Nucleation kinetics of folded chain crystals of polyethylene on active centers

Kinetic equations describing formation of nuclei of a new phase on active centers from supercooled melt taking into account exhaustion of active centers are solved numerically. Basic characteristics of nucleation process (total number of nuclei and nucleation rate) of folded chain crystals of polyethylene at low supercooling are determined and compared with measured data. Our model gives good coincidence with experimental measurements of the total number density of nuclei. Nucleation rate reaches some quasistationary limit at sufficiently long time, which is approximately 20% of the stationary nucleation rate determined by the standard way.     

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.     

Homogeneous nucleation of droplets from supersaturated vapor in a closed system

Kinetic equations describing homogeneous nucleation kinetics within standard model are solved numerically under the condition of a constant number of molecules in the considered system. It has consequences to decrease the supersaturation of the supersaturated vapor during the process of the formation of small droplets of a new phase. The decrease of supersaturation occurs in a short time and reaches some value which remains unchanged for a relatively long time (quasistationary regime), especially at lower initial supersaturations. This time interval decreases with increasing value of the initial supersaturation. In the quasistationary regime the nucleation rate reaches its stationary value. At higher initial supersaturation, the rate of formation of nuclei goes to some maximum value corresponding to the stationary nucleation rate and then decreases with time due to the decrease of supersaturation.     

Influence of vapor depletion on nucleation rate

During condensation in finite systems part of molecules is transformed from supersaturated mother phase to a new one and depletion of the mother phase occurs. Kinetic equations describing homogeneous nucleation process including decrease of supersaturation are solved numerically. It is shown that dependency of nucleation rate on nucleus size reaches some maximum, which decreases with time and moves to higher nucleus sizes. Nucleation rate is negative for undercritical size of nuclei.     

Vapor phase nucleation on neutral and charged nanoparticles: condensation of supersaturated trifluoroethanol on Mg nanoparticles

A new technique is described to study the condensation of supersaturated vapors on nanoparticles under well-defined conditions of vapor supersaturation, temperature, and carrier gas pressure. The method is applied to the condensation of supersaturated trifluoroethanol (TFE) vapor on Mg nanoparticles. The nanoparticles can be activated to act as condensation nuclei at supersaturations significantly lower than those required for homogeneous nucleation. The number of activated nanoparticles increases with increasing the vapor supersaturation. The small difference observed in the number of droplets formed on positively and negatively charged nanoparticles is attributed to the difference in the mobilities of these nanoparticles. Therefore, no significant charge preference is observed for the condensation of TFE vapor on the Mg nanoparticles.     

Free-energy landscape of nucleation with an intermediate metastable phase studied using capillarity approximation

Capillarity approximation is used to study the free-energy landscape of nucleation when an intermediate metastable phase exists. The critical nucleus that corresponds to the saddle point of the free-energy landscape as well as the whole free-energy landscape can be studied using this capillarity approximation, and various scenarios of nucleation and growth can be elucidated. In this study, we consider a model in which a stable solid phase nucleates within a metastable vapor phase when an intermediate metastable liquid phase exists. We predict that a composite critical nucleus that consists of a solid core and a liquid wetting layer as well as pure liquid and pure solid critical nuclei can exist depending not only on the supersaturation of the liquid phase relative to that of the vapor phase but also on the wetting behavior of the liquid surrounding the solid. The existence of liquid critical nucleus indicates that the phase transformation from metastable vapor to stable solid occurs via the intermediate metastable liquid phase, which is quite similar to the scenario of nucleation observed in proteins and colloidal systems. By studying the minimum-free-energy path on the free-energy landscape, we can study the evolution of the composition of solid and liquid within nuclei which is not limited to the critical nucleus.     

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.     

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.     

Statistico-probabilistic approach to taking account of the vapor depletion in the kinetics of homogeneous nucleation: a free-molecular regime of droplet growth

We propose a statistico-probabilistic approach to investigate the process of homogeneous formation of droplets in a vapor phase in the presence of an already formed and growing droplet under free-molecular regime of droplet growth after the instantaneous creation of initial vapor supersaturation. We find the probability density for the formation of a new, nearest (neighbor) droplet in the vicinity of an initially formed droplet. The mean distance between two neighboring droplets is also determined, as well as the average time lag for the formation of the nearest (neighbor) droplet; the latter quantity serves as an estimate for the duration of the nucleation stage. An estimate for the average number of droplets forming in unit volume by the end of the nucleation stage is also given. Our results are compared with the predictions of classical nucleation theory which assumes the density uniformity of a metastable phase. Where the proposed approach is applicable, there is observed qualitative agreement between the results. The underlying cause of this agreement is analyzed and the limits of applicability of the uniformity approximation are clarified.     

Crisis of stability of hydration shell of Na<Superscript>+</Superscript> ion in condensing water vapor

The Gibbs energy and equilibrium work of the formation of nuclei of the condensed phase on sodium ions are calculated on the molecular level by a Monte Carlo simulation using a detailed interaction model. The stationary rate of nucleation is estimated based on the data obtained. The presence of ionic impurities only substantially affects the rate of nucleation at strong vapor supersaturation. The nucleus losses its thermodynamic stability with an increase in the size of the nucleus and the barrier is formed depending on the work of formation on the size of the nucleus. An abrupt loss of stability is accompanied by pushing the ion off of the microdroplet surface and the restoration of the network of hydrogen bonds. The effect of pushing an ion to the surface of a cluster greatly depends on many-particle polarization interactions.     

Experimental studies of the vapor phase nucleation of refractory compounds. VI. The condensation of sodium

In this paper we discuss the condensation of sodium vapor and the formation of a sodium aerosol as it occurs in a gas evaporation condensation chamber. A one-dimensional model describing the vapor transport to the vapor/aerosol interface was employed to determine the onset supersaturation, in which we assume the observed location of the interface is coincident with a nucleation rate maximum. We then present and discuss the resulting nucleation onset supersaturation data within the context of nucleation theory based on the liquid droplet model. Nucleation results appear to be consistent with a cesium vapor-to-liquid nucleation study performed in a thermal diffusion cloud chamber.     

Gradient theory computation of the radius-dependent surface tension and nucleation rate for n-nonane clusters

The Van der Waals-Cahn-Hilliard gradient theory (GT) is applied to determine the structure and the work of formation of clusters in supersaturated n-nonane vapor. The results are analyzed as functions of the difference of pressures of the liquid phase and vapor phase in chemical equilibrium, which is a measure for the supersaturation. The surface tension as a function of pressure difference shows first a weak maximum and then decreases monotonically. The computed Tolman length is in agreement with earlier results [L. Granasy, J. Chem. Phys. 109, 9660 (1998)] obtained with a different equation of state. A method based on the Gibbs adsorption equation is developed to check the consistency of GT results (or other simulation techniques providing the work of formation and excess number of molecules), and to enable an efficient interpolation. A cluster model is devised based on the density profile of the planar phase interface. Using this model we analyze the dependency of the surface tension on the pressure difference. We find three major contributions: (i) the effect of asymmetry of the density profile resulting into a linear increase of the surface tension, (ii) the effect of finite thickness of the phase interface resulting into a negative quadratic term, and (iii) the effect of buildup of a low-density tail of the density profile, also contributing as a negative quadratic term. Contributions (i)-(iii) fully explain the dependency of the surface tension on the pressure difference, including the range relevant to nucleation experiments. Contributions (i) and (ii) can be predicted from the planar density profile. The work of formation of noncritical clusters is derived and the nucleation rate is computed. The computed nucleation rates are closer to the experimental nucleation rate results than the classical Becker-D?ring theory, and also the dependence on supersaturation is better predicted.     

Adsorption Isotherms for Concentrated Aqueous-Organic Solutions (CAOS)

From an engineering point of view the present authors have proposed the simple model of a homogeneous nucleation relationship in the liquid phase that can predict the number concentration of nucleated particles in various operating conditions. Experiments of liquid-phase nucleation in which the precursor monomers were generated by several methods have successfully confirmed the predictions of the model. In the present paper, our previous model of homogeneous nucleation is extended to the case in which the precursor monomers are generated in a gas-phase system. First, a relationship between number concentration and mean volume diameter of nucleated aerosol particles and operating conditions is obtained considering the free molecular regime around the critical nuclei, which is the main difference with the liquid phase. Second, the validity of the relation lies in experimental use of dioctyl sebacate particles generated by evaporation-condensation. As a result, the predictions are in excellent agreement with the experimental results after considering substantial losses of monomers and particles to the walls of the experimental system because in the gasphase the diffusion velocity and the critical supersaturation ratio of monomer are higher than those in the liquid phase. Copyright 2000 Academic Press.     

Statistical Approach to the Description of Homogeneous Nucleation in the Vapor-Gas Medium: Effects of the Heat of Phase Transition

The effect of the heat release of phase transition is included in the formalism of the statistical approach in order to allow for the depletion of a substance comprising the metastable phase in the kinetics of the homogeneous nucleation of supersaturated vapor. The diffusion regime of the exchange of molecules between the vapor and the growing droplets is considered on the assumption of instantaneous creation of the initial vapor supersaturation. The time-dependent boundary condition on the surface of a sphere with a fixed radius and a center coinciding with the center of a growing droplet is used in the problem of heat conductivity that allowed us to provide the heat balance of phase transition. The main characteristics of the nucleation stage are calculated for the representative vapor-gas systems. It is shown that the allowance for the effects of the heat release of phase transition resulted in a rather notable change in the kinetics of nucleation even at a sufficiently high concentration of the passive gas.     

On the Problem of Nucleation in a Supersaturated Vapor in the Presence of Heterogeneous Centers of Different Natures

The rules for constructing the analytical approximation are formulated for the law of the accumulation of the condensing substance by droplets that have emerged at centers of different natures. The accuracy of the suggested approximation is tested for the limiting cases of homogeneous nucleation and nucleation at centers of the same type. A general algorithm is constructed for describing the nucleation stage at centers of different natures after instantaneous creation of vapor supersaturation. Numerical calculations are conducted for the numbers of droplets formed at positively and negatively charged univalent ions in water vapors.     

The Kinetics of Crystal Growth

The fundamentals of the theory of crystal growth comprise the two concepts of growth via two-dimensional nuclei and growth with the aid of dislocations. Measurements of the relationship between rate of growth and supersaturation are particularly suitable for testing theoretical predictions. Experimental results are compared with predictions for crystal growth from the vapor phase, from solution, and from the melt. Depending on the experimental conditions, examples of both types of growth are met irrespective of the nature of the mother phase.     

Kinetics of droplet condensation through a double free-energy barrier

Results are presented for the kinetics of nucleation of liquid droplets from a one-component vapor phase on a planar lyophobic substrate patterned with a large number of easily wettable (lyophilic) circular domains. If the wettability of these lyophilic domains is characterized by a contact angle smaller than pi2, for intermediate values of the supersaturation, the condensation of a droplet on a lyophilic domain occurs through a free-energy barrier with two maxima, that is, through a double barrier. A simple model is proposed for the kinetics of droplet condensation through a double barrier that combines Kramers's [Physica (Utrecht) 7, 284 (1940)] transition rate theory with known results of nucleation theory. In the framework of this model, the solution is derived for the steady-state limit of the nucleation process. The number of lyophilic domains available for droplet condensation reduces with time as domains are occupied by droplets. The problem of droplet condensation through a double barrier is solved taking into account the effect of the time-dependent depletion in the number of available lyophilic domains.     

The stage of nonisothermal nucleation of supercritical particles of a new phase under nonstationary conditions of particle diffusion growth and heat transfer to a medium

An analytical theory has been formulated for the stage of nonisothermal nucleation of supercritical particles in a metastable medium with instantaneously generated initial supersaturation. The theory takes into account the nonuniformities of metastable substance concentration and temperature, which result from the nonstationary diffusion of the substance to growing particles and the nonstationary transfer of the heat of the phase transition from the particles to the medium. The formulated theory extends the approach based on the concept of excluded volume that has recently been used in the theory of the stage of nucleation under isothermal conditions. This approach implies that the nucleation intensity of new particles is suppressed in spherical diffusion regions with certain sizes that surround previously nucleated supercritical particles and remaining unchanged in the rest of the medium. It has been shown that, when self-similar solutions are used for nonstationary equations of substance diffusion to particles and heat transfer from the particles, the ratio between the excluded volume and the particle volume is independent of particle size, thereby enabling one to analytically solve the integral equation for the excluded volume throughout a system as a time function at the stage of nucleation. The main characteristics of the phase transition have been found for the end of the stage of nucleation. Comparison has been carried out with the characteristics obtained in terms of the isothermal and nonisothermal nucleation theory upon uniform vapor consumption and heat dissipation (the mean-field approximation of vapor supersaturation and temperature).     

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.