flux is evaluated by applying mass balance at full thermodynamic equilibrium of the system under study. This approach eliminates both the need for evaluating cluster properties in an unstable constrained-equilibrium state and ambiguity in the normalization constant required in the nucleation-rate expression. Moreover, it naturally spans the entire composition range between the two pure monomers. The cluster fluxes derived using this new model are similar in form to those of classical derivations, so previously developed methods for evaluation of the net nucleation rate can be applied directly to the new formulation.     

Homogeneous nucleation: classical formulas as asymptotic limits of the Cahn-Hilliard approach

Classical expressions for the critical cluster work of formation approximate the nonclassical expressions based in the density functional theory of capillarity for the limit of low supersaturation degrees. However, the ratio between classical and nonclassical expressions for nucleation rates grows as the supersaturation degree decreases. Here, with the aim to obtain simple and more accurate expressions that approximate the modern nucleation rate formulas, an asymptotic expansion of the Cahn-Hilliard expression of the critical work of formation is developed within the limit of low supersaturation. In such asymptotic expansion, terms up to third order are retained. The ratios between the corrected classical expressions and the nonclassical ones are now decreasing for supersaturation degrees tending to zero. However, the corrected approximate formulas are as difficult to handle as the exact Cahn-Hilliard expressions. When only the two first low-order terms of the asymptotic expansion are retained, a simpler corrected classical expression is obtained but it can only approximate nonclassical expressions up to order unity. Finally, using a Becker-Doring model of nucleation, the kinetic prefactor of the critical nuclei rate of formation is modeled consistently with the Cahn-Hilliard approach to the critical work of formation.     

Accounting for fluctuations in cluster parameters upon the description of nucleation in supersaturated vapor

A theory is proposed for stationary homogeneous nucleation in supersaturated vapor in which a modified expression for the rate of cluster evaporation was used to calculate the equilibrium distribution over the nucleus sizes and the rates of their formation. This rate was determined by the extrapolation to the region of small sizes of the corresponding expression for the macroscopic droplet derived according to thermodynamic notions that take fluctuations into account. Modified dependences of the size of critical nucleus and the rate of nucleation on the supersaturation and the temperature are determined and compared with the data of the classical theory of nucleation and experimental results.     

Kinetic model for binary homogeneous nucleation in the H2O-H2SO4 system: comparison with experiments and classical theory of nucleation

A kinetic model to predict nucleation rates in the sulfuric acid-water system is presented. It allows calculating steady-state nucleation rates and the corresponding time lag, using a direct solution of a system of kinetic equations that describe the populations of sub- and near-critical clusters. This kinetic model takes into account cluster-cluster collisions and decay of clusters into smaller clusters. The model results are compared with some predictions obtained with the classical nucleation theory (CNT) and also with available measurement data obtained in smog chambers or flow tubes. It is shown that in the case of slow nucleation processes, the kinetic model and the CNT as used by Shugard et al. [J. Chem. Phys. 75, 5298 (1974)] give the same results. However, in the case of intensive nucleation, a large part of the nucleation flux is due to cluster-cluster collisions and the CNT underestimates the nucleation rates.     

Nucleation versus spinodal decomposition in confined binary solutions

Basic features of spinodal decomposition, on one side, and nucleation, on the other side, and the transition between both mechanisms are analyzed within the framework of a generalized thermodynamic cluster model based on the generalized Gibbs approach. Hereby the clusters, representing the density or composition variations in the system, may change with time both in size and in their intensive state parameters (density and composition, for example). In the first part of the analysis, we consider phase separation processes in dependence on the initial state of the system for the case when changes of the state parameters of the ambient system due to the evolution of the clusters can be neglected as this is the case for cluster formation in an infinite system. As a next step, the effect of changes of the state parameters on cluster evolution is analyzed. Such depletion effects are of importance both for the analysis of phase formation in confined systems and for the understanding of the evolution of ensembles of clusters in large (in the limit infinite) systems. The results of the thermodynamic analysis are employed in both cases to exhibit the effect of thermodynamic constraints on the dynamics of phase separation processes.     

Nucleation of NaCl nanoparticles in supercritical water: molecular dynamics simulations

Formation of NaCl nanoparticles in supercritical water is studied using molecular dynamics simulation method. We have simulated particle nucleation and growth in NaCl-H2O fluids, with salt concentration of 5.1 wt %, in the temperature and density range of 673-1073 K and 0.17-0.34 g/cm(3), respectively. The cluster size distributions, the size of critical nuclei and cluster lifetimes are reported. The size distribution of emerging clusters shows a very strong dependence on the system's density, with larger clusters forming at lower densities. Clusters consisting of approximately 14-24 ions appear critical for the thermodynamic states examined. The local structures of critical clusters are found to be amorphous. The lifetime values for clusters containing more than 20 ions are in the range of 10-50 ps. We have calculated the NaCl nucleation rates, which appear to be on the order of 10(28) cm(-3) s(-1).     

Complete thermodynamically consistent kinetic model of particle nucleation and growth: numerical study of the applicability of the classical theory of homogeneous nucleation

A complete thermodynamically consistent elementary reaction kinetic model of particle nucleation and growth from supersaturated vapor was developed and numerically evaluated to determine the conditions for the steady-state regime. The model treats all processes recognized in the aerosol science (such as nucleation, condensation, evaporation, agglomerationcoagulation, etc.) as reversible elementary reactions. It includes all possible forward reactions (i.e., of monomers, dimers, trimers, etc.) together with the thermodynamically consistent reverse processes. The model is built based on the Kelvin approximation, and has two dimensionless parameters: S0-the initial supersaturation and Theta-the dimensionless surface tension. The time evolution of the size distribution function was obtained over the ranges of parameters S0 and Theta. At low initial supersaturations, S0, the steady state is established after a delay, and the steady-state distribution function corresponds to the predictions of the classical nucleation theory. At high initial supersaturations, the depletion of monomers due to condensation on large clusters starts before the establishing of the steady state. The steady state is never reached, and the classical nucleation theory is not applicable. The boundary that separates these two regimes in the two dimensionless parameter space, S0 and Theta, was determined. The model was applied to several experiments on water nucleation in an expansion chamber [J. Wolk and R. Strey, J. Phys. Chem. B 105, 11683 (2001)] and in Laval nozzle [Y. J. Kim et al., J. Phys. Chem. A 108, 4365 (2004)]. The conditions of the experiments performed using Laval nozzle (S0=40-120) were found to be close to the boundary of the non-steady-state regime. Additional calculations have shown that in the non-steady-state regime the nucleation rate is sensitive to the rate constants of the initial steps of the nucleation process, such as the monomer-monomer, monomer-dimer, etc., reactions. This conclusion is particularly important for nucleation from supersaturated water vapor, since these processes for water molecules at and below the atmospheric pressure are in the low pressure limit, and the rate constants can be several orders of magnitude lower than the gas kinetic. In addition, the impact of the thermodynamic inconsistency of the previously developed partially reversible kinetic numerical models was assessed. At typical experimental conditions for water nucleation, S0=10 and Theta=10 (T=250 K), the error in the particle nucleation rate introduced by the thermodynamic inconsistency exceeds one order of magnitude.     

Tests of the homogeneous nucleation theory with molecular-dynamics simulations. I. Lennard-Jones molecules

Two kinds of the homogeneous nucleation theory exist at the present: the classical nucleation theory and the semiphenomenological model. To test them, we performed molecular-dynamics (MD) simulations of nucleation from vapor to liquid with 5000-20,000 Lennard-Jones-type molecules. Simulations were done for various values of supersaturation ratios (from 2 to 10) and temperatures (from 80 to 120 K). We compared the size distribution of clusters in MD simulations with those in the theoretical models because the number density of critical clusters governs the nucleation rate. We found that the semiphenomenological model achieves excellent agreements in size distributions of the clusters with all MD simulations we done. The classical theory underestimates the number density of the clusters in the temperature range of 80-100 K, but overestimates in 100-120 K. The semiphenomenological model also predicts well the nucleation rate in MD simulations, while the classical nucleation theory does not. Our results confirmed the validity of the semiphenomenological model for Lennard-Jones-type molecules.     

Quasi-unary homogeneous nucleation of H2SO4-H2O

We show that the binary homogeneous nucleation (BHN) of H2SO4-H2O can be treated as quasi-unary nucleation of H2SO4 in equilibrium with H2O vapor. A scheme to calculate the evaporation coefficient of H2SO4 molecules from H2SO4-H2O clusters is presented and a kinetic model to simulate the quasi-unary nucleation of H2SO4-H2O is developed. In the kinetic model, the growth and evaporation of sulfuric acid clusters of various sizes are explicitly simulated. The kinetic quasi-unary nucleation model does not have two well-recognized problems associated with the classical BHN theory (violation of the mass action law and mismatch of the cluster distribution for monomers) and is appropriate for the situations where the assumption of equilibrium cluster distribution is invalid. The nucleation rates predicted with our quasi-unary kinetic model are consistent with recent experimental nucleation experiments in all the cases studied, while the most recent version of the classical BHN model systematically overpredicts the nucleation rates. The hydration of sulfuric acid clusters, which is not considered in the classical model but is accounted for implicitly in our kinetic quasi-unary model, is likely to be one of physical mechanisms that lead to lower nucleation rates. Further investigation is needed to understand exactly what cause the difference between the kinetic quasi-unary model and the classical BHN model.     

Mean-field kinetic nucleation theory

A new semiphenomenological model of homogeneous vapor-liquid nucleation is proposed in which the cluster kinetics follows the "kinetic approach to nucleation" and the thermodynamic part is based on the revised Fisher droplet model with the mean-field argument for the cluster configuration integral. The theory is nonperturbative in a cluster size and as such is valid for all clusters down to monomers. It contains two surface tensions: macroscopic (planar) and microscopic. The latter is a temperature dependent quantity related to the vapor compressibility factor at saturation. For Lennard-Jones fluids the microscopic surface tension possesses a universal behavior with the parameters found from the mean-field density functional calculations. The theory is verified against nucleation experiments for argon, nitrogen, water, and mercury, demonstrating very good agreement with experimental data. Classical nucleation theory fails to predict experimental results when a critical cluster becomes small.     

Nucleation on active centers in confined volumes

Kinetic equations describing nucleation on active centers are solved numerically to determine the number of supercritical nuclei, nucleation rate, and the number density of nuclei for formation both of droplets from vapor and also crystalline phase from vapor, solution, and melt. Our approach follows standard nucleation model, when the exhaustion of active centers is taken into account via the boundary condition, and thus no additional equation (expressing exhaustion of active centers) is needed. Moreover, we have included into our model lowering of supersaturation of a mother phase as a consequence of the phase transition process within a confined volume. It is shown that the standard model of nucleation on active centers (Avrami approach) gives faster exhaustion of active centers as compared with our model in all systems under consideration. Nucleation rate (in difference to standard approach based on Avrami model) is equal to the time derivative of the total number of nuclei and reaches some maximum with time. At lower nucleation barrier (corresponding to higher initial supersaturation or lower wetting angle of nucleus on the surface of active center) the exhaustion of active centers is faster. Decrease in supersaturation of the mother phase is faster at higher number of active centers.     

Recent developments in the kinetic theory of nucleation

A review of recent progress in the kinetics of nucleation is presented. In the conventional approach to the kinetic theory of nucleation, it is necessary to know the free energy of formation of a new-phase particle as a function of its independent variables at least for near-critical particles. Thus the conventional kinetic theory of nucleation is based on the thermodynamics of the process. The thermodynamics of nucleation can be examined by using various approaches, such as the capillarity approximation, density functional theory, and molecular simulation, each of which has its own advantages and drawbacks. Relatively recently a new approach to the kinetics of nucleation was proposed [Ruckenstein E, Nowakowski B. J Colloid Interface Sci 1990;137:583; Nowakowski B, Ruckenstein E. J Chem Phys 1991;94:8487], which is based on molecular interactions and does not employ the traditional thermodynamics, thus avoiding such a controversial notion as the surface tension of tiny clusters involved in nucleation. In the new kinetic theory the rate of emission of molecules by a new-phase particle is determined with the help of a mean first passage time analysis. This time is calculated by solving the single-molecule master equation for the probability distribution function of a surface layer molecule moving in a potential field created by the rest of the cluster. The new theory was developed for both liquid-to-solid and vapor-to-liquid phase transitions. In the former case the single-molecule master equation is the Fokker-Planck equation in the phase space which can be reduced to the Smoluchowski equation owing to the hierarchy of characteristic time scales. In the latter case, the starting master equation is a Fokker-Planck equation for the probability distribution function of a surface layer molecule with respect to both its energy and phase coordinates. Unlike the case of liquid-to-solid nucleation, this Fokker-Planck equation cannot be reduced to the Smoluchowski equation, but the hierarchy of time scales does allow one to reduce it to the Fokker-Plank equation in the energy space. The new theory provides an equation for the critical radius of a new-phase particle which in the limit of large clusters (low supersaturations) yields the Kelvin equation and hence an expression for the macroscopic surface tension. The theory was illustrated with numerical calculations for a molecular pair interaction potential combining the dispersive attraction with the hard-sphere repulsion. The results for the liquid-to-solid nucleation clearly show that at given supersaturation the nucleation rate depends on the cluster structure (for three cluster structures considered-amorphous, fcc, and icosahedral). For both the liquid-to-solid and vapor-to-liquid nucleation, the predictions of the theory are consistent with the results of classical nucleation theory (CNT) in the limit of large critical clusters (low supersaturations). For small critical clusters the new theory provides higher nucleation rates than CNT. This can be accounted for by the fact that CNT uses the macroscopic interfacial tension which presumably overpredicts the surface tension of small clusters, and hence underpredicts nucleation rates.