New approach to the kinetics of heterogeneous unary nucleation on liquid aerosols of a binary solution

The formation of a droplet on a hygroscopic center may occur either in a barrierless way via Kohler activation or via nucleation by overcoming a free energy barrier. Unlike the former, the latter mechanism of this process has been studied very little and only in the framework of the classical nucleation theory based on the capillarity approximation whereby a nucleating droplet behaves like a bulk liquid. In this paper the authors apply another approach to the kinetics of heterogeneous nucleation on liquid binary aerosols, based on a first passage time analysis which avoids the concept of surface tension for tiny droplets involved in nucleation. Liquid aerosols of a binary solution containing a nonvolatile solute are considered. In addition to modeling aerosols formed through the deliquescence of solid soluble particles, the considered aerosols constitute a rough model of "processed" marine aerosols. The theoretical results are illustrated by numerical calculations for the condensation of water vapor on binary aqueous aerosols with nonvolatile nondissociating solute molecules using Lennard-Jones potentials for the molecular interactions.     

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.     

Homogeneous nucleation at high supersaturation and heterogeneous nucleation on microscopic wettable particles: A hybrid thermodynamic/density-functional theory

Homogeneous nucleation at high supersaturation of vapor and heterogeneous nucleation on microscopic wettable particles are studied on the basis of Lennard-Jones model system. A hybrid classical thermodynamics and density-functional theory (DFT) approach is undertaken to treat the nucleation problems. Local-density approximation and weighted-density approximation are employed within the framework of DFT. Special attention is given to the disjoining pressure of small liquid droplets, which is dependent on the thickness of wetting film and radius of the wettable particle. Different contributions to the disjoining pressure are examined using both analytical estimations and numerical DFT calculation. It is shown that van der Waals interaction results in negative contribution to the disjoining pressure. The presence of wettable particles results in positive contribution to the disjoining pressure, which plays the key role in the heterogeneous nucleation. Several definitions of the surface tension of liquid droplets are discussed. Curvature dependence of the surface tension of small liquid droplets is computed. The important characteristics of nucleation, including the formation free energy of the droplet and nucleation barrier height, are obtained.     

Multicomponent nucleation: thermodynamically consistent description of the nucleation work

A thermodynamically consistent formula is derived for the nucleation work in multicomponent homogeneous nucleation. The derivation relies on the conservative dividing surface which defines the nucleus as having specific surface energy equal to the specific surface energy sigma0 of the interface between the macroscopically large new and old phases at coexistence. Expressions are given for the radius of the nucleus defined by the conservative dividing surface and by the surface of tension. As a side result, the curvature dependence of the surface tension sigmaT of the nucleus defined by the surface of tension is also determined. The analysis is valid for nuclei of any size, i.e., for nucleation in the whole range of conditions between the binodal and the spinodal of the metastable old phase provided the inequality sigmaT < or = sigma0 is satisfied. It is found that under the conditions of validity of the analysis the nucleation rate is higher than the nucleation rate given by the classical nucleation theory. The general results are applied to nucleation of unary liquids or solids in binary gaseous, liquid or solid mixtures.     

Molecular dynamics study of structural and thermodynamic characteristics of nanodroplets of simple fluid

Equilibrium configurations of Lennard-Jones nanodroplets composed of 10–15000 spherically symmetric molecules placed in the center of a spherical container are studied at constant temperature by the molecular dynamics method. The distribution of local density is found and size dependences of density in the center of droplet, first coordination number, and energy surface tension coinciding for equimolecular dividing surface with specific excess free energy of droplet are studied. Radial distribution function is also determined. It is established that the passage of structural characteristics to their macroscopic values is observed for droplets containing as little as about 300 molecules, while, for energy surface tension, analogous passage for energy surface tension occurs for droplets containing 700–6000 molecules.     

Monte Carlo simulation study of droplet nucleation

A new rigorous Monte Carlo simulation approach is employed to study nucleation barriers for droplets in Lennard-Jones fluid. Using the gauge cell method we generate the excess isotherm of critical clusters in the size range from two to six molecular diameters. The ghost field method is employed to compute the cluster free energy and the nucleation barrier with desired precision of (1-2)kT. Based on quantitative results obtained by Monte Carlo simulations, we access the limits of applicability of the capillarity approximation of the classical nucleation theory and the Tolman equation. We show that the capillarity approximation corrected for vapor nonideality and liquid compressibility provides a reasonable assessment for the size of critical clusters in Lennard-Jones fluid; however, its accuracy is not sufficient to predict the nucleation barriers for making practical estimates of the rate of nucleation. The established dependence of the droplet surface tension on the droplet size cannot be approximated by the Tolman equation for small droplets of radius less than four molecular diameters. We confirm the conclusion of ten Wolde and Frenkel [J. Chem. Phys. 109, 9901 (1998)] that integration of the normal component of the Irving-Kirkwood pressure tensor severely underestimates the nucleation barriers for small clusters.     

Thermodynamics of Nucleation on the Particles of Salts-Strong Electrolytes: The Allowance for Ion Adsorption in the Droplet Surface Layer

The contributions dependent on ionic specificity (including those related to the differences in polarizabilities of cations and anions) to the surface tension and ion adsorption on the boundary between an aqueous strong electrolyte solution and the vapor-gas phase are taken into account. The role of these contributions in the thermodynamics of vapor condensation on salt particles completely dissolved in droplets that are emerged on these particles from the vapor is studied. Consistent domains of the applicability of the analytical theory suggesting the complete dissolution of a salt particle and the dissociation of substance comprising this particle into ions, as well as the ideal behavior of solution in a droplet and the linearization with respect to excess electric potentials near the droplet surface, are established in the approximation of a quasi-planar interface. Formulas are derived for the threshold values of the chemical potential of vapor molecules upon the barrierless condensation and for critical vapor supersaturations upon the barrier condensation on salt particles. In the explicit form, these formulas express the dependence of these values on the initial size of salt particles, physicochemical parameters of solution in droplets, and the charge of formed ions. Calculations for water condensation on NaCl, Na₂SO₄, and MgCl₂ particles are performed using these formulas.     

Thermodynamics of heterogeneous crystal nucleation in contact and immersion modes

One of the most intriguing problems of heterogeneous crystal nucleation in droplets is its strong enhancement in the contact mode (when the foreign particle is presumably in some kind of contact with the droplet surface) compared to the immersion mode (particle immersed in the droplet). Heterogeneous centers can have different nucleation thresholds when they act in contact or immersion modes. The underlying physical reasons for this enhancement have remained largely unclear. In this paper we present a model for the thermodynamic enhancement of heterogeneous crystal nucleation in the contact mode compared to the immersion one. To determine if and how the surface of a liquid droplet can thermodynamically stimulate its heterogeneous crystallization, we examine crystal nucleation in the immersion and contact modes by deriving and comparing with each other the reversible works of formation of crystal nuclei in these cases. The line tension of a three-phase contact gives rise to additional terms in the formation free energy of a crystal cluster and affects its Wulff (equilibrium) shape. As an illustration, the proposed model is applied to the heterogeneous nucleation of hexagonal ice crystals on generic macroscopic foreign particles in water droplets at T = 253 K. Our results show that the droplet surface does thermodynamically favor the contact mode over the immersion one. Surprisingly, the numerical evaluations suggest that the line tension contribution (from the contact of three water phases (vapor-liquid-crystal)) to this enhancement may be of the same order of magnitude as or even larger than the surface tension contribution.     

Model description of contact angles in electrowetting on dielectric layers

The electrowetting on dielectric (EWOD) technique has considerable potential for microfluidic and biomedical applications. The Lippmann-Young model based on the force balance concept has long been used to predict the contact angles of droplets under electrowetting. However, recent experimental evidence has indicated that this model fails to provide accurate predictions of the lower contact angles associated with saturation conditions at higher electric potentials. Hence, the study simulates the internal flow in an actuated droplet and treats it as stagnation-point flow. This kinetic energy is then taken into consideration while calculating the contact angles using an energy balance model. The energy of an actuated droplet is contributed by the combination of the side surface tension energy, the base tension energy, the dielectric energy, and the kinetic energy when deriving the energy balance model. Consequently, the new energy balance model modifies the Lippmann-Young equation, thereby providing enhanced reasonable predictions of the droplet contact angle across the higher electric potential where the contact angles are close to the saturated condition.     

Chemistry of Interfaces,M. J. Jaycock and G. D. Parfitt. Halstead Press,John Wiley and Sons,New York, 1981. 279 pp. $90. Paul Becher,Wilmington

A unique physical model is proposed for relating the dimensions and properties of droplets in aqueous diesel fuel invert mlcroemulsions to the measured water vapor pressures over such systems. The model assumes discrete droplets containing surfactant-sheathed liquid cores. A dynamic equilibrium condition is visualized wherein a closed mass transfer cycle e3tists, involving the movement of water molecules from the droplet interior, through the surfactant sheath into the continuous medium and vapor space above the pool. The flat-surface fugacity of the liquid water in the aqueous core would be reduced relative to that of normal water because of Increased intermolecular association stemming from high pressure in the aqueous core caused by surface tension forces. The possible presence of dissolved surfactant constituents would reduce this fugacity even further. The mass transfer cycle is assumed to be completed by the absorption of water vapor into transitory, flat surfaces of reduced fugacity, droplet core water exposed by collapsing droplets at the pool surface. These are assumed to be continually reforming into submerged microemulsion droplets as additional droplets collapse at the pool surface.

Analytical relationships based upon the described model allowed calculation of droplet core and sheath dimensions and droplet external interfacial tension. The efficacy of the proposed model is supported by the congruity of the thus derived values.     

Kinetic theory of binary nucleation based on a first passage time analysis

The binary classical nucleation theory (BCNT) is based on the Gibbsian thermodynamics and applies the macroscopic concept of surface tension to nanosize clusters. This leads to severe inconsistencies and large discrepancies between theoretical predictions and experimental results regarding the nucleation rate. We present an alternative approach to the kinetics of binary nucleation which avoids the use of classical thermodynamics for clusters. The new approach is an extension to binary mixtures of the kinetic theory previously developed by Narsimhan and Ruckenstein and Ruckenstein and Nowakowski [J. Colloid Interface Sci. 128, 549 (1989); 137, 583 (1990)] for unary nucleation which is based on molecular interactions and in which the rate of emission of molecules from a cluster is determined via a mean first passage time analysis. This time is calculated by solving the single-molecule master equation for the probability distribution of a "surface" molecule moving in a potential field created by the cluster. The starting master equation is a Fokker-Planck equation for the probability distribution of a surface molecule with respect to its phase coordinates. Owing to the hierarchy of characteristic time scales in the evolution of the molecule, this equation can be reduced to the Smoluchowski equation for the distribution function involving only the spatial coordinates. The new theory is combined with density functional theory methods to determine the density profiles. This is essential for nucleation in binary systems particularly when one of the components is surface active. Knowing these profiles, one can determine the potential fields created by the cluster, its rate of emission of molecules, and the nucleation rate more accurately than by using the uniform density approximation. The new theory is illustrated by numerical calculations for a model binary mixture of Lennard-Jones monomers and rigidly bonded dimers of Lennard-Jones atoms. The amphiphilic character of the dimer component (i.e., its surface activity) is induced by the asymmetry in the interaction between a monomer and the two different sites of a dimer. The inconsistencies of the BCNT are avoided in the new theory.     

Surface tension of different sized single-component droplets,according to macroscopic data obtained using the lattice gas model and the critical droplet size during phase formation

Size dependences of the surface tension of spherical single-component droplets are calculated using equations of the lattice gas model for 19 compounds. Parameters of the model are found from experimental data on the surface tension of these compounds for a macroscopic planar surface. The chosen low-molecular compounds satisfy the law of corresponding states. To improve agreement with the experimental data, Lennard-Jones potential parameters are varied within 10% deviations. The surface tensions of different sized equilibrium droplets are calculated at elevated and lowered temperatures. It is found that the surface tension of droplets grows monotonically as the droplet size increases from zero to its bulk value. The droplet size R ₀ corresponding to zero surface tension corresponds to the critical size of the emergence of a new phase. The critical droplet sizes in the new phase of the considered compounds are estimated for the first time.     

Sn-Pb合金颗粒异质成核及其冷却凝固行为预测

The Sn-5%Pb droplets with sizes of 150 and 185 μm were generated by Uniform Droplet Spray（UDS） under N2 atmosphere with 1. 36 μmol / L oxygen. The appearance of Sn-5%Pb droplets under the optical microscope showed that the droplets are uniform and spherical. The method employed non-adiabatic calorimetry to determine the nucleation point and undercooling of droplets. The fraction covered by oxide was calculated as a function of time and temperature. The model for heterogeneous nucleation catalyzed by oxidation on the droplet surface was developed,which was a reasonable expression of the heterogeneous nucleation and solidification behavior of Sn-Pb droplets. The CCT curves were computed using above experimental results under the heterogeneous surface nucleation of droplet. The same model can be applied to predict the heterogeneous nucleation behavior of the droplets for any type of cooling schedule. The calculation results are very reliable based on the experimental data.     

Effects of lateral diffusion on morphology and dynamics of a microscopic lattice-gas model of pulsed electrodeposition

The influence of nearest-neighbor diffusion on the decay of a metastable low-coverage phase (monolayer adsorption) in a square lattice-gas model of electrochemical metal deposition is investigated by kinetic Monte Carlo simulations. The phase-transformation dynamics are compared to the well-established Kolmogorov-Johnson-Mehl-Avrami theory. The phase transformation is accelerated by diffusion, but remains in accord with the theory for continuous nucleation up to moderate diffusion rates. At very high diffusion rates the phase-transformation kinetic shows a crossover to instantaneous nucleation. Then, the probability of medium-sized clusters is reduced in favor of large clusters. Upon reversal of the supersaturation, the adsorbate desorbs, but large clusters still tend to grow during the initial stages of desorption. Calculation of the free energy of subcritical clusters by enumeration of lattice animals yields a quasiequilibrium distribution which is in reasonable agreement with the simulation results. This is an improvement relative to classical droplet theory, which fails to describe the distributions, since the macroscopic surface tension is a bad approximation for small clusters.     

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.     

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.     

Ion-induced nucleation in solution: promotion of solute nucleation in charged levitated droplets

We have investigated the nucleation and growth of sodium chloride in both single quiescent charged droplets and charged droplet populations that were levitated in an electrodynamic levitation trap (EDLT). In both cases, the magnitude of a droplet's net excess charge (ions(DNEC)) influenced NaCl nucleation and growth, albeit in different capacities. We have termed the phenomenon ion-induced nucleation in solution. For single quiescent levitated droplets, an increase in ions(DNEC) resulted in a significant promotion of NaCl nucleation, as determined by the number of crystals observed. For levitated droplet populations, a change in NaCl crystal habit, from regular cubic shapes to dome-shaped dendrites, was observed once a surface charge density threshold of -9 x 10(-4) e.nm(-2) was surpassed. Although promotion of NaCl nucleation was observed for droplet population experiments, this can be attributed in part to the increased rate of solvent evaporation observed for levitated droplet populations having a high net charge. Promotion of nucleation was also observed for two organic acids, 2,4,6-trihydroxyacetophenone monohydrate (THAP) and alpha-cyano-4-hydroxycinnamic acid (CHCA). These results are of direct relevance to processes that occur in both soft-ionization techniques for mass spectrometry and to a variety of industrial processes. To this end, we have demonstrated the use of ion-induced nucleation in solution to form ammonium nitrate particles from levitated droplets to be used in in vitro toxicology studies of ambient particle types.     

Key thermodynamic characteristics of nucleation on charged and neutral cores of molecular sizes in terms of the gradient density functional theory

The gradient density functional theory and the Carnahan–Starling model formulated for describing the contribution of hard spheres have been used to calculate the profiles of condensate density in small critical droplets formed via homogeneous nucleation, as well as in stable and critical droplets formed via heterogeneous nucleation on solid charged and neutral condensation cores of molecular sizes. The calculations performed for water and argon at different values of condensate chemical potential have yielded the heights of the activation barriers for homoand heterogeneous nucleation as functions of vapor supersaturation at preset system temperatures. The interaction of condensate molecules with a solid core has been described by the resultant potential of molecular attractive forces. In the case of a charged core, the long-range Coulomb potential of electric forces has additionally been taken into account. Dielectric permittivities have been calculated as known functions of the local density of the fluid and temperature. The radius of the equimolecular droplet surface has been chosen as a variable describing the droplet size. Dependences of the chemical potential of condensate molecules in a droplet on its size have been plotted for water and argon with allowance for the action of capillary, electrostatic, and molecular forces. It has been shown that the role of the molecular force potential in heterogeneous nucleation increases with the size of condensation cores.     

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.