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).     

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.     

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.     

The role of temperature in nucleation processes

Heat and mass transfers are coupled processes, also in nucleation. In principle, a nucleating cluster would have a different temperature compared to the surrounding supersaturated old phase because of the heat release involved with attaching molecules to the cluster. In turn a difference in temperature across the cluster surface is a driving force for the mass transfer to and from the cluster. This coupling of forces in nonisothermal nucleation is described using mesoscopic nonequilibrium thermodynamics, emphasizing measurable heat effects. An expression was obtained for the nonisothermal nucleation rate in a one-component system, in the case where a temperature difference exists between a cluster distribution and the condensed phase. The temperature is chosen as a function of the cluster size only, while the temperature of the condensed phase is held constant by a bath. The generally accepted expression for isothermal stationary nucleation is contained as a limiting case of the nonisothermal stationary nucleation rate. The equations for heat and mass transport were solved for stationary nucleation with the given cluster distribution and with the temperature controlled at the boundaries. A factor was defined for these conditions, determined by the heat conductivity of the surrounding phase and the phase transition enthalpy, to predict the deviation between isothermal and nonisothermal nucleation. For the stationary state described, the factor appears to give small deviations, even for primary nucleation of droplets in vapor, making the nonisothermal rate smaller than the isothermal one. The set of equations may lead to larger and different thermal effects under different boundary conditions, however.     

Influence of Vapor–Gas Medium Heating due to the Condensation Heat on the Stage of Multicomponent Droplet Nucleation

Colloid Journal - The article describes the effect of phase transition heat on the temperature of a closed multicomponent vapor–gas metastable phase and on growing supercritical droplets and...     

Nucleation on active centers in confined volumes

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

Nucleation in hydrophobic cylindrical pores: a lattice model

We consider the nucleation process associated with capillary condensation of a vapor in a hydrophobic cylindrical pore (capillary evaporation). The liquid-vapor transition is described within the framework of a simple lattice model. The phase properties are characterized both at the mean-field level and with Monte Carlo simulations. The nucleation process for the liquid to vapor transition is then specifically considered. Using umbrella sampling techniques, we show that nucleation occurs through the condensation of an asymmetric vapor bubble at the pore surface. Even for highly confined systems, good agreement is found with macroscopic considerations based on classical nucleation theory. The results are discussed in the context of recent experimental work on the extrusion of water in hydrophobic pores.     

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.     

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.     

Chemically assisted release of transition metals in graphite vaporizers for atomic spectrometry

The processes associated with the vaporization of microgram samples and modifiers in a graphite tube ET AAS were investigated by the example of transition metals. The vapor absorption spectra and vaporization behavior of μg-amounts Cd, Zn, Cu, Ag, Au, Ni, Co, Fe, Mn and Cr were studied using the UV spectrometer with CCD detector, coupled with a continuum radiation source. The pyrocoated, Ta or W lined tubes, with Ar or He as internal gases, and filter furnace were employed in the comparative experiments. It was found that the kinetics of atomic vapor release changed depending on the specific metal–substrate–gas combination; fast vaporization at the beginning was followed by slower ‘tailing.’ The absorption continuum, overlapped by black body radiation at longer wavelengths, accompanied the fast vaporization mode for all metals, except Cd and Zn. The highest intensity of the continuum was observed in the pyrocoated tube with Ar. For Cu and Ag the molecular bands overlapped the absorption continuum; the continuum and bands were suppressed in the filter furnace. It is concluded that the exothermal interaction of sample vapor with the material of the tube causes the energy evolution in the gas phase. The emitted heat is dispersed near the tube wall in the protective gas and partially transferred back to the surface of the sample, thus facilitating the vaporization. The increased vapor flow causes over-saturation and gas-phase condensation in the absorption volume at some distance from the wall, where the gas temperature is not affected by the reaction. The condensation is accompanied by the release of phase transition energy via black body radiation and atomic emission. The particles of condensate and molecular clusters cause the scattering of light and molecular absorption; slow decomposition of the products of the sample vapor–substrate reaction produces the ‘tailing’ of atomic absorption signal. The interaction of graphite with metal vapor or oxygen, formed in the decomposition of metal oxide, is the most probable source of chemical energy, which facilitates the vaporization. Intensity of the process depends on chemical properties of the sample and substrate and efficiency of mass and heat transfer by the protective gas. The discussed mechanism of chemically assisted vapor release signifies the energy exchange between all participants of the vaporization process in ET AAS including the matrix, modifier, purge gas and analyte. The finding contributes in the ET AAS theory regarding the mechanisms of vaporization and mass transfer in the presence of matrix and modifiers.     

Long-time growth kinetics of first order phase transitions in the presence of a boundary layer

The late stage growth mechanism for a first order phase transition, either through nucleation growth or spinodal decomposition, is well understood to be an Ostwald ripening or coarsening process, in which larger domains grow at the expense of smaller ones. The growth kinetics in this regime was shown by Lifshitz and Slyozov to follow at(1/3) law. However, the kinetics is altered if there exists a barrier ahead of the growth front, irrespective of the physical origin of the boundary layer. We present an analytic calculation for the growth kinetics in the presence of a boundary layer, showing that in the limit of barrier-dominated growth, the domains grow with at(1/2) law. This result holds true in the dilute regime independent of whether the growing nuclei are spherical or cylindrical.     

A new methodology for the simulation of solid state phase transition kinetics by combination of nucleation and nuclei growth processes

A new methodology for the simulation of solid state phase transition kinetics has been developed by combining the influence of nucleation rate, nuclei growth rate and the power p characterizing the contact area between the growing particles. The equations used in this methodology were well known, and have been used previously for creating some of the most popular solid-state kinetic equations. The developed methodology made possible calculations of separate rate constants for two processes affecting the rate of phase transition—nucleation (described with K ₁) and nuclei growth (described with K ₂). Similar phase transitions were also approximated with the well-known single constant Avrami–Erofeev equation, but we successfully calculated both constants according to the new methodology, which allowed a separate evaluation of these two processes and explained the different induction periods. The effects of empirically adjusted constants on theoretically calculated kinetic curves were thus determined.     

Growth and motion of heterogeneous water droplets in laminar diffusion chambers: 2. High nanoparticle concentrations

The heterogeneous condensation of water vapor on nanoparticles and the growth of formed droplets are numerically studied under conditions of laminar diffusion chamber (LDC) of a new type with a hot porous wall. The main attention is focused on the growth of heterogeneous droplets in the gas flow at high number densities of nanoparticles exceeding 10⁸ droplet/m³. Under these conditions, there is an interrelation between the growth of droplets and processes of heat and mass transfer in LDC due to vapor depletion and the release of latent heat of phase transition on growing droplets. The efficiency of the coverage of nanoparticles with water film is studied under LDC nonuniform conditions. It is shown that, at initial number densities of nanoparticles (N _d > 10¹¹ droplet/m³), heterogeneous droplets do not grow to optically detected sizes.     

Competing evaporation and condensation processes during the boiling of methane

The atomistic mechanism of the boiling of methane is explored from molecular dynamics simulations. The liquid --> vapor transition is initiated by local density fluctuations resulting in a nanometer-sized domain that exhibits both liquid and vapor characteristics. Though the rates of evaporation and condensation events increase dramatically in this area, the overall balance exhibits only a marginal net rate of evaporation. Growth of the precritical domain leads to the nucleation of a vapor phase in which isolated methane molecules are confined by a liquid-vapor interface. After crossing the transition state, the system experiences progressive destabilization of the liquid phase and the evaporation processes clearly outnumber the condensation events.     

Kinetics of nucleation in highly polar substances in the presence of ions

Based on of a previously proposed model for describing the effect of the electric field of the ion on the vapor-liquid phase transition, expressions for the work of formation of the critical nucleus in highly polar substances were derived. Analytical expressions for the nucleation frequency were obtained and used to calculate the heterogeneous nucleation frequency in a supersaturated water vapor.     

Kinetics of supersaturated vapor nucleation on molecular condensation nuclei

The kinetics of nucleation is calculated for a supersaturated vapor containing molecular condensation nuclei, that is, foreign molecules able to induce the formation of viable nuclei of a condensed phase by themselves. In contrast to the previous calculation, the possibility of the escape of molecular condensation nuclei from very small clusters containing a few condensed vapor molecules is taken into account. More exact equations are derived for the rate of steady-state nucleation and the concentration of aerosol particles in a quasisteady-state regime of nucleation. The calculation demonstrates that, at a high probability of the escape of a molecular condensation nucleus, the predominating mechanism of cluster formation is the attachment of a molecular condensation nucleus to a cluster formed from vapor molecules rather than their condensation on the nucleus. At the same time, allowances for the possible escape of molecular condensation nuclei from clusters slightly affect the rate of nucleation and the concentration of aerosol particles being formed.     

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.     

Computational modelling of nematic phase ordering by film and droplet growth over heterogeneous substrates

This paper presents a computational study of defect nucleation associated with the kinetics of the isotropic-to-nematic phase ordering transition over heterogeneous substrates, as it occurs in new liquid crystal biosensor devices, based on the Landau-de Gennes model for rod-like thermotropic nematic liquid crystals. Two regimes are identified due to interfacial tension inequalities: (i) nematic surface film nucleation and growth normal to the heterogeneous substrate, and (ii) nematic surface droplet nucleation and growth. The former, known as wetting regime, leads to interfacial defect shedding at the moving nematic-isotropic interface. The latter droplet regime, involves a moving contact line, and exhibits two texturing mechanisms that also lead to interfacial defect shedding: (a) small and large contact angles of drops spreading over a heterogeneous substrate, and (b) small drops with large curvature growing over homogeneous patches of the substrate. The numerical results are consistent with qualitative defect nucleation models based on the kinematics of the isotropic-nematic interface and the substrate-nematic-isotropic contact line. The results extend current understanding of phase ordering over heterogeneous substrates by elucidating generic defect nucleation processes at moving interfaces and moving contact lines.