Experimental Nucleation Rates Research Articles

Empirical function for homogeneous water nucleation rates

Very recently, Wölk and Strey [J. Phys. Chem. B 105, 11683 (2001)] presented empirical temperature correction functions for calculating homogeneous nucleation rates J of H2O and D2O (1<J/cm−3 s−1<1020) from classical nucleation theory over an extended range of temperature T (200<T/K<310) and supersaturations S (5<S<200). Here, we critically test the correction functions to the Becker–Döring nucleation rate equation JBD against an extensive set of experimental data, and find that the equations distinctly improve the agreement between theory and experiment for very little extra work. The success of the corrected nucleation rate functions is surprising, given that they were developed based on experimental nucleation rates measured in a nucleation pulse chamber over a limited nucleation rate range 105<J/cm−3 s−1<1010, supersaturation range 6<S<22, and temperature range 220<T/K<260.

Void nucleation at elevated temperatures under cascade-damage irradiation

showed reasonable agreement between theory and experiment. This application also predicts correctly the temporal development of large-scale spatial heterogeneous microstructure during the void nucleation stage. Comparison between calculated and experimental void nucleation rates in neutron-irradiated molybdenum at temperatures where vacancy emission from voids is negligible showed reasonable agreement as well. It was clearly demonstrated that the athermal shrinkage of relatively large voids experimentally observable in molybdenum at such temperatures may be easily explained in the framework of the present theory.

Kinetics of homogeneous binary condensation at the stage of formation of the bulk of the liquid phase

The second stage of homogeneous binary condensation is considered. It is during this stage, following the stage of nucleation, that the bulk of the liquid phase is formed. A kinetic theory of homogeneous binary condensation at this stage is developed. Taking into account the depletion of the initial vapor mixture, transcendental equations are found which determine the time dependence both of composition of droplets and of activities of vapor mixture components. The problem of finding the spectrum of linear sizes of droplets is reduced to the solution of analogous problem in the kinetics of homogeneous unary condensation and is solved with the help of iteration procedure. A quantitative definition of the concepts of the “first” and “second” stages of binary condensation is given, and the estimates for the durations of these stages are obtained. The duration of the first stage (usually called “nucleation”) is important to correctly calculate experimental nucleation rates. As shown by numerical evaluations, the use of the first stage duration instead of the duration of the “nucleation pulse” in calculating experimental nucleation rates can, under some circumstances, improve their agreement with predictions of the classical nucleation theory.

Homogeneous nucleation of n-pentanol in a laminar flow diffusion chamber

Nucleation rate isotherms of n-pentanol were measured in laminar flow diffusion chamber. n-pentanol was chosen for nucleating vapor and helium for carrier gas as a part of a world wide joint experiment on homogeneous nucleation. Experimental temperature range was from 260 to 290 K. Experimental nucleation rate range was from 103 to 107 cm−3 s−1. The results were compared to the classical nucleation theory and experimental data found in literature. Experimental results were three orders of magnitude higher than predicted by the theory. The difference was constant over the whole experimental range. The saturation ratio dependency of nucleation rate was well predicted by the theory. The number of molecules in the critical clusters was quite consistent with the theory. The results were in reasonable agreement with data found in literature.

Analysis of water–ethanol nucleation rate data with two component nucleation theorems

We generalize the second nucleation theorem to multicomponent systems. Nucleation theorems are used to extract the molecular composition and excess internal energy of the critical cluster from experimental nucleation rates in a water–ethanol mixture. The excess internal energy is found to depend only weakly on temperature and to be almost solely a function of the molecular numbers of water and ethanol in the cluster. We estimate the contribution of the kinetic pre-factor to our analysis, and find that it is small in the case of the first theorem, but significant for the second theorem. We find that capillarity approximation fails to predict the experimental critical size and excess energy in this highly nonideal system.

Energetics of small n-pentanol clusters from droplet nucleation rate data

We use nucleation theorems to extract the excess internal energy of small molecular clusters of n-pentanol from experimental droplet nucleation rates. Corrections to the theorems are derived, in order to take into account the nonideality of the vapor phase, but these have only a small effect on our results. Experimental datasets from different groups provide information about clusters of different sizes at a range of temperatures. The analysis shows that there are significant and intriguing deviations from the predictions of the capillarity approximation.

Nucleation at high pressure. II. Wave tube data and analysis

Nucleation rate data, obtained from expansion wave tube experiments, are reported for several vapor–gas mixtures at high pressure. Results are given for water–vapor in the presence of helium and nitrogen gas, and for n-nonane in helium and methane. For all these mixtures, carrier gas pressures of 10, 25, and 40 bar have been applied, with temperatures ranging from 230 to 250 K. An extended form of the nucleation theorem (in terms of the derivative of the nucleation rate with respect to carrier gas pressure) is derived, which appears to be very helpful in the interpretation of high pressure data. It can be used to obtain the carrier gas content of the critical nucleus directly from the pressure dependence of experimental nucleation rates. Combining this method with the theoretical considerations of part I of this paper [J. Chem. Phys. 111, 8524 (1999), preceding paper]: the nucleation behavior of water at high pressures of both helium and nitrogen can quantitatively be understood. For n-nonane in helium our “pressure perturbation approach” is also valid. For n-nonane in methane, however, this approach fails because of the high methane solubility in the liquid phase.

Cahn–Hilliard-type density functional calculations for homogeneous ice nucleation in undercooled water

The free energy of ice nuclei and the nucleation rate in undercooled water are calculated in the framework of a single-order parameter Cahn–Hilliard theory. The coefficients of the quartic free energy-order parameter relationship and of the square-gradient term are chosen in order to reproduce the measured Gibbs free energy difference, the ice–water interfacial free energy, and the interface thickness predicted by molecular dynamics simulations. Without adjustable parameters, the model reproduces fairly the experimental nucleation rates down to −40°C. A simple one-parameter model is adopted to describe the specific heat anomaly of liquid water at lower temperatures. The range of nucleation rates allowed by thermodynamically consistent choices of this parameter encloses extreme experimental values (∼1030m−3s−1) Bartell and Huang obtained at deep undercoolings (∼− 73°C). The temperature and size-dependencies of the interfacial properties (free energy, Tolman-length, etc.) are discussed.

Determination of homogeneous nucleation rates from laminar-flow diffusion chamber data

MADMAcS 1 is a 2D algorithm for the simulation of coupled fluid flow, heat and mass transport, and aerosol dynamics based on the modal aerosol dynamics modelling technique ( Whitby, 1989, 1996; Wilck and Stratmann, 1997; Whitby and McMurry, 1997). It is applied to measurements of nucleation of dibutyl phtalate in a laminar flow diffusion chamber presented by Hämeri et al. (1996). A “linear range” is specified inside of which the relation between measured particle number concentrations and nucleation rates is easy and allows the determination of experimental nucleation rates. Based on this approach, a general inversion method for laminar flow diffusion chamber measurements is introduced that is insensitive against many simulation parameters. Moreover, the method can be carried out with simpler models than MADMAcS once the linear range is determined. In a preliminary sensitivity analysis, the thermodynamical properties of the involved substances, particularly the vapor pressure, are shown to be of crucial importance for the reliability of nucleation measurements.

Relationship of phase diagrams and surfaces of new phase nucleation rates

Experimental and theoretical investigations of vapor nucleation began about 100 years ago. Until the 1980s, experiments generally measured only critical supersaturation values. Since then, measurement procedures have substantially improved and nucleation rates can now be measured as a function of temperature, vapor activities, and pressure with high accuracy. Nucleation theory has made obvious progress, but the understanding of nucleation phenomenon is far from complete. New approaches to conceptualizing nucleation are necessary in order to identify possible new directions for further improvement of nucleation theory. One such approach is the analysis of the topology of nucleation rate surfaces. The creation of a nucleation rate surface is based on knowledge of phase equilibrium diagrams, limited experimental nucleation results, and a few plausible assumptions. In this article, the surfaces of the nucleation rates as a function of pressure or activity for single and binary systems for nucleation from metastable vapor, liquid, and crystalline states are constructed. By using surface topology analysis, some problems in nucleation theory are more clearly formulated and future directions for improvement can be found. Currently, it is not possible to create a universal nucleation theory based only on first principles. Partial theoretical success can be obtained only in the case of systems with well-known molecular interaction potentials. By scaling the experimental nucleation rate surfaces for portions of the phase diagrams that are identical to one another, a semiempirical nucleation rate surface for an unknown system can be created from its phase diagram. Scaling should give quantitative nucleation rates. Phase diagrams must be more fully incorporated into the interpretation of experimental nucleation results and nucleation theory development.

Nucleation in oxide glasses: comparison of theory and experiment

Experimental nucleation rates on eight oxide glasses crystallizing polymorphously are analysed in terms of non–classical nucleation theories, including phenomenological and density functional approaches. It is demonstrated that of the six non–classical models considered here, only the semiempirical density functional approach and the phenomenological diffuse interface theory are fully consistent with the data. The other models (the self–consistent classical theory, the detailed molecular theories and a Ginzburg–Landau/Cahn–Hilliard model) did not provide a satisfactory fit with experiment. The failure/success of these approaches is discussed in terms of the applied approximations.

Measurement of the molecular content of binary nuclei. IV. Use of the nucleation rate surfaces for the n-nonane-n-alcohol series

In a previous paper the molecular content of binary water-n-alcohol nuclei has been determined from nucleation rate measurements. A strong mutual enhancement of water and alcohol in forming the nuclei was observed, although macroscopically the higher alcohols are only partially miscible with water. In this paper we replace water by n-nonane, that is, we examine n-nonane-CiH2i+1OH systems with i=2–6. Using the nucleation pulse technique nucleation rates in the range 105<J/cm−3 s−1<109 are measured. Ranging from pure n-nonane to pure n-alcohol the n-nonane and n-alcohol activities, a1 and a2, respectively, are varied for each system with about eight intermediate compositions at a constant temperature of T=230 K. A rather reluctant conucleation of the n-alcohols with n-nonane is found, the most stubborn being ethanol. However, one observes that with increasing alcohol chain length the nucleation process tends to become more ideal. We present the full experimental nucleation rate surface J(a1,a2) for n-nonane-n-propanol as an example. From the nucleation rate surface for each system the onset activities corresponding to a reference nucleation rate of J0=107 cm−3 s−1 are determined. From the slopes of the nucleation rate surfaces one obtains the individual numbers of molecules in the critical cluster ni*≈∂ ln J/∂ ln ai. As noted previously, determining the molecular content this way does not involve any particular theoretical model, nor does it depend on the structure of the critical cluster. Accordingly, the average composition of the critical clusters is obtained. An alcohol depletion of the nuclei at low alcohol activity fraction is found for all alcohols examined, the degree diminishing with increasing alcohol chain length. Macroscopically all alcohols are miscible with nonane. Similarly, a depletion of n-nonane in the nuclei is observed at low n-nonane activity fractions. The approach towards a macroscopic miscibility gap for the shorter alcohols is reflected in quantitative but no qualitative changes of the composition of the microscopic nuclei.

A test of the Hrubÿ parameter to estimate glass-forming ability

Experimental nucleation and growth rates for some glasses that nucleate in the bulk were used to calculate critical cooling rates for glass formation ( R c) by the TTT method. Resulting critical cooling rates were consistent with laboratory melting and quenching and also with experimental data of critical cooling rates for lithium disilicate glass. Strong correlation between the Hrubÿ parameter of glass stability ( K gl) and the glass-forming ability, evaluated by R c, was found. Therefore, K gl can be used to estimate the glass-forming ability for glasses that exhibit volume nucleation.

Homogeneous nucleation in a laminar flow diffusion chamber: The operation principles and possibilities for quantitative rate measurements

The laminar flow diffusion chamber technique for experimental homogeneous nucleation rate measurements is presented. In this type of chamber the flow containing vapor is rapidly cooled and critical supersaturations for homogeneous nucleation are obtained. Here the main operation principles as well as data reduction methods are investigated and the results are applied to dibutyl phthalate (DBP) vapor, which works well in this type of chamber. The nucleation rate has to be calculated using the profiles of saturation ratio and temperature, and using a theoretical expression for the nucleation rate. We have used three different nucleation theories when investigating the experimental nucleation rate: classical nucleation theory, self-consistent correction to classical theory, and the Dillmann–Meier theory. According to investigations in this work the determination of nucleation rate is general and rather independent of the theory used, and therefore the results can be considered reliable. The volume inside which the nucleation event is calculated to take place is examined experimentally. The comparison between model calculations of the actual nucleation volume and experiments show reasonable agreement.

Condensation of supersaturated vapors of hydrogen bonding molecules: Ethylene glycol, propylene glycol, trimethylene glycol, and glycerol

The critical supersaturations required for the homogeneous nucleation (rate of 1 drop cm−3 s−1) of ethylene glycol, propylene glycol, trimethylene glycol and glycerol vapors have been measured over wide temperature ranges (e.g., 280–400 K) using an upward thermal diffusion cloud chamber. At lower temperatures the experimental nucleation rates are much higher than the predictions of the classical nucleation theory. Glycerol shows the best agreement between experiment and theory in the temperature range of 340–370 K. An apparent increase in the critical supersaturation of glycerol is observed with increasing carrier gas (helium) pressure and this effect is more pronounced at lower temperatures. The results from corresponding states and scaled nucleation models indicate that the nucleation behavior of glycerol is quite different from other glycols. Glycerol requires higher critical supersaturations compared to the other glycols at the same reduced temperatures. This implies quite small critical clusters for glycerol (20–50 molecules) in the temperature range 300–380 K. The discrepancy between experiment and theory at lower temperatures may be explained by considering that the surface tension of the critical clusters is lower than the bulk surface tension. It is, however, surprising that a Tolman type correction for the curvature dependent surface tension could be applicable for such small critical clusters. Further theoretical work is required in order to fully understand the observed higher nucleation rates at lower temperatures of glycols and glycerol.

Classical growth of hard-sphere colloidal crystals.

The classical theory of nucleation and growth of crystals is examined for concentrated suspensions of hard-sphere colloidal particles. The work of Russel is modified, extended, and evaluated, explicitly. Specifically, the Wilson-Frenkel growth law is modified to include the Gibbs-Thomson effect and is evaluated numerically. The results demonstrate that there is a critical nucleus radius below which crystal nuclei will not grow. A kinetic coefficient determines the maximum growth velocity possible. For large values of this coefficient, quenches to densities above the melting density show interface limited growth with the crystal radius increasing linearly with time. For quenches into the coexistence region the growth is diffusion limited, with the crystal radius increasing as the square root of elapsed time. Smaller values of the kinetic coefficient produce long lived transients which evidence quasi-power-law growth behavior with exponents between one half and unity. The smaller kinetic coefficients also lead to larger crystal compression. Crystal compression and nonclassical exponents have been observed in recent experiments. The theory is compared to data from small angle scattering studies of nucleation and growth in suspensions of hard colloidal spheres. The experimental nucleation rate is much larger than the theoretically predicted value as the freezing point is approached but shows better agreement near the melting point. The crystal growth with time is described reasonably well by the theory and suggests that the experiments are observing long lived transient rather than asymptotic growth behavior. (c) 1995 The American Physical Society

Measurement of the molecular content of binary nuclei. III. Use of the nucleation rate surfaces for the water-n-alcohol series

In two preceding papers the molecular content of binary ethanol-hexanol and water-ethanol nuclei, respectively, was determined from nucleation rate measurements. While nucleation of ethanol-hexanol mixtures behaved nearly ideal, a strong mutual nucleation enhancement for water-ethanol was observed. Here we extend the investigations to include the longer chain alcohols, that is water −CiH2i+1OH systems with i=2 to 6. Using the nucleation pulse technique developed in the past few years nucleation rates in the range 105<J/cm−3 s−1<109 are measured. Ranging from pure water to pure alcohol the water and alcohol activities, a1 and a2, respectively, are varied for each system with about ten intermediate compositions at constant temperature T=260 K. Aside from a remarkably similar behavior of the various alcohols, one observes that with increasing alcohol chain length the mutual nucleation enhancement decreases. Since all water-alcohol systems behave qualitatively similar, we confine ourselves to present the full experimental nucleation rate surface J(a1,a2) for one system, water-n-pentanol, as an example. From the nucleation rate surface for each system the onset activities corresponding to a reference nucleation rate of J0=107 cm−3 s−1 are determined. From the slopes of the nucleation rate surfaces one obtains the individual numbers of molecules in the critical cluster ni* because ∂ ln J/∂ ln ai≊ni*. As noted previously, determining the molecular content this way does not involve any particular theoretical model, nor does it depend on the structure of the critical cluster. Accordingly, the average composition of the critical clusters can be obtained. An alcohol enrichment of the nuclei at low alcohol activity fraction is found for all alcohols examined, the degree diminishing with increasing alcohol chain length. The appearance of a macroscopic miscibility gap for the higher alcohols is not reflected in any qualitative change of the composition of the microscopic nuclei.

Competition between vitrification and crystallization of methanol at high pressure

We have studied methanol at high pressure up to 33 GPa at room temperature with x-ray diffraction, optical (polarization) microscopy, Raman spectroscopy, and detection of hydrostaticity. A competition between crystallization and vitrification is observed when methanol is superpressed beyond the freezing pressure of 3.5 GPa: between 5.0 and 10.5 GPa crystals can nucleate, but if this region is surpassed quickly enough (within a few seconds), methanol remains amorphous. For the first time the nucleation rate and the crystal growth velocity have been studied as a function of pressure. These kinetic properties can be described by classical nucleation theory in agreement with, respectively, Turnbull–Fisher and Wilson–Frenkel type behavior using one and the same activated hard-sphere diffusion coefficient. The experimental nucleation rate and the crystal growth velocity are both effectively reduced to zero above 10.5 GPa, because the diffusion is suppressed. At these pressures methanol is compressed into a glass.

Measurement of the molecular content of binary nuclei. II. Use of the nucleation rate surface for water–ethanol

In a preceding paper we determined the molecular contents of binary nuclei from homogeneous nucleation rate measurements studing ethanol–hexanol as a nearly ideal model system. Here we report the analogous measurements for the nonideal water–ethanol system known from previous investigations to display surface enrichment of ethanol in the nuclei. The nucleation pulse technique applied allows accurate measurements of homogeneous nucleation rates in the range 105<J/cm−3 s−1<109. We determined the homogeneous nucleation rates of mixed droplets in supersaturated water–ethanol vapor mixtures with argon as the carrier gas. The experiments were performed as functions of the water and ethanol gas phase activities, a1 and a2, respectively, ranging from pure water to pure ethanol with ten intermediate activity ratios at T=260 K. The experimental data shape a nucleation rate surface in three-dimensional J–a1–a2 space from which the critical activities for a rate of Jc=107 cm−3 s−1 were determined. The critical activities are compared with the predictions of the classical binary nucleation theory. The observed large discrepancies can be attributed to very water-rich compositions of the nuclei predicted by the theory. The results support previous findings for the water-n-propanol system, where even larger discrepancies were observed. In a next step the individual numbers of ethanol and water molecules in the nuclei are directly determined from the slopes of the experimental nucleation rate surface. The observed variations of the molecular numbers with vapor phase composition differ substantially from that of an ideal mixture. Comparing the measured nucleus compositions to the predictions of a recently proposed explicit cluster model one observes an almost quantitative agreement.

Construction and test of laminar flow diffusion chamber: homogeneous nucleation of DBP and n-hexanol

A laminar flow diffusion chamber was constructed for nucleation studies. The performance of the chamber was tested by investigating the homogeneous nucleation of DBP and n-hexanol vapours. The observed experimental nucleation rates varied from 10 2 to 10 8 s −1 cm −3. The experimental nucleation rates for DBP were observed to be from 3 to 5 orders of magnitude higher than the nucleation rates predicted by the classical nucleation theory. The experimental results were compared with the self-consistent theory by Girschick, S., Chiu, C.-P. and McMurry, P. (1990) Time-dependent aerosol models and homogeneous nucleation rates. Aerosol Sci. Technol. 13, 465 and with the Dillmann-Meier nucleation theory. The experimental nucleation rates were also compared with other experimental studies. The results of the nucleation rate of DBP agree quite well with the earlier experimental results. For n-hexanol the experimental nucleation rates are not as reliable as for DBP due to different losses. However, the experimental nucleation rates of n-hexanol agree well with the classical theory and are several orders of magnitude smaller than the earlier experimental results.