Binary Nucleation Kinetics. II. Numerical Solution of the Birth-Death Equations

We numerically solve the complete set of coupled differential equations describing transient binary nucleation kinetics for vapor-to-liquid phase transitions. We investigate binary systems displaying both positive and negative deviations from ideality in the liquid phase and obtain numerical solutions over a wide range of relative rates of monomer impingement. We emphasize systems and conditions that either have been or can be investigated experimentally. In almost every case, we find behavior consistent with Stauffer\u27s idea that the major particle flux passes through the saddle point with an orientation angle that depends on the rates of monomer impingement. When this is true, the exact numerical steady state nucleation rates are within 10%-20% of the predictions of Stauffer\u27s analytical theory. The predictions of Reiss\u27 saddle point theory also agree with the numerical results over a wide range of relative monomer impingement rates as long as the equilibrium vapor pressures of the two pure components are similar, but Stauffer\u27s theory is more generally valid. For systems with strong positive deviations from ideality, we find that the saddle point approximation can occasionally fail for vapor compositions that put the system on the verge of partial liquid phase miscibility. When this situation occurs for comparable monomer impingement rates, we show that the saddle point approximation can be rescued by evaluating an appropriately modified nucleation rate expression. When the two impingement rates differ significantly, however, the major particle flux may bypass the saddle point and cross a low ridge on the free energy surface. Even in these rare cases, the analytical saddle point result underpredicts the numerical result by less than a factor of 10. Finally, we study the transition from binary to unary nucleation by progressively lowering the vapor concentration of one component. Both Reiss\u27 and Stauffer\u27s rate expressions fail under these conditions, but our modified rate prescription remains within 10%-20% of the exact numerical rate

Binary Nucleation Kinetics. III. Transient Behavior and Time Lags

Transient binary nucleation is more complex than unary because of the bidimensionality of the cluster formation kinetics. To investigate this problem qualitatively and quantitatively, we numerically solved the birth-death equations for vapor-to-liquid phase transitions. Our previous work [J. Chem. Phys 103, 1137 (1995)] showed that the customary saddle point and growth path approximations are almost always valid in steady state gas phase nucleation and only fail if the nucleated solution phase is significantly nonideal. The current work demonstrates that in its early transient stages, binary nucleation rarely, if ever, occurs via the saddle point. This affects not only the number of particles forming but their composition and may be important for nucleation in glasses and other condensed mixtures for which time scales are very long. Before reaching the state of saddle point nucleation, most binary systems pass through a temporary stage in which the region of maximum flux extends over a ridge on the free energy surface. When ridge crossing nucleation is the steady state solution, it thus arises quite naturally as an arrested intermediate state that normally occurs in the development of saddle point nucleation. While the time dependent and steady state distributions of the fluxes and concentrations for each binary system are strongly influenced by the gas composition and species impingement rates, the ratio of nonequilibrium to equilibrium concentrations has a quasiuniversal behavior that is determined primarily by the thermodynamic properties of the liquid mixture. To test our quantitive understanding of the transient behavior, we directly calculated the time lag for the saddle point flux and compared it with the available analytical predictions. Although the analytical results overestimate the time lag by factors of 1.2-5, they should be adequate for purposes of planning experiments. We also found that the behavior of the saddle point time lag can indicate when steady state ridge crossing nucleation will occur

Binary Nucleation Kinetics. I. Self-Consistent Size Distribution

Using the principle of detailed balance, we derive a new self-consistency requirement, termed the kinetic product rule, relating the evaporation coefficients and equilibrium cluster distribution for a binary system. We use this result to demonstrate and resolve an inconsistency for an idealized Kelvin model of nucleation in a simple binary mixture. We next examine several common forms for the equilibrium distribution of binary clusters based on the capillarity approximation and ideal vapor behavior. We point out fundamental deficiencies for each expression. We also show that each distribution yields evaporation coefficients that formally satisfy the new kinetic product rule but are physically unsatisfactory because they depend on the monomer vapor concentrations. We then propose a new form of the binary distribution function that is free of the deficiencies of the previous functions except for its reliance on the capillarity approximation. This new self-consistent classical (SCC) size distribution for binary clusters has the following properties: It satisfies the law of mass action; it reduces to an SCC unary distribution for clusters of a single component; and it produces physically acceptable evaporation rate coefficients that also satisfy the new kinetic product rule. Since it is possible to construct other examples of similarly well-behaved distributions, our result is not unique in this respect, but it does give reasonable predictions. As an illustrative example, we calculate binary nucleation rates and vapor activities for the ethanol-hexanol system at 260 K using the new SCC distribution and compare them to experimental results. The theoretical rates are uniformly higher than the experimental values over the entire vapor composition range. Although the predicted activities are lower, we find good agreement between the measured and theoretical slope of the critical vapor activity curve at a constant nucleation rate of 107 cm-3 s-2

Small Angle Neutron Scattering from Nanodroplet Aerosols

We report the first measurements of small angle neutron scattering from an aerosol. The aerosol was produced by expanding a D2O-N2 vapor mixture in a supersonic Laval nozzle. The neutron wavelength (0.5 nm) is less than the typical particle size, and we can therefore derive the average particle size (5-8 nm), number density (~1012 cm-3), and polydispersity of the size distribution directly from the experimental data rather than by inferring them from complex models of particle formation and growth. We also predict and observe a Doppler shift-induced anisotropy in the scattering pattern due to the directed motion of the aerosol in the nozzle. Further applications of this new technique are discussed

Effect of Carrier Gas Pressure on Condensation in a Supersonic Nozzle

Supersonic nozzle experiments were performed with a fixed water or ethanol vapor pressure and varying amounts of nitrogen to test the hypothesis that carrier gas pressure affects the onset of condensation. Such an effect might occur if nonisothermal nucleation were important under conditions of excess carrier gas in the atmospheric pressure range, as has been suggested by Ford and Clement [J. Phys. A 22, 4007 (1989)]. Although a small increase was observed in the condensation onset temperature as the stagnation pressure was reduced from 3 to 0.5 atm, these changes cannot be attributed to any nonisothermal effects. The pulsed nozzle experiments also exhibited two interesting anomalies: (1) the density profiles for the water and ethanol mixtures were shifted in opposite directions from the dry N2 profile; (2) a long transient period was required before the nozzle showed good pulse-to-pulse repeatability for condensible vapor mixtures. To theoretically simulate the observed onset behavior, calculations of nucleation and droplet growth in the nozzle were performed that took into account two principal effects of varying the carrier gas pressure: (1) the change in nozzle shape due to boundary layer effects and (2) the variation in the heat capacity of the flowing gas. Energy transfer limitations were neglected in calculating the nucleation rates. The trend of the calculated results matched that of the experimental results very well. Thus, heat capacity and boundary layer effects are sufficient to explain the experimental onset behavior without invoking energy transfer limited nucleation. The conclusions about the rate of nucleation are consistent with those obtained recently using an expansion cloud chamber, but are at odds with results from thermal diffusion cloud chamber measurements

Binary Condensation in a Supersonic Nozzle

We present data from the first systematic studies of binary condensation in supersonic nozzles. The apparatus used to conduct the experiments is described in detail, and the important issues of stability and reproducibility of the experiments are discussed. Experiments were conducted with water, ethanol, propanol, and binary mixtures of these compounds. Onset was determined in the temperature range of 190-215 K, and for each mixture composition the pressures of the condensible species at an onset temperature of 207 K were determined. For the ideal ethanol-propanol mixtures, the onset pressures at constant temperature vary almost linearly between those of the pure components. In contrast the isothermal onset pressures for the nonideal water-ethanol and water-propanol mixtures lie below the straight line joining the pure component values. This large reduction in the total pressure of condensible at onset for the aqueous alcohol mixtures is indicative of a strong mutual enhancement in the particle formation process

Doppler Shift Anisotropy in Small Angle Neutron Scattering

The two-dimensional patterns in our small angle neutron scattering (SANS) experiments from rapidly moving aerosols are anisotropic. To test the kinematic theory of two-body scattering that describes the anisotropy, we conducted SANS experiments using a constant source of D2O aerosol with droplets moving at ~440 m/s, and varied the neutron velocity from 267 to 800 m/s. The theoretically predicted anisotropy of the laboratory scattering intensities agrees well with the experimental results. Based on an analysis of the scattering intensity in the Guinier region, we also determined the particle velocity. The results are in very good agreement with independent velocity estimates based on supersonic flow measurements

Aspects of homogeneous nucleation

NOTE: Text or symbols not renderable in plain ASCII are indicated by [...]. Abstract is included in .pdf document. Experimental investigations of vapor phase binary nucleation were carried out for both the methanesulfonic acid-water and the sulfuric acid-water systems. A rapid mixing device produced acid-water aerosols under isothermal conditions and at relative acidities (Ra), 0.04 < Ra < 0.65, relative humidities (Rh), 0.01 < Rh < 0.65, and temperatures, T = 20, 25 and 30°C. The number concentration of the aerosol at the exit of the nucleation and growth tube is extremely sensitive to the binary nucleation rate. Thus at low particle concentrations, when condensation did not significantly change the saturation levels the binary nucleation rates were estimated from the number concentration data as a function of Ra, Rh and temperature. Particle size distributions were also measured and found to vary with the amount of acid and water present. An integral model considering both nucleation and growth simulated the experimental system and predicted the total number of particles, the total mass in the aerosol phase, and the mass average diameter at the exit of the nucleation and growth tube. The simulations reproduced the experimental results quite well for the methansulfonic acid-water binary, if the nucleation rate was adjusted by a temperature dependent correction factor which ranged from [...] to [...]. Further analysis showed that the ratio of experimental to theoretical nucleation rates for both acid-water systems was a strong function of the predicted number of acid molecules in the critical nucleus. Classical homogeneous nucleation theory was extended to nonisothermal conditions by simultaneously solving cluster mass and energy balances. In vapor phase nucleation, the steady state nucleation rate was lower than the corresponding isothermal rate and this discrepancy increased as the pressure of the background gas decreased. After the initial temperature transients decayed, subcritical clusters were found to have temperatures elevated with respect to that of the background gas

The Structure of Aqueous-Alkane Nanodroplets

We conduct in situ small angle X-ray scattering (SAXS) experiments on D2O-nonane nanodroplets produced in a supersonic nozzle. Scattering from standard shapes like well-mixed spheres and core-shell structures does not agree with the experimental data. The \u27lens-on-sphere\u27 models suggested by molecular dynamics simulations fit the scattering data well, but the amount of D2O condensed, based on the SAXS fitting parameters, is only half of that measured by infrared absorption spectroscopy