A study of sulfur homogeneous nucleation from supersaturated vapor. Determination of surface tension of sulfur nanoparticles

Homogeneous nucleation in sulfur vapor is studied in a laminar-flow chamber. Concentration and size distribution of resulting aerosol particles are measured with a diffusion spectrometer of aerosols and a PK.GTA-0,3-002 photoelectric particle counter. The crystal structure of the formed particles is studied by X-ray diffraction analysis. The rate of sulfur evaporation from a boat and the profile of a deposit on the chamber wall along the axial coordinate are determined by gravimetry. Axial and radial temperature profiles are measured using a chromel-alumel thermocouple. The vapor concentration distribution in the chamber is found and the supersaturation is calculated from the solution of the mass-transfer problem. An experimental low-laborious method is developed for the supersaturation cutoff. This method enables one to rapidly deter-mine the position of the zone in which the nucleation proceeds at the highest rate. The position of the zone of nucleation found by this method is in good agreement with the results of calculations based on experimental data and theoretical calculation of the rate of nucleation by an exact formula that has been recently derived based on the works by Kusaka and Reiss, as well as the Frenkel liquid kinetics theory. The surface tension of critical sulfur nuclei resulting from the nucleation is calculated based on this formula and experimental data on the nucleation. It is established that, in a temperature range of 312–319 K, the critical nuclei have tension surface radius R _s ～ 10.6 Å and surface tension σ = 72.5 ± 1.1 dyn/cm. The surface tension of critical sulfur nuclei in this temperature range is constant and approximately 5% higher than that of a planar surface.     

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.     

Evaluation of surface tension and Tolman length as a function of droplet radius from experimental nucleation rate and supersaturation ratio: metal vapor homogeneous nucleation

Zinc and silver vapor homogeneous nucleations are studied experimentally at the temperature from 600 to 725 and 870 K, respectively, in a laminar flow diffusion chamber with Ar as a carrier gas at atmospheric pressure. The size, shape, and concentration of aerosol particles outcoming the diffusion chamber are analyzed by a transmission electron microscope and an automatic diffusion battery. The wall deposit is studied by a scanning electron microscope (SEM). Using SEM data the nucleation rate for both Zn and Ag is estimated as 10(10) cm(-3) s(-1). The dependence of critical supersaturation on temperature for Zn and Ag measured in this paper as well as Li, Na, Cs, Ag, Mg, and Hg measured elsewhere is analyzed. To this aim the classical nucleation theory is extended by the dependence of surface tension on the nucleus radius. The preexponent in the formula for the vapor nucleation rate is derived using the formula for the work of formation of noncritical embryo [obtained by Nishioka and Kusaka [J. Chem. Phys. 96, 5370 (1992)] and later by Debenedetti and Reiss [J. Chem. Phys. 108, 5498 (1998)]] and Reiss replacement factor. Using this preexponent and the Gibbs formula for the work of formation of critical nucleus the dependence of surface tension on the radius R(S) of the surface of tension is evaluated from the nucleation data for above-mentioned metals. For the alkali metals and Ag the surface tension was determined to be a strong function of R(S). For the bivalent metals (Zn, Hg, and Mg) the surface tension was independent of radius in the experimental range. A new formula for the Tolman length delta as a function of surface tension and radius R(S) is derived by integration of Gibbs-Tolman-Koenig equation assuming that delta is a monotonic function of radius. The formula derived is more correct than the Tolman formula and convenient for the elaboration of experimental data. Using this formula the values of delta are determined as a function of R(S) from the experimental nucleation data. It is determined that all the metals considered are characterized by strong dependence of delta on radius; for the bivalent metals delta changes sign.     

A study of homogeneous nucleation of ibuprofen in a flow chamber. Determination of the surface tension of critical nuclei

Homogeneous nucleation of ibuprofen vapor is studied in a nucleation flow chamber, a horizontal quartz tube equipped with an external heater. The area of the chamber where the nucleation proceeds most efficiently is determined, and the volume of this area is estimated. The temperature and supersaturation are determined and the homogeneous nucleation rate is calculated for this area. Saturation vapor pressure over liquid ibuprofen is measured in a temperature range of 353–383 K. Using an exact formula that has recently been derived for the nucleation rate based on the works by Kusaka, Reiss, as well as the Frenkel liquid-kinetics theory, surface tension and the radius of surface of tension of a critical nucleus σ= 25.9 mN/m and R _s = 1.6 nm, respectively, are calculated at 318 K. The measurement of the surface tension of an ibuprofen planar surface shows that, at 318 K, σ_∞ = 24.38 mN/m; i.e., σ is higher than σ_∞ by 6%. A critical nucleus is established as containing nearly 36 ibuprofen molecules.     

Surface tension of water droplets upon homogeneous droplet nucleation in water vapor

A method has been proposed for determining interfacial free energy from the data of molecular dynamics simulation. The method is based on the thermodynamic integration procedure and is distinguished by applicability to both planar interfaces and those characterized by a high curvature. The workability of the method has been demonstrated by the example of determining the surface tension for critical nuclei of water droplets upon condensation of water vapor. The calculation has been performed at temperatures of 273–373 K and a pressure of 1 atm, thus making it possible to determine the temperature dependence of the surface tension for water droplets and compare the results obtained with experimental data and the simulation results for a “planar” vapor–liquid interface.     

Molecular-dynamics simulation of argon nucleation from supersaturated vapor in the NVE ensemble

The possibility to conduct simulations of homogeneous nucleation of argon from a supersaturated vapor phase using a microcanonical or NVE ensemble is evaluated (NVE: number of particles N, volume V, and energy E are constant). In order to initiate a phase separation kinetic energy is removed from the system in one step which transfers the system into a supersaturated state. After this temperature jump the simulation is continued in a NVE ensemble. The simulations are performed for different initial-state points and different temperature jumps. The cluster formation and growth over the course of the adiabatic simulations are analyzed. The progression of the temperature being related to the cluster size in NVE systems is traced. Also the influence of the size of the simulation system is investigated. For a certain range of low supersaturation a dynamic coexistence between two states has been found. Furthermore, the obtained nucleation rates are correlated with two simple functions. By applying the nucleation theorems to these functions the size and excess energy of the critical cluster are estimated. The results are consistent with other theoretical data and experimental data available in the literature.     

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.     

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.     

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.     

Accounting for fluctuations in cluster parameters upon the description of nucleation in supersaturated vapor

A theory is proposed for stationary homogeneous nucleation in supersaturated vapor in which a modified expression for the rate of cluster evaporation was used to calculate the equilibrium distribution over the nucleus sizes and the rates of their formation. This rate was determined by the extrapolation to the region of small sizes of the corresponding expression for the macroscopic droplet derived according to thermodynamic notions that take fluctuations into account. Modified dependences of the size of critical nucleus and the rate of nucleation on the supersaturation and the temperature are determined and compared with the data of the classical theory of nucleation and experimental results.     

Quasihomogeneous nucleation of amyloid beta yields numerical bounds for the critical radius, the surface tension, and the free energy barrier for nucleus formation

Amyloid aggregates are believed to grow through a nucleation mediated pathway, but important aggregation parameters, such as the nucleation radius, the surface tension of the aggregate, and the free energy barrier toward aggregation, have remained difficult to measure. Homogeneous nucleation theory, if applicable, can directly relate these parameters to measurable quantities. We employ fluorescence correlation spectroscopy to measure the particle size distribution in an aggregating solution of Alzheimer's amyloid beta molecule (Abeta(1-40)) and analyze the data from a homogeneous nucleation theory perspective. We observe a reproducible saturation concentration and a critical dependence of various aspects of the aggregation process on this saturation concentration, which supports the applicability of the nucleation theory to Abeta aggregation. The measured size distributions show a valley between two peaks ranging from 5 to 50 nm, which defines a boundary for the value of the nucleation radius. By carefully controlling the conditions to inhibit heterogeneous nucleation, we can hold off nucleation in a 25 times supersaturated solution for at least up to 3 h at room temperature. This quasi-homogeneous kinetics implies that at room temperature, the surface energy of the Abeta/water interface is > or =4.8 mJ/m(2), the free energy barrier to nucleation (at 25 times supersaturation) is > or =1.93x10(-19) J, and the number of monomers in the nucleus is > or =29.     

Study of sulfur heterogeneous nucleation from supersaturated vapor on tungsten oxide and sodium chloride seed particles. Determination of contact angle of critical sulfur nuclei

A procedure has been developed for determining the contact angle of a critical nucleus formed on seed particles during the heterogeneous nucleation of a vapor in a flow chamber. The procedure comprises the determination of the fraction of enlarged particles, as well as the selective separation of nanoparticles over sizes to locate the zone of intense nucleation. The concentration and size distribution of aerosol particles have been measured with a diffusion spectrometer of aerosols. Vapor concentration distributions and supersaturation fields have been determined by solving the mass-transfer problem. The calculated supersaturation fields are in good agreement with the location of the intense nucleation zone experimentally found with the help of selective separation. The fractions of enlarged particles have been determined as functions of supersaturation in the chamber. A formula has been derived for calculating the fraction and size distribution function of enlarged particles at known supersaturation and temperature fields and a preset contact angle. The contact angles are selected in a manner such that the calculated fraction of enlarged particles coincides with that measured experimentally. It has been revealed that the contact angle of critical sulfur nuclei formed on tungsten oxide seed particles with average radii 〈R _p〉 ≈ 5.8?4.4 nm is in a range of 21.2?20.5°, while, in the case of sodium chloride seed particles with 〈R _p〉 ≈ 6.0?4.4 nm, the contact angle is 20.4?17.4°. The size of a critical nucleus has been found to be proportional to calculated average radius of a seed particle 〈R _p〉 in both cases.     

Precipitation from homogeneous solution applied to the drop technique for the study of nucleation

The technique of precipitation from homogeneous solution has been successfully coupled with the drop technique for study of the nucleation process.     

Surface energies of chain molecules from homogeneous nucleation

A model for the interpretation of homogeneous nucleation data for chain molecules is presented. The two surface energies σ_s and σ_e are related to interchain and intrachain bonding. Surface energies calculated from experimental data on n-alkanes from octane to dotriacontane and polyethylene agree with estimated values. The results are discussed in relation to surface energies measured from spherulite growth rates in polymers but these values are not known with sufficient reliability to provide a good basis for comparison.