Non-Steady-State Temperature Field of the Vapor-Gas Mixture in the Vicinity of a Growing Droplet: Balance of the Heat of Phase Transition

The non-steady-state temperature field of the vapor-gas medium in the vicinity of a droplet growing in supersaturated vapor is constructed. In the conduction problem, a time-dependent boundary condition is used which ensures the fulfillment of the balance condition of the heat of phase transition. The resultant temperature field is compared with the one obtained in the heat conduction problem with the equilibrium boundary condition on the surface of a droplet of a fixed radius. Although the solution with the equilibrium boundary condition does not ensure the balance between the heat released on the growing droplet and the heat distributed due to heat conduction in the vapor-gas medium, the difference between the two solutions is not very large. This difference is important for describing the homogeneous nucleation of supersaturated vapor in the vicinity of a growing droplet, as is indicated by comparison of the vapor supersaturation fields constructed with and without allowance for thermal effects, as well as with the use of solutions to the diffusion and heat conduction problems with various boundary conditions.__________Translated from Kolloidnyi Zhurnal, Vol. 67, No. 3, 2005, pp. 333–341.Original Russian Text Copyright © 2005 by Grinin, Zhuvikina, Gor.     

Experimental Determination of the Surface Energy of Critical Nuclei During the Nucleation of Supersaturated Vapor

Theoretical substantiation of the empirical method for determining the surface (or excess) energy of critical nuclei was performed within the framework of thermodynamic approach. Characteristics of critical nuclei were determined based on the studies of the nucleation of supersaturated vapors of glycerol in the vicinity of its melting point and the critical temperature of carrier gas. The effect of the specific features of physicochemical parameters of the studied substance and carrier gas on the parameters of critical nuclei was revealed. Experimental values of the surface energy of critical nuclei were compared with those calculated by the droplet model. The necessity for the allowance for the temperature dependence of the surface energy of critical nuclei was demonstrated. It was noted that the largest deviation from the predictions of droplet model arises at small (about 10) numbers of molecules in a critical nucleus; as the size of a nucleus increases, the surface energy, in the limiting case, tends to physically correct value.     

A competitive particle nucleation mechanism in the polymerization of homogenized styrene emulsions

The effects of concentrations of surfactant (sodium lauryl sulfate [SLS]) and initiator (sodium persulfate [SPS]) on the polymerization of homogenized styrene emulsions, stabilized by SLS/lauryl methacrylate (LMA) or SLS/stearyl methacrylate (SMA), were studied. The rate of polymerization increases with increasing [SLS] or [SPS]. In addition to monomer droplet nucleation, the formation of particle nuclei in the aqueous phase (homogeneous nucleation) plays a crucial role in the polymerization kinetics. In comparison with the LMA containing polymerization system, monomer droplet nucleation becomes more important when the more hydrophobic SMA was used as the costabilizer. Furthermore, the degree of homogeneous nucleation increases with increasing [SPS].     

Universal Asymptotics of the Thermodynamic Characteristics of the Nucleation on Small Macroscopic Nuclei of Soluble Surfactants

Asymptotic behavior of thermodynamic characteristics of nucleation on small macroscopic nuclei of soluble surfactants at their complete dissolution in a nucleating droplet is studied. It is taken into account that, in the region of small sizes of nuclei and corresponding small sizes of critical nuclei of liquid phase, the chemical potential of condensate and the work of droplet formation are affected by the presence of dense surfactant adsorption monolayer on the droplet surface. It is shown that, as the limiting surface area per surfactant molecule in adsorption monolayer increases, the behavior of thermodynamic characteristics of nucleation in the region of small nucleus sizes is characterized by the transition from asymptotics at the adsorption of almost all substance comprising nucleus in a monolayer to the asymptotics at constant adsorption. The study performed is not limited by the selection of specific adsorption isotherms; therefore, the obtained asymptotic dependences of thermodynamic characteristics on the nucleus size can be considered as universal for the heterogeneous nucleation on the nuclei of soluble surfactants.     

The Rate of Spontaneous Formation of Microscopic Nuclei in Supersaturated Vapor

The equilibrium nucleus-size distribution determined by the method of statistical physics has been analyzed. The analysis has shown that nuclei composed of 1000 or fewer molecules are microscopic objects. They are described by partition functions and cannot be described by thermodynamic methods. An approach has been proposed that makes it possible to determine a partition function over internal degrees of freedom of a nucleus and express the aforementioned distribution via commonly accepted thermodynamic parameters. The solution of the problem is reduced to the determination of the evaporation rate of clusters by extrapolating the evaporation rate, which has been calculated for a macroscopic droplet of an incompressible liquid in terms of thermodynamic concepts with allowance for fluctuations, to the sizes of nuclei. As a result, a theory has been formulated for homogeneous stationary nucleation. The comparison of the proposed theory with experimental data has shown that the calculated sizes of critical nuclei coincide with the measured ones and that the theoretical nucleation rates either coincide with the measured rates or agree with them within one or two decimal orders of magnitude.     

Statistics of the Nucleation Process under the Conditions of Gradual Creation of the Metastable State in Vapor

Statistics of the nucleation process at the gradual creation of the metastable state in vapor under the conditions when sufficiently large number of droplets arises in a vapor–gas system by the end of nucleation stage is studied on the assumption of randomness of the droplet nucleation. The probability distribution for a given number of droplets created in the unit volume of the vapor–gas system by the end of the nucleation stage is constructed. The distribution has the Gaussian form with the dispersion depending on the regime of matter exchange between droplets and vapor. The mean number of nucleated droplets under given conditions coincides (with an accuracy of up to the first-order small values with respect to the relative fluctuations of this value) with the predictions of the theory dealing with the average nucleation rate.     

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.     

Statistical Characteristics of the Process of Homogeneous Nucleation in the Vapor–Gas Medium during Free Molecular Regime of the Growth of Forming Droplets

Statistical approach to the study of the process of homogeneous nucleation of droplets in the vapor–gas medium in the presence of originally generated growing droplet at free molecular regime of droplet growth after the instantaneous creation of initial vapor supersaturation is proposed. The probability density of the creation of a new droplet in the vicinity of originally generated droplet is found. The mean distance between two neighboring droplets and the relative scatter of this distance are determined. The mean expectation time for the appearance of neighboring droplet estimating the duration of the droplet nucleation stage is found. The average number of droplets in a unit volume of the vapor–gas medium by the end of the droplet nucleation stage is estimated. The results obtained are compared with the predictions of the theory based on the assumption of the homogeneity of metastable phase.     

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.     

Stresses inside critical nuclei

The usual derivation of classical nucleation theory is inappropriate for crystal nucleation. In particular, it leads to a seriously flawed estimate of the pressure inside a critical nucleus. This has consequences for the prediction of possible metastable phases during the nucleation process. In this paper, we reanalyze the theory for crystal nucleation based on the thermodynamics of small crystals suspended in a liquid, due to Mullins (J. Chem. Phys. 1984, 81, 1436). As an illustration of the difference between the classical picture and the present approach, we consider a numerical study of crystal nucleation in binary mixtures of hard spherical colloids with a size ratio of 1:10. The stable crystal phase of this system can be either dense or expanded. We find that, in the vicinity of the solid-solid critical point where the crystallites are highly compressible, small crystal nuclei are less dense than large nuclei. This phenomenon cannot be accounted for by either classical nucleation theory or by the Gibbsian droplet model.     

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.     

Dynamics of the droplet size distribution function during the gradual creation of a supersaturation state under different regimes of droplet growth

The method of the direct numerical integration of kinetic equations of droplet size distribution functions that was previously proposed by the authors is employed to solve the problem of condensation relaxation in a vapor-gas mixture during the creation of a supersaturation state at a finite rate. Two relaxation regimes are considered. In the static regime, the mixture is expanded at a constant rate until a preset supersaturation ratio is achieved; in the dynamic regime, the expansion is continued. Solutions are obtained for argon-cesium and argon-ethane mixtures, thus making it possible to study the dependence of the process character on the Knudsen number. The effects of the rate of the supersaturation creation and the relaxation regime on the droplet size distribution function are analyzed.     

Size Distributions of Silver and Copper Crystals as a Function of Electrocrystallization Conditions on Glassy Carbon

Specific features and regularities of size distributions of crystalline nuclei are studied using the electrocrystallization of copper and silver on glassy carbon as an example. For a potentiostatic nucleation mode, with increasing overvoltage, a nonuniform size distribution approaches a uniform one. For a galvanostatic nucleation mode, the most probable nuclei size decreases with increasing current density and decreasing concentration of copper ions in solution, and the nuclei size distribution becomes more nonuniform due to a more pronounced non-steady-state nature of galvanostatic nucleation.     

Molecular dynamics investigation of the transient regime in the freezing of salt clusters

Molecular dynamics (MD) investigations of the freezing of supercooled liquids can identify nuclei far smaller than can be detected in laboratory experiments, to date, and consequently can provide information about nucleation so far inaccessible to experiment. In a recent MD study of the freezing of clusters of SeF6, a new method of recording nucleation events was introduced. It involved the observation of times of first appearance of nuclei of the size of n. An advantage of the new approach is that it provides information about the size of the critical nucleus. For nuclei smaller than the critical size, it also avoids the overshoots of nucleation rates that precluded the application of the Miloshev-Wu method in the subcritical region. Kinetic information in the transient regime can be characterized by three parameters, the time lag, the reduced moment, and the steady state nucleation rate. To get some idea of how general the new approach is, a very different system was investigated, that of clusters of NaCl. Two different fitting functions were used to analyze the results. The first one adopted the log-normal probability distribution of Wu. The second function was a modification of Shneidman's analytical solution appropriate for large nuclei. The second function gave a rather good account of MD data for all nuclear sizes and temperatures and gave more stable results in the subcritical region. Several inferences of the sizes of critical nuclei were made. Applying the criterion for n* based on the Zeldovich solution of the Becker-D?ring equations, we estimated the critical nucleus sizes to be 14, 18, and 24 ions for quench temperatures of 640, 690, and 740 K, respectively. Even though the interionic interactions initiating nucleation in salt are very different from the van der Waals interactions in clusters of SeF6, the characteristic aspects of the transient regimes of the two systems were quite similar.     

Power-law stage of slow relaxation in solutions with spherical micelles

General (independent of models selected for surfactant molecular aggregates) analytical relations are derived to describe the initial stage of slow relaxation in micellar solutions with spherical micelles. This stage precedes the final stage of the relaxation occurring via an exponential decay of disturbances with time. The relations obtained are applicable throughout the interval of micellar solution concentrations from the first to the second critical micellization concentration. It is shown that the initial stage is characterized by power laws of variations in the concentrations of monomers and micelles with time, these laws being different for the relaxation processes proceeding from above and below toward equilibrium values of micellar solution parameters. Relations are derived for the duration of this stage, and the effect of initial conditions is studied. Characteristic times of the power-law stage are determined and compared with the characteristic time of the final exponent-law relaxation stage. The behavior of these times is investigated at surfactant solution concentrations in the vicinity of, and noticeably above, the first critical micellization concentration. On the basis of the droplet and quasi-droplet thermodynamic models of surfactant molecular aggregates, numerical solutions are found for nonlinearized equations of slow relaxation for the time dependence of surfactant monomer concentrations at all stages of the slow relaxation. Numerical results obtained from the models are compared with the results of a general analytical study.     

Analysis of the Results of Numerical Simulation of the Supersaturated Vapor Condensation Relaxation

The process of bulk vapor condensation from the vapor-gas mixture after the fast creation of the supersaturated state as a result of vapor expansion is discussed. Scaling relations are derived for the time of condensation relaxation and droplet number density. The principal possibility of the experimental determination of nucleation rate is demonstrated based on the results obtained. The effect of initial phase, frequency, and pulsation amplitude on the condensation relaxation is analyzed for the case, when thermodynamic parameters of a mixture in the process of condensation are subjected to perturbations in the form of low-amplitude harmonic pulsations. The domain of applicability of results obtained on the scale of pulsation frequencies is determined.     

A real-time AFM study on crystal nucleation and growth in nanodroplets of isotactic polypropylene

Isotactic polypropylene (iPP) nanodroplets were prepared by using the classical droplet method in this study. The formation of nanodroplets allowed the controlled observation of polymer nucleation as well as access to crystal growth at exceptionally high supercooling in iPP. Three cases including the heterogeneous nucleation and fast crystallization in iPP droplets, the formation of multiple independent homogeneous nuclei within a single droplet and a single nucleus within a single droplet were detected by using atomic force microscopy (AFM) during gradually cooling after remelting the nanodroplets. Moreover, it is found that when the volume of droplet is larger than the value of ca. 130000 nm3, the first case was observed. Otherwise, the latter two cases appeared. The temperature at which the onset of nucleation was observed in individual droplets was found to be mainly dependent on height of the droplets when the size scale of the droplet is comparable to the size of the critical nucleus in at least one dimension, which indicates the nucleation behavior under confinement.     

Spinodal for the solution-to-crystal phase transformation

The formation of crystalline nuclei from solution has been shown for many systems to occur in two steps: the formation of quasidroplets of a disordered intermediate, followed by the nucleation of ordered crystalline embryos within these droplets. The rate of each step depends on a respective free-energy barrier and on the growth rate of its near-critical clusters. We address experimentally the relative significance of the free-energy barriers and the kinetic factors for the nucleation of crystals from solution using a model protein system. We show that crystal nucleation is 8-10 orders of magnitude slower than the nucleation of dense liquid droplets, i.e., the second step is rate determining. We show that at supersaturations of three or four k(B)T units, crystal nuclei of five, four, or three molecules transform into single-molecule nuclei, i.e., the significant nucleation barrier vanishes below the thermal energy of the molecules. We show that the main factor, which determines the rate of crystal nucleation, is the slow growth of the near-critical ordered clusters within the quasidroplets of the disordered intermediate. Analogous to the spinodal in supersaturated fluids, we define a solution-to-crystal spinodal from the transition to single-molecule crystalline nuclei. We show that heterogeneous nucleation centers accelerate nucleation not only because of the wettinglike effects that lower the nucleation barrier, as envisioned by classical theory, but by helping the kinetics of growth of the ordered crystalline embryos.     

Miniemulsion polymerization of styrene using alkyl methacrylates as the reactive cosurfactant

 Stable styrene miniemulsions were prepared by using alkyl methacrylates as the reactive cosurfactant. Like conventional cosurfactants (e.g., cetyl alcohol (CA) and hexadecane (HD)), alkyl methacrylates (e.g., dodecyl methacrylate (DMA) and stearyl methacrylate (SMA)) may act as a cosurfactant in stabilizing the homogenized miniemulsions. Furthermore, the methacrylate group may be chemically incorporated into latex particles in subsequent miniemulsion polymerization. The data of the monomer droplet size, creaming rate and phase separation of monomer as a function of time were used to evaluate the shelf-life of miniemulsions stabilized by sodium dodecyl sulfate in combination with various cosurfactants. Polystyrene latex particles were produced via both monomer droplet nucleation and homogeneous nucleation in the miniemulsion polymerization using CA or DMA as the cosurfactant, with the result of a quite broad particle size distribution. On the other hand, the miniemulsion polymerization with HD or SMA showed a predominant monomer droplet nucleation. The resultant particle size distribution was relatively narrow. In miniemulsion polymerization, the less hydrophobic DMA is similar to CA, whereas the more hydrophobic SMA is similar to HD. Received: 19 November 1996 Accepted: 20 February 1997     

AFM studies of metal deposition: instantaneous nucleation and the growth of cobalt nanoparticles on boron-doped diamond electrodes.

In situ atomic force microscopy (AFM) is used to study the growth of cobalt nuclei on a boron doped diamond electrode under potentiostatic control. The rate of growth of the nuclei at the electrode surface is monitored using AFM as a function of time at different deposition potentials. The nucleation of cobalt nuclei is found to be "instantaneous" and the growth of the nuclei is shown to be kinetically rather than diffusionally controlled over periods of tens and hundreds of seconds. At very short times (<10 seconds) the kinetics of nucleation are apparent.