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.     

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.     

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.     

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.     

Size-dependent phase stability of a molecular nanocrystal: a proxy for investigating the early stages of crystallization

We make the link between the size-dependent phase stability of a nanocrystal and the phase-transition behavior of emerging crystallites during the earliest stages of crystallization, by using the former as a proxy for the latter. We outline an extension of the classical nucleation theory to describe crystal nucleation and subsequent transformations of competing polymorphic phases that characterize Ostwald's rule of stages. The theoretical framework reveals that the relative stability of the competing phases is a function of cluster size, which in turn varies with time, and therefore explains the complex transformation behavior observed for some systems. We investigated the stability of a nanocrystal of dl-norleucine by means of molecular simulation as a proxy for post-nucleation phase-transformation behavior in emerging crystallites. The simulations reveal that, for nanocrystals, the surface energy of the transition state of a transformation can dominate the barrier to phase change, thus causing metastable phases to be stabilized, not because they are thermodynamically stable, but rather due to kinetic hindering. Therefore, in the context of the earliest stages of crystal growth, not only does phase stability vary as a function of cluster size, and hence time, but thermodynamically feasible transformations are also prone to kinetic hindering.     

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.     

氟化钾团簇的相变、成核和重结晶的分子动力学模拟

我们利用Born-Mayer-Huggins相互作用势函数对(KF)_N(N=108，256，500和864)团簇进行了分子动力学(MD)模拟。为了避免周期性边界条件对相变、成核和重结晶的干扰作用，对体系采用了自由边界。基于MD模拟结果，对团簇的熔化温度、熔化焓、扩散系数、成核速率、固液界面自由能、临界核大小等进行了计算和讨论。在对(KF)₈₆₄双晶团簇的热退火模拟中，观察到了固态的重结晶和晶粒的生长。经典的成核理论成功地解释了(KF)₈₆₄双晶团簇的重结晶MD模拟结果。     

A study of nucleation in supersaturated ibuprofen vapor in a flow diffusion chamber

Isothermal nucleation of supersaturated ibuprofen racemate vapor has been experimentally studied in a flow diffusion chamber at 293.3 and 301.2 K. Nucleation rates have been measured in the range of 10⁴?10⁴ cm^?3 s^?1 as functions of supersaturation. According to the first nucleation theorem, the numbers of molecules in critical nuclei have been found and used to determine the nucleation rate and supersaturation values as depending on the sizes of critical nuclei. The comparison of the experimental data with theoretical predictions has shown that the nucleation rates measured as functions of the numbers of molecules in critical nuclei are higher than the rates predicted by the classical theory by six to seven decimal orders of magnitude but, within one order of magnitude, coincide with the rates predicted by a theory previously proposed in a work by one of the authors, in which nucleation clusters were considered to be microscopic objects.     

Nitrogen nucleation in a cryogenic supersonic nozzle

We follow the vapor-liquid phase transition of N(2) in a cryogenic supersonic nozzle apparatus using static pressure measurements. Under our operating conditions, condensation always occurs well below the triple point. Mean field kinetic nucleation theory (MKNT) does a better job of predicting the conditions corresponding to the estimated maximum nucleation rates, J(max) = 10(17±1) cm(-3) s(-1), than two variants of classical nucleation theory. Combining the current results with the nucleation pulse chamber measurements of Iland et al. [J. Chem. Phys. 130, 114508-1 (2009)], we use nucleation theorems to estimate the critical cluster properties. Both the theories overestimate the size of the critical cluster, but MKNT does a good job of estimating the excess internal energy of the clusters.     

Galvanostatic nucleation and growth under diffusion control

A theory is proposed for galvanostatic three-dimensional nucleation and growth of new phase clusters controlled by diffusion of depositing ions in the bulk of the electrolyte to the clusters surface. The mathematical model is used to calculate the time dependences of the overpotential, number, and size of clusters formed on the electrode.     

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.     

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.     

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.     

Multicomponent dynamical nucleation theory and sensitivity analysis

Vapor to liquid multicomponent nucleation is a dynamical process governed by a delicate interplay between condensation and evaporation. Since the population of the vapor phase is dominated by monomers at reasonable supersaturations, the formation of clusters is governed by monomer association and dissociation reactions. Although there is no intrinsic barrier in the interaction potential along the minimum energy path for the association process, the formation of a cluster is impeded by a free energy barrier. Dynamical nucleation theory provides a framework in which equilibrium evaporation rate constants can be calculated and the corresponding condensation rate constants determined from detailed balance. The nucleation rate can then be obtained by solving the kinetic equations. The rate constants governing the multistep kinetics of multicomponent nucleation including sensitivity analysis and the potential influence of contaminants will be presented and discussed.     

The formation and kinetics of Ionized Cluster Beams

The formation and kinetics of large vapourized-material cluster beams (large size metal clusters) are discussed. The clusters are formed by injecting the vapour of solid state materials into a high vacuum region through a nozzle of a heated crucible. The conditions under which metal clusters form are analysed using nucleation theory. Computer simulation by combining the nucleation and flow equations has also been made. The results show that the theory can be useful in predicting qualitative dependences of metal cluster formation on operation conditions. Several experimental results are also presented, which support the finding that a large size metal cluster is formed by homogeneous nucleation and growth. The advantageous characteristics of ionized cluster beam for thin film formation are also discussed.     

On the Kinetics of Nonisothermal Nucleation

The kinetics of supersaturated vapor nucleation is considered with allowance for the possible difference between the temperature of the clusters and that of the gaseous phase. The resultant equation for the steady-state nucleation rate has the same form as the equation of the classical theory but contains the work of the thermodynamically irreversible droplet formation. This equation relates the nucleation rate to the cluster temperatures. It is transformed into the equation of the classical theory if the clusters have the same temperature as the gaseous phase. On the basis of the thermal balance for the clusters, an equation relating their temperature and concentration to the nucleation rate is derived. The form of this equation is similar to the equation of heat propagation in a moving medium in the presence of heat sources and sinks.     

Effects of translational and rotational diffusion on the association in kinetic model of nucleation

Kinetics of an association and dissociation of single elements with the effects of translational and rotational diffusion and angular limitations is discussed. Separated clusters embedded in a solution of orientable single elements are considered.Steady-state positional and angular distribution of single elements is calculated from the equation of translational-rotational diffusion and the boundary conditions proposed for orientation-limited association. Although spherical orientable elements are assumed, the model can be used for non-spherical particles with aspect ratios close to unity.Diffusion-limited rate constants of association and dissociation are proposed which depend on translational and rotational diffusion constants of single elements, the tolerance angle of the association, and the cluster size.Effective concentration of single elements and effective rate constants are expressed by the equilibrium and diffusion-limited rate constants. Effects of finite diffusion rates and finite tolerance angle are discussed.The equations of the kinetic model of nucleation are modified due to the diffusion-limited rate of the association.     

Particle nucleation in emulsion polymerization. I. A theory for homogeneous nucleation

This paper is the first in a series intended to clarify the particle nucleation mechanisms in emulsion polymerization. The theory for particle nucleation by precipitation of oligomeric radicals from the water phase is discussed and a model based on the diffusion, propagation and termination steps is presented. The physical factors that influence the capture rate of oligomers in particles are discussed, and qualitative expressions for the electrostatic repulsion and reversible diffusion are derived. These factors are shown to be able to explain the relatively slow absorption rate of oligomers in particles and micelles. A kinetic model for simultaneous particle nucleation and limited flocculation is presented. Numerical integration of this model shows that the particle number goes through a maximum and that simultaneous nucleation and flocculation of primary particles may take place after Interval I in an emulsion polymerization is finished.