Homogeneous Nucleation Rates Research Articles

Multidimensional Kinetics of Nucleation in Liquid-Vapor Systems

A metastable system is described as an ensemble of locally isolated statistically independent centers, with a possibility of one and only one nucleus of a close-to- critical radius emerging on each one of these centers. It is assumed that this event occurs as a result of fluctuations in a heterophase subsystem and leads to the formation of a viable nucleus at a given point in space. The process of the emergence of this nucleus is treated as the first crossing of the potential barrier by a Brownian particle. Proceeding from the principles of nonequilibrium thermodynamics, dynamic equations of bubble (droplet) growth are derived, which correspond to the Onsager relations. These formulas are used as Langevin equations in multidimensional phase space and are related to the respective Fokker-Planck equation whose solution enables one to determine the local rate of emergence of a viable nucleus and, as a consequence, the rate of its emergence in the entire system. An alternative expression is given for the rate of homogeneous steady-state nucleation, which differs from the classical expression by the pre-exponential factor and, in the case where one parameter (radius) may be sufficient, gives close limits of attainable superheat (supersaturation). Given the expression for the nonequilibrium work of bubble (droplet) and the distribution of heterogeneous centers, the obtained result may be readily generalized to the case of heterogeneous nucleation.

Monte Carlo simulations of critical cluster sizes and nucleation rates of water.

We have calculated the critical cluster sizes and homogeneous nucleation rates of water at temperatures and vapor densities corresponding to experiments by Wolk and Strey [J. Phys. Chem B 105, 11683 (2001)]. The calculations have been done with an expanded version of a Monte Carlo method originally developed by Vehkamaki and Ford [J. Chem. Phys. 112, 4193 (2000)]. Their method calculates the statistical growth and decay probabilities of molecular clusters. We have derived a connection between these probabilities and kinetic condensation and evaporation rates, and introduce a new way for the calculation of the work of formation of clusters. Three different interaction potential models of water have been used in the simulations. These include the unpolarizable SPC/E [J. Phys. Chem. 91, 6269 (1987)] and TIP4P [J. Chem. Phys. 79, 926 (1983)] models and a polarizable model by Guillot and Guissani [J. Chem. Phys. 114, 6720 (2001)]. We show that TIP4P produces critical cluster sizes and a temperature and vapor density dependence for the nucleation rate that agree well with the experimental data, although the magnitude of nucleation rate is constantly overestimated by a factor of 2 x 10(4). Guissani and Guillot's model is somewhat less successful, but both the TIP4P and Guillot and Guissani models are able to reproduce a much better experimental temperature dependency of the nucleation rate than the classical nucleation theory. Using SPC/E results in dramatically too small critical clusters and high nucleation rates. The water models give different average binding energies for clusters. We show that stronger binding between cluster molecules suppresses the decay probability of a cluster, while the growth probability is not affected. This explains the differences in results from different water models.

Homogeneous nucleation rates of higher n-alcohols measured in a laminar flow diffusion chamber.

Nucleation rate isotherms of n-butanol, n-pentanol, n-hexanol, n-heptanol, and n-octanol were measured in a laminar flow diffusion chamber using helium as carrier gas. The measurements were made at 250-310 K, corresponding to reduced temperatures of 0.43-0.50, and at atmospheric pressure. Experimental nucleation rate range was from 10(3) to 10(7) cm(-3) s(-1). The expression and accuracy of thermodynamic parameters, in particular equilibrium vapor pressure, were found to have a significant effect on calculated nucleation rates. The results were compared to the classical nucleation theory (CNT), the self-consistency corrected classical theory (SCC) and the Hale's scaled model of the CNT. The average ratio between the experimental and theoretical nucleation rates for all alcohols used was 1.5x10(3) when the CNT was used, and 0.2x10(-1) when the SCC was used and 0.7x10(-1) when the Hale's scaled theory was used. The average values represent all the alcohols used at the same reduced temperatures. The average ratio was about the same throughout the temperature range, although J(exp)/J(the) calculated with the Hale's scaled theory increased slightly with increasing temperature. The saturation ratio dependency was predicted closest to experiment with the classical nucleation theory. The nucleation rates were compared to those found in the literature. The measurements were in reasonable agreement with each other. The molecular content of critical alcohol clusters was between 35 and 80 molecules. At a fixed reduced temperature, the number of molecules in a critical cluster decreased as a function of alcohol carbon chain length. The number of molecules in critical clusters was compared to those predicted by the Kelvin equation. The theory predicted the critical cluster sizes well.

A study on the variations in the density of nucleated particles with the maximum crystallization rate and with the corresponding temperature, by using differential scanning calorimetry

An analysis is carried out on two procedures which have been proposed for the determination of the temperature dependence of homogeneous crystal nucleation rates in glassy solids by using differential scanning calorimetry (DSC) measurements. The first procedure is based upon the hypothesis that the density of nucleated particles will increase monotonically as the reciprocal of the temperature corresponding to the maximum crystallization rate increases. The second procedure is based on the observation that the maximum crystallization rate increases as the corresponding temperature grows. The validity of both procedures is assessed for glassy solids by considering two specific crystal growth models.

Simulation of Ideal Crystal Superheating and Decay

Molecular dynamics (MD) method is used to investigate time features of the transition of a superheated ideal crystal into liquid. Different levels of statistical description of the process are used. An approach to the calculation of the lifetime of a metastable crystal is developed. The homogeneous nucleation rate dependence on temperature is calculated. The decay under conditions of the constant rate heating is simulated as well. The limits of predictability of the MD method arising from chaotic properties of many-particle systems are shown for the problem concerned.

Isothermal Nucleation Rates in Supersonic Nozzles and the Properties of Small Water Clusters

We make direct measurements of stationary, homogeneous nucleation rates, J = N/Δt, in supersonic Laval nozzles. We determine the number densities, N, of droplets formed from small-angle neutron scattering (SANS) experiments and the time intervals during which nucleation occurs, Δt ≈10 μs, from static pressure measurements along the axis of the nozzle. Applying these techniques to nozzles with different expansion rates, we obtain the first isothermal nucleation rate measurements as a function of supersaturation for these devices with a relatively small error margin in J of ±50%. At temperatures T of 210, 220, and 230 K, the maximum nucleation rates for D2O lie in the range 4 × 1016< J/cm-3s-1 < 3 × 1017 for supersaturations S ranging from 46 to 143. At the highest temperature, the predictions of classical nucleation theory lie slightly below the experimental points but are still within experimental error. At the lower temperatures, the classical predictions lie well below the measured values. The discrepancy increases as the temperature is lowered and exceeds the measurement error bars. In contrast, the predictions of the empirical temperature correction function to the classical theory proposed by Wölk and Strey (Wölk, J.; Strey, R. J. Phys. Chem. B 2001, 105, 11683) agree quite well with the experimental data points over the entire supersaturation and temperature ranges. Finally, we apply the first and second nucleation theorems to the data and directly estimate the number of molecules in the critical cluster n* and the excess internal energy Ex(n*), respectively. The agreement between these values and the classical values predicted assuming that the critical cluster is a compact spherical object is really quite good even though under our conditions n* is less than 10. The good agreement for the classical values of the excess internal energy implies that the poor temperature dependence of the classical rate predictions arises from the classical theory's failure to treat correctly the excess internal entropy of the critical cluster.

Homogeneous nucleation of supersaturated KCl-solutions from single levitated microdroplets

Single levitated microdroplets of aqueous potassium chloride (KCl) solutions are investigated as a function of temperature and humidity of their surroundings. Supersaturated solution droplets are obtained from the evaporation of water. The phase transition into solid microparticles is investigated as a function of temperature and humidity, where the experimental conditions favor homogeneous nucleation. The heat of solution of supersaturated KCl-solution droplets containing variable salt concentrations is derived, which is compared to previous results on dilute and saturated KCl-solutions. It is found to be negative, contrary to saturated and dilute solutions. This gives evidence for the formation of KCl clusters in supersaturated solutions prior to nucleation. The rates of homogeneous nucleation from supersaturated KCl solutions are reported, where the temporal evolution of homogeneous nucleation is investigated. Processes occurring on different time scales are identified. These are discussed along with previous works on homogeneous nucleation of pure water and salt solutions.

Diffusion Control of Homogeneous Crystallization in Nanoconfined Domains of Block Copolymers

Confined crystallization in a poly(oxyethylene)‐b‐poly(oxybutylene)/poly(oxybutylene) blend (E115B103/B28, φE = 0.14) with bcc morphology and in a polystyrene‐b‐poly (oxyethylene)‐b‐polystyrene (S‐E‐S) triblock copolymer (S40E136S40, φE = 0.407) with lamellar morphology was studied using differential scanning calorimetry (DSC). Two types of confined crystallization with different characteristic Avrami exponents were identified in both systems. At higher crystallization temperature (T c), the Avrami exponent is 1.0 and the overall crystallization rate is controlled by the homogeneous nucleation rate. At lower T c, the Avrami exponent is 0.5, which is attributed to diffusion‐controlled confined crystallization. This shows that diffusion has a great influence on the overall crystallization rate when chain mobility is reduced, which can be caused either by lower T c or by constrained microstructure.

Control of Nucleation in Solution Growth of Anatase TiO2 on Glass Substrate

Low-temperature preparation of ceramic thin films in aqueous solution is an important new field in the synthesis of functional materials. This work examines control of nucleation in the growth of anatase titania (TiO2) on a common glass substrate at 55−80 °C using an inorganic precursor, TiF4. It is found that TiO2 on fused silica is first formed as single-crystalline nuclei during an initial heterogeneous nucleation. These TiO2 crystallites elongate along the [001] direction with their [110] axis perpendicular to the glass substrate; the elongated nuclei later grew into a round flowerlike morphology owing to a significant increase in homogeneous nucleation rate during the continued growth. The flowerlike polycrystalline TiO2 islands, either as discrete crystallites or as lined crystallite arrays, subsequently develop into a planar crystallite monolayer. In general, low-pH condition suppresses heterogeneous nucleation and growth; this allows a self-alignment of thin threadlike crystallites. In contrast, h...

Limit of overheating and the threshold behavior in laser ablation.

Constant temperature and pressure molecular-dynamics simulations in conjunction with constant pressure and enthalpy simulations, designed to examine the threshold behavior in laser ablation, demonstrate that the rate of homogeneous nucleation (explosive boiling) increases sharply in a very narrow temperature range at approximately 90% of the critical temperature. Moreover, the homogeneous nucleation is sufficiently rapid to prevent the superheated liquid from entering the spinodal region at densities greater than the critical density.

Asymptotic evaluation of probability density functions for mean aerosol particle formation rates by homogeneous nucleation in turbulent gas jets

Mean aerosol particle formation rates resulting from homogeneous nucleation in turbulent gas jets are evaluated using a probability density function (pdf) for temperature and vapor concentration fluctuations in the gas phase. Nucleation rates are assumed to depend sensitively on the gas-phase conditions, allowing asymptotic analysis to be used for evaluating mean particle formation rates. The asymptotic analysis shows that to leading order, the overall shape of the pdf does not matter significantly when calculating mean particle formation rates. Instead, mean particle formation rates are essentially linearly proportional to the value of the pdf where an exponential term in the expression for particle nucleation rates is maximized. Homogeneous nucleation rates of particles in turbulent gas jets may thus potentially be controlled by suitable modifications of the value of the pdf only in the region of the pdf where particle nucleation rates are maximized. Asymptotic analyzes for mean particle formation rates compare favorably with results obtained using numerical integration.

SAXS/WAXS/DSC studies on crystallization of a polystyrene- b-poly(ethylene oxide)- b-polystyrene triblock copolymer with lamellar morphology and low glass transition temperature

Crystallization of a polystyrene- b-poly(ethylene oxide)- b-polystyrene (S–EO–S) triblock copolymer, S 40EO 136S 40, with lamellar morphology in the melt and low glass transition temperature ( T g=47 °C) of the S block was studied. The triblock copolymer was cooled from ordered melt and isothermal crystallization was conducted at crystallization temperatures ( T c) near the T g of the S block. It is found that crystallization behavior of S 40EO 136S 40 strongly depends on T c. When T c is far below T g, an Avrami exponent n=0.5 is observed, which is attributed to diffusion-controlled confined crystallization. As T c slightly increases, the Avrami exponent is 1.0, indicating that crystallization is confined and crystallization rate is determined by the rate of homogeneous nucleation. When T c is just below the T g of the S block, crystallization tends to become breakout and accordingly Avrami exponent changes from 1.0 to 3.2. Crystallinity and melting temperature of the EO block in breakout crystallization are slightly higher than those in confined crystallization. Time-resolved small and wide angle X-ray scattering (SAXS/WAXS) were used to monitor isothermal crystallization of S 40EO 136S 40. It shows that the long period is constant in confined crystallization, but it gradually increases during breakout crystallization. WAXS result reveals that confined or breakout crystallization has no effect on the crystal structure of the EO block.

Homogeneous nucleation rates for D2O in a supersonic Laval nozzle

We measured the homogeneous nucleation rates of D2O in a supersonic nozzle. Small angle neutron scattering (SANS) experiments characterized the nanodroplet aerosols and yielded the number density N of particles formed. In these new SANS measurements the number densities were in the range of 4×1011&amp;lt;N/cm−3&amp;lt;2×1012. We then determined the characteristic time interval corresponding to the peak nucleation rate ΔtJmax from conventional pressure profile measurements in the nozzle. The sensitive time is typically (11±3) μs. Combining these two pieces of information we obtained the nucleation rate J=NNZ/ΔtJmax, where NNZ is the number density of the aerosol in the nucleation zone. In this nozzle, the peak nucleation rate ranges between 6×1016 and 1.2×1017 cm−3 s−1 and is quite insensitive to the initial conditions of the expansion.

Nucleation and particle growth in supercooled flows of SF6/N2 and UF6/N2 mixtures

Abstract Nucleation and particle growth are examined for supercooled QF 6 vapor streams diluted in a carrier gas like H 2 , He, N 2 , Ar, or Xe under conditions commonly met in molecular laser isotope separation. Critical cluster radii r * are determined by the usual Kelvin surface-energy correction for a droplet’s curvature, but in addition account is taken of a reduction in the intrinsic heat of vaporization of a small cluster due to a decreased average monomer extrication energy. Critical embryo populations are calculated using a new theory published earlier. The results are believed to be an improvement over earlier Farkas–Becker–Doring–Volmer–Zeldovich relations. Time-dependent equations for combined nucleation and particle growth rates in adiabatically cooled gases are formulated, and explicit approximate solutions are derived. As expected, particle growth rates are found to quickly overtake homogeneous nucleation rates, once growth commences. The theory is applied to supersonically expanded UF 6 /N 2 and SF 6 /N 2 flows, and appears generally in agreement with experiment.

Review: Nucleation in solutions revisited

AbstractExisting and new results in nucleation in solutions are outlined from a unified point of view. The thermodynamics of the process is considered and expressions are given for the supersaturation, the nucleation work and the size of the nucleus in homogeneous or heterogeneous nucleation. It is shown how the nucleation theorem can be used for a model‐independent determination of the nucleus size from experimental data. The mechanism and kinetics of nucleation are also considered and formulae are presented for the supersaturation dependence of the monomer attachment frequency and the stationary rate of homogeneous or heterogeneous nucleation. General expressions for the induction time and the critical supersaturation ratio for crystallization are given. An approximate formula is derived for estimating the width of the metastable zone with the help of data for the solubility of the substance crystallized. Existing experimental data are used for verification of the validity of some of the presented theoretical dependences.

Enhanced particle formation and growth due to mixing processes in the tropopause region

Enhancement of the binary homogeneous nucleation rate of H 2SO 4 and H 2O and condensation of H 2O and HNO 3 on liquid sulphate aerosol particles due to mixing processes in the tropopause region were investigated by conducting theoretical studies and by using a section aerosol box model. If two air parcels with a large initial temperature and humidity difference are mixed with each other the nucleation rate within the resulting air parcel will be enhanced. This is due to the curvature of the saturation vapour pressure curve. The theoretical studies show that in the tropopause region enhancements of the nucleation rate of up to five orders of magnitude can occur. Since the mixing causes strong supersaturations also the condensation rate will be enhanced, especially for HNO 3. The simulation with the sectional aerosol box model shows that after the mixing of two air parcels up to 40 wt% of HNO 3 is taken up by the smallest aerosol particles. Since the initial air parcels are initialized with 0 wt% HNO 3 this corresponds to a very strong enhancement. Measurements during the STREAM 1998 campaign show an entrainment of stratospheric air into the troposphere during the flight on 15 July leading to a production of new particles. Box model studies reproduce fairly well the measured number of newly formed particles.

Recent studies of internal and surface nucleation in silicate glasses

This article reviews recent findings on internal and surface nucleation in silicate glasses. The internal homogeneous nucleation rates sharply decrease and the induction times increase with the T g/ T L ratio ( T g is the glass–transition temperature and T L is the liquidus temperature). Only systems that have T g/ T L < 0.58 display measurable internal nucleation rates on a laboratory time–scale. Numerous tests of the classical nucleation theory have demonstrated that the theory fails to describe nucleation rates in glasses quantitatively. Possible explanations for this failure are tested and discussed. Surface nucleation depends strongly on the surface quality, e.g. tips, cracks and scratches, elastic stresses, foreign particles and surrounding atmosphere. The mechanisms of surface nucleation are still not fully understood, but some of the key factors are now known and the surface–nucleation density can thus be controlled for the development of sintered glasses or glass ceramics.

Response to: On methods of determination of homogeneous nucleation rates of protein crystals

In their letter, Galkin and Vekilov suggest that our analysis [Dixit et al., Colloids Surf. A 190, 47 (2001)] of their experimental technique [Galkin and Vekilov, J. Phys. Chem. B 103, 10965 (1999); J. Am. Chem. Soc. 122, 156 (2000)] for estimating protein crystal nucleation rates contains inconsistencies. Here, we examine their criticisms and find that our analysis does not contain inconsistencies. We conclude by restating our claim in [Dixit et al., Colloids Surf. A 190, 47 (2001)] that Galkin and Vekilov underestimate protein crystal nucleation rates by several orders of magnitude because of the way their experimental data is interpreted.

On the methods of determination of homogeneous nucleation rates of protein crystals

A recently published analysis of an experimental method to directly determine the homogeneous rates of nucleation of protein crystals or other solid phases of proteins inferred that the method underestimates the homogeneous nucleation rates by several orders of magnitude. Here, we consider the presented arguments and find that the analysis contains certain inconsistencies. We conclude that the homogeneous nucleation rate data produced using the direct determination technique are, most likely, correct.

An improved parameterization for sulfuric acid–water nucleation rates for tropospheric and stratospheric conditions

In this paper we present parameterized equations for calculation of sulfuric acid–water critical nucleus compositions, critical cluster radii and homogeneous nucleation rates for tropospheric and stratospheric conditions. The parameterizations are based on a classical nucleation model. We used an improved model for the hydrate formation relying on ab initio calculations of small sulfuric acid clusters and on experimental data for vapor pressures and equilibrium constants for hydrate formation. The most rigorous nucleation kinetics and the thermodynamically consistent version of the classical binary homogeneous nucleation theory were used. The parameterized nucleation rates are compared with experimental ones, and at room temperature and relative humidities above 30% they are within experimental error. At lower temperatures and lower humidities the agreement is somewhat poorer. Overall, the values of nucleation rates are increased compared to a previous parameterization and are within an order of magnitude compared with theoretical values for all conditions studied. The parameterized equations will reduce the computing time by a factor 1/500 compared to nonparameterized nucleation rate calculations and therefore are in particular useful for large‐scale models. The parameterized formulas are valid at temperatures between 230.15 K and 305.15 K, relative humidities between 0.01% and 100%, and sulfuric acid concentrations from 104 to 1011 cm−3. They can be used to extrapolate the classical results down to 190 K. The parametrization is limited to cases where nucleation rates are between 10−7 and 1010 cm−3s−1, and the critical cluster contains at least four molecules.