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.     

Kinetics of NaCl nucleation in supercritical water investigated by molecular dynamics simulations

At system pressures between 17 MPa and 25 MPa the nucleation and growth of NaCl nanoparticles in water at supercritical conditions was investigated by molecular dynamics simulations at different system temperatures and system densities. Our results show that particle formation takes place within a few hundred picoseconds after the jump from ambient to supercritical conditions. After nucleation a phase of growth by adding monomers is followed by growth via cluster-cluster collisions. We present results on the time development of distributions of cluster sizes, cluster compositions, and cluster temperatures as well as radial distribution functions and nucleation rates.     

Homogeneous nucleation of n-nonane and n-propanol mixtures: a comparison of classical nucleation theory and experiments

The homogeneous nucleation rates for n-nonane-n-propanol vapor mixtures have been calculated as a function of vapor-phase activities at 230 K using the classical nucleation theory (CNT) with both rigorous and approximate kinetic prefactors and compared to previously reported experimental data. The predicted nucleation rates resemble qualitatively the experimental results for low n-nonane gas phase activity. On the high nonane activity side the theoretical nucleation rates are about three orders of magnitude lower than the experimental data when using the CNT with the approximate kinetics. The accurate kinetics improves the situation by reducing the difference between theory and experiments to two orders of magnitude. Besides the nucleation rate comparison and the experimental and predicted onset activities, the critical cluster composition is presented. The total number of molecules is approximated by CNT with reasonable accuracy. Overall, the classical nucleation theory with rigorous kinetic prefactor seems to perform better. The thermodynamic parameters needed to calculate the nucleation rates are revised extensively. Up-to-date estimates of liquid phase activities using universal functional activity coefficient Dortmund method are presented together with the experimental values of surface tensions obtained in the present study.     

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.     

Crystal growth and classical nucleation theory

Calculations were performed of the crystal growth rates in lithium disilicate glass in the low-temperature regime where homogeneous nucleation is observed. The computations were executed using the gain-loss (Becker–Doring) equations that form the framework of Classical Nucleation Theory (CNT). The growth rates were obtained in several different ways, using various choices for the kinetic model, the generalized diffusion coefficient, and the physical input data. The results of these calculations are compared with recently obtained experimental values of the growth rates.     

Improved quasi-unary nucleation model for binary H2SO4-H2O homogeneous nucleation

Aerosol nucleation events have been observed at a variety of locations worldwide, and may have significant climatic and health implications. Binary homogeneous nucleation (BHN) of H2SO4 and H2O is the foundation of recently proposed nucleation mechanisms involving additional species such as ammonia, ions, and organic compounds, and it may dominate atmospheric nucleation under certain conditions. We have shown in previous work that H2SO4-H2O BHN can be treated as a quasi-unary nucleation (QUN) process involving H2SO4 in equilibrium with H2O vapor, and we have developed a self-consistent kinetic model for H2SO4-H2O nucleation. Here, the QUN approach is improved, and an analytical expression yielding H2SO4-H2O QUN rates is derived. Two independent measurements related to monomer hydration are used to constrain the equilibrium constants for this process, which reduces a major source of uncertainty. It is also shown that the capillarity approximation may lead to a large error in the calculated Gibbs free energy change for the evaporation of H2SO4 molecules from small H2SO4-H2O clusters, which affects the accuracy of predicted BHN nucleation rates. The improved QUN model-taking into account the recently measured energetics of small clusters-is thermodynamically more robust. Moreover, predicted QUN nucleation rates are in better agreement with available experimental data than rates calculated using classical H2SO4-H2O BHN theory.     

Atomistic theory of amyloid fibril nucleation

We consider the nucleation of amyloid fibrils at the molecular level when the process takes place by a direct polymerization of peptides or protein segments into β-sheets. Employing the atomistic nucleation theory (ANT), we derive a general expression for the work to form a nanosized amyloid fibril (protofilament) composed of successively layered β-sheets. The application of this expression to a recently studied peptide system allows us to determine the size of the fibril nucleus, the fibril nucleation work, and the fibril nucleation rate as functions of the supersaturation of the protein solution. Our analysis illustrates the unique feature of ANT that the size of the fibril nucleus is a constant integer in a given supersaturation range. We obtain the ANT nucleation rate and compare it with the rates determined previously in the scope of the classical nucleation theory (CNT) and the corrected classical nucleation theory (CCNT). We find that while the CNT nucleation rate is orders of magnitude greater than the ANT one, the CCNT and ANT nucleation rates are in very good quantitative agreement. The results obtained are applicable to homogeneous nucleation, which occurs when the protein solution is sufficiently pure and/or strongly supersaturated.     

Molecular dynamics simulations of nucleation from vapor to solid composed of Lennard-Jones molecules

We performed molecular dynamics (MD) simulations of nucleation from vapor at temperatures below the triple point for systems consisting of 10(4)-10(5) Lennard-Jones (L-J) type molecules in order to test nucleation theories at relatively low temperatures. Simulations are performed for a wide range of initial supersaturation ratio (S(0) ? 10-10(8)) and temperature (kT = 0.2-0.6ε), where ε and k are the depth of the L-J potential and the Boltzmann constant, respectively. Clusters are nucleated as supercooled liquid droplets because of their small size. Crystallization of the supercooled liquid nuclei is observed after their growth slows. The classical nucleation theory (CNT) significantly underestimates the nucleation rates (or the number density of critical clusters) in the low-T region. The semi-phenomenological (SP) model, which corrects the CNT prediction of the formation energy of clusters using the second virial coefficient of a vapor, reproduces the nucleation rate and the cluster size distributions with good accuracy in the low-T region, as well as in the higher-T cases considered in our previous study. The sticking probability of vapor molecules onto the clusters is also obtained in the present MD simulations. Using the obtained values of sticking probability in the SP model, we can further refine the accuracy of the SP model.     

A kinetic model for nonisothermic homogeneous nucleation

A model for isothermal homogeneous nucleation is proposed that improves the classical model. A quasiequilibrium distribution of clusters was calculated on a basis of the Frenkel’-Lothe-Pound theory. The dependence of the free energy of clusters on their size was represented by an interpolation formula relating the free energy of dimers and large clusters to which a notion of macroscopic surface tension is applicable. The nucleation rate and the dependence of the cluster temperature on their size were calculated by balance equations describing the heating of from a cluster due to the condensation of monomers and its cooling due to collisions with an ambient gas. It is shown that the nucleation rate in excess buffer gas is higher than for the pure condensing gas by approximately two orders of magnitude. The model adequately describes the experimental data for the nucleation of methanol supersaturated vapor.     

Homogeneous nucleation rates of higher n-alcohols measured in a laminar flow diffusion chamber

Nucleation rate isotherms of n-butanol, n-pentanol, n-hexanol, n-heptanol, and n-octanol were measured in a laminar flow diffusion chamber using helium as carrier gas. The measurements were made at 250-310 K, corresponding to reduced temperatures of 0.43-0.50, and at atmospheric pressure. Experimental nucleation rate range was from 10(3) to 10(7) cm(-3) s(-1). The expression and accuracy of thermodynamic parameters, in particular equilibrium vapor pressure, were found to have a significant effect on calculated nucleation rates. The results were compared to the classical nucleation theory (CNT), the self-consistency corrected classical theory (SCC) and the Hale's scaled model of the CNT. The average ratio between the experimental and theoretical nucleation rates for all alcohols used was 1.5x10(3) when the CNT was used, and 0.2x10(-1) when the SCC was used and 0.7x10(-1) when the Hale's scaled theory was used. The average values represent all the alcohols used at the same reduced temperatures. The average ratio was about the same throughout the temperature range, although J(exp)/J(the) calculated with the Hale's scaled theory increased slightly with increasing temperature. The saturation ratio dependency was predicted closest to experiment with the classical nucleation theory. The nucleation rates were compared to those found in the literature. The measurements were in reasonable agreement with each other. The molecular content of critical alcohol clusters was between 35 and 80 molecules. At a fixed reduced temperature, the number of molecules in a critical cluster decreased as a function of alcohol carbon chain length. The number of molecules in critical clusters was compared to those predicted by the Kelvin equation. The theory predicted the critical cluster sizes well.     

Rate of two-dimensional nucleation: verifying classical and atomistic theories by Monte Carlo simulation

We present kinetic Monte Carlo simulation data for the stationary rate J of two-dimensional nucleation of monolayer-thick crystal clusters in the growth of an atomically smooth (100) face of single-component Kossel crystal. The data are over a wide range of supersaturations s and effective broken-bond energies omega of nearest-neighbor atoms, and the J values span about 15 orders of magnitude. The simulation reveals that, in the s,omega range studied, the ln J vs s curve is smooth but with nearly linear portions connected with rather sharply curved segments when the omega value is sufficiently great. The simulation J(s) data are used for verification of the classical (CNT) and atomistic (ANT) nucleation theories without free parameters. It turns out that whereas J is overestimated by CNT, it is underestimated by ANT. The disagreement between theory and simulation is much greater for CNT than for ANT and, with increasing omega, it increases for the former but almost disappears for the latter. However, the ANT prediction about broken linear ln J vs s dependence is not confirmed by the simulation in the s,omega range studied.     

Diffusion coefficients for crystal nucleation and growth in deeply undercooled glass-forming liquids

We calculate, employing the classical theory of nucleation and growth, the effective diffusion coefficients controlling crystal nucleation of nanosize clusters and the subsequent growth of micron-size crystals at very deep undercoolings, below and above Tg, using experimental nucleation and growth data obtained for stoichiometric Li2O.2SiO2 and Na2O.2CaO.3SiO2 glasses. The results show significant differences in the magnitude and temperature dependence of these kinetic coefficients. We explain this difference showing that the composition and/or structure of the nucleating critical clusters deviate from those of the stable crystalline phase. These results for diffusion coefficients corroborate our previous conclusion for the same glasses, based on different experiments, and support the view that, even for the so-called case of stoichiometric (polymorphic) crystallization, the nucleating phase may have a different composition and/or structure as compared to the parent glass and the evolving macroscopic crystalline phase. This finding gives a key to explain the discrepancies between calculated (by classical nucleation theory) and experimentally observed nucleation rates in these systems, in particular, and in deeply undercooled glass-forming liquids, in general.     

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.     

Monte Carlo simulations of critical cluster sizes and nucleation rates of water

We have calculated the critical cluster sizes and homogeneous nucleation rates of water at temperatures and vapor densities corresponding to experiments by Wolk and Strey [J. Phys. Chem B 105, 11683 (2001)]. The calculations have been done with an expanded version of a Monte Carlo method originally developed by Vehkamaki and Ford [J. Chem. Phys. 112, 4193 (2000)]. Their method calculates the statistical growth and decay probabilities of molecular clusters. We have derived a connection between these probabilities and kinetic condensation and evaporation rates, and introduce a new way for the calculation of the work of formation of clusters. Three different interaction potential models of water have been used in the simulations. These include the unpolarizable SPC/E [J. Phys. Chem. 91, 6269 (1987)] and TIP4P [J. Chem. Phys. 79, 926 (1983)] models and a polarizable model by Guillot and Guissani [J. Chem. Phys. 114, 6720 (2001)]. We show that TIP4P produces critical cluster sizes and a temperature and vapor density dependence for the nucleation rate that agree well with the experimental data, although the magnitude of nucleation rate is constantly overestimated by a factor of 2 x 10(4). Guissani and Guillot's model is somewhat less successful, but both the TIP4P and Guillot and Guissani models are able to reproduce a much better experimental temperature dependency of the nucleation rate than the classical nucleation theory. Using SPC/E results in dramatically too small critical clusters and high nucleation rates. The water models give different average binding energies for clusters. We show that stronger binding between cluster molecules suppresses the decay probability of a cluster, while the growth probability is not affected. This explains the differences in results from different water models.     

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.     

Sensitivity of nucleation phenomena on range of interaction potential

Theoretical and computational investigations of nucleation have been plagued by the sensitivity of the phase diagram to the range of the interaction potential. As the surface tension depends strongly on the range of interaction potential and as the classical nucleation theory (CNT) predicts the free energy barrier to be directly proportional to the cube of the surface tension, one expects a strong sensitivity of nucleation barrier to the range of the potential; however, CNT leaves many aspects unexplored. We find for gas-liquid nucleation in Lennard-Jones system that on increasing the range of interaction the kinetic spinodal (KS) (where the mechanism of nucleation changes from activated to barrierless) shifts deeper into the metastable region. Therefore the system remains metastable for larger value of supersaturation and this allows one to explore the high metastable region without encountering the KS. On increasing the range of interaction, both the critical cluster size and pre-critical minima in the free energy surface of kth largest cluster, at respective kinetic spinodals, shift towards smaller cluster size. In order to separate surface tension contribution to the increase in the barrier from other non-trivial factors, we introduce a new scaling form for surface tension and use it to capture both the temperature and the interaction range dependence of surface tension. Surprisingly, we find only a weak non-trivial contribution from other factors to the free energy barrier of nucleation.     

Binary nucleation of water and sodium chloride

Nucleation processes in the binary water-sodium chloride system are investigated in the sense of the classical nucleation theory (CNT). The CNT is modified to be able to handle the electrolytic nature of the system and is employed to investigate the acceleration of the nucleation process due to the presence of sodium chloride in the steam. This phenomenon, frequently observed in the Wilson zone of steam turbines, is called early condensation. Therefore, the nucleation rates of the water-sodium chloride mixture are of key importance in the power cycle industry.     

Test of classical nucleation theory on deeply supercooled high-pressure simulated silica

We test classical nucleation theory (CNT) in the case of simulations of deeply supercooled, high density liquid silica, as modeled by the van Beest-Kramer-van Santen potential. We find that at density rho=4.38 gcm(3), spontaneous nucleation of crystalline stishovite occurs in conventional molecular dynamics simulations at temperature T=3000 K, and we evaluate the nucleation rate J directly at this T via "brute force" sampling of nucleation events in numerous independent runs. We then use parallel, constrained Monte Carlo simulations to evaluate DeltaG(n), the free energy to form a crystalline embryo containing n silicon atoms, at T=3000, 3100, 3200, and 3300 K. By comparing the form of DeltaG(n) to CNT, we test the ability of CNT to reproduce the observed behavior as we approach the regime where spontaneous nucleation occurs on simulation time scales. We find that the prediction of CNT for the n dependence of DeltaG(n) fits reasonably well to the data at all T studied. Deltamu, the chemical potential difference between bulk liquid and stishovite, is evaluated as a fit parameter in our analysis of the form of DeltaG(n). Compared to directly determined values of Deltamu extracted from previous work, the fitted values agree only at T=3300 K; at lower T the fitted values increasingly overestimate Deltamu as T decreases. We find that n(*), the size of the critical nucleus, is approximately ten silicon atoms at T=3300 K. At 3000 K, n(*) decreases to approximately 3, and at such small sizes methodological challenges arise in the evaluation of DeltaG(n) when using standard techniques; indeed even the thermodynamic stability of the supercooled liquid comes into question under these conditions. We therefore present a modified approach that permits an estimation of DeltaG(n) at 3000 K. Finally, we directly evaluate at T=3000 K the kinetic prefactors in the CNT expression for J, and find physically reasonable values; e.g., the diffusion length that Si atoms must travel in order to move from the liquid to the crystal embryo is approximately 0.2 nm. We are thereby able to compare the results for J at 3000 K obtained both directly and based on CNT, and find that they agree within an order of magnitude. In sum, our work quantifies how certain predictions of CNT (e.g., for Deltamu) break down in this deeply supercooled limit, while others [the n dependence of DeltaG(n)] are not as adversely affected.