Nucleation Theorem Research Articles

Theory of Anomalous Critical-Cluster Content in High-Pressure Binary Nucleation

Nucleation experiments in binary (a-b) mixtures, when component a is supersaturated and b (carrier gas) is undersaturated, reveal that for some mixtures at high pressures the a content of the critical cluster dramatically decreases with pressure contrary to expectations based on classical nucleation theory. We show that this phenomenon is a manifestation of the dominant role of the unlike interactions at high pressures resulting in the negative partial molar volume of component a in the vapor phase beyond the compensation pressure. The analysis is based on the pressure nucleation theorem for multicomponent systems which is invariant to a nucleation model.

Volumes of critical bubbles from the nucleation theorem

A corollary of the nucleation theorem due to Kashchiev [Nucleation: Basic Theory with Applications (Butterworth-Heinemann, Oxford, 2000)] allows the volume V(*) of a critical bubble to be determined from nucleation rate measurements. The original derivation was limited to one-component, ideal gas bubbles with a vapor density much smaller than that of the ambient liquid. Here, an exact result is found for multicomponent, nonideal gas bubbles. Provided a weak density inequality holds, this result reduces to Kashchiev's simple form which thus has a much broader range of applicability than originally expected. Limited applications to droplets are also mentioned, and the utility of the p(T,x) form of the nucleation theorem as a sum rule is noted.

Forms and applications of the nucleation theorem

The nucleation theorem is a general relation between the nucleation work, the nucleus size, and the supersaturation or other thermodynamic parameters of the old phase. The theorem appears in different forms, depending not only on the chosen set of independent variables describing the nucleation work but also on which of these variables is changed while the others are held fixed. This paper gives a rigorous, systematic, and comprehensive presentation of various forms of the nucleation theorem and shows how some of them can be applied to concrete cases of nucleation. Both theoretical and experimental applications of the theorem to nucleation in unary, binary, or ternary phases are considered.

Cluster activation theory as an explanation of the linear dependence between formation rate of 3nm particles and sulphuric acid concentration

Abstract. According to atmospheric observations new particle formation seems to be a function of sulphuric acid concentration to the power from one to two. The nucleation theorem then predicts that the critical cluster contains one to two sulphuric acid molecules. However, existing nucleation theories predicts that the power is more (or equal) than 2. Here we present an activation theory, which can explain the observed slope. In cluster activation the clusters containing one sulphuric acid molecule will activate for further growth due to heterogeneous nucleation, heterogeneous chemical reactions including polymerization or activation of soluble clusters. In the activation process organic vapours are typically needed as condensing agents.

Molecular dynamics investigation of homogeneous nucleation and cluster growth ofplatinum clusters from supersaturated vapour

The formation of platinum nanoparticles from a supersaturated vapour phase isinvestigated by molecular dynamics simulations. Argon is added as carrier gas, removingthe condensation heat from the nucleating system. The interactions between the platinumatoms are modelled by the multi-body embedded atom method. The nucleation rates inhighly supersaturated systems are estimated as well as properties of the critical clustersfrom the nucleation theorems. Furthermore, the coalescence of platinum nanoparticles isinvestigated and a coalescence activated structural transition in platinum nanoparticles isdescribed.

Determining the nucleation rate from the dimer growth probability

A new method is proposed for the determination of the stationary one-component nucleation rate J with the help of data for the growth probability P2 of a dimer which is the smallest cluster of the nucleating phase. The method is based on an exact formula relating J and P2, and is readily applicable to computer simulations of nucleation. Using the method, the dependence of J on the supersaturation s is determined by kinetic Monte Carlo simulations of two-dimensional (2D) nucleation of monolayers on the (100) face of Kossel crystal. The change of J over nearly 11 orders of magnitude is followed and it is found that the classical nucleation theory overestimates the simulation J values by an s-dependent factor. The 2D nucleus size evaluated via the nucleation theorem is described satisfactorily by the classical Gibbs-Thomson equation and its corrected version accounting for the spinodal limit of 2D nucleation.

Homogeneous nucleation rate measurements of 1-butanol in helium: A comparative study of a thermal diffusion cloud chamber and a laminar flow diffusion chamber

Isothermal homogeneous nucleation rates of 1-butanol were measured both in a thermal diffusion cloud chamber and in a laminar flow diffusion chamber built recently at the Institute of Chemical Process Fundamentals, Academy of Sciences of the Czech Republic, Prague, Czech Republic. The chosen system 1-butanol-helium can be studied reasonably well in both devices, in the overlapping range of temperatures. The results were compared with those found in the literature and those measured by Lihavainen in a laminar flow diffusion chamber of a similar design. The same isotherms measured with the thermal diffusion cloud chamber occur at highest saturation ratios of the three devices. Isotherms measured with the two laminar flow diffusion chambers are reasonably close together; the measurements by Lihavainen occur at lowest saturation ratios. The temperature dependences observed were similar in all three devices. The molecular content of critical clusters was calculated using the nucleation theorem and compared with the Kelvin equation. Both laminar flow diffusion chambers provided very similar sizes slightly above the Kelvin equation, whereas the thermal diffusion cloud chamber suggests critical cluster sizes significantly smaller. The results found elsewhere in the literature were in reasonable agreement with our results.

Homogeneous nucleation of iron from supersaturated vapor investigated by molecular dynamics simulation

The nucleation of particles from a supersaturated iron vapor is investigated by molecular dynamics simulations. The supersaturated iron vapor is embedded in an argon heat bath which removes the heat generated during the nucleation process. The nucleation rates are calculated from the cluster size statistics with a method proposed by Yasuoka and Matsumoto. Within this analysis no assumptions concerning the size of the critical cluster are required. The obtained values are consistent with the experimental data of iron nucleation rates. By applying the first nucleation theorem the size of the critical cluster is determined from nucleation rate data. It is in the order of one to three atoms which agrees with the observations of the cluster size fluctuations during the nucleation process. The excess energy of the critical cluster is estimated by the second nucleation theorem.

Nucleation near the spinodal: limitations of mean field density functional theory.

We investigate the diverging size of the critical nucleus near the spinodal using the gradient theory (GT) of van der Waals and Cahn and Hilliard and mean field density functional theory (MFDFT). As is well known, GT predicts that at the spinodal the free energy barrier to nucleation vanishes while the radius of the critical fluctuation diverges. We show numerically that the scaling behavior found by Cahn and Hilliard for these quantities holds quantitatively for both GT and MFDFT. We also show that the excess number of molecules Deltag satisfies Cahn-Hilliard scaling near the spinodal and is consistent with the nucleation theorem. From the latter result, it is clear that the divergence of Deltag is due to the divergence of the mean field isothermal compressibility of the fluid at the spinodal. Finally, we develop a Ginzburg criterion for the validity of the mean field scaling relations. For real fluids with short-range attractive interactions, the near-spinodal scaling behavior occurs in a fluctuation dominated regime for which the mean field theory is invalid. Based on the nucleation theorem and on Wang's treatment of fluctuations near the spinodal in polymer blends, we infer a finite size for the critical nucleus at the pseudospinodal identified by Wang.

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.

Mesoscopic nonequilibrium kinetics of nucleation processes

This article reviews our recent efforts to provide a rigorous description of the kinetics of nucleation processes from a mesoscopic and nonequilibrium perspective. Its usefulness to analyze different important aspects of the process, such as the validity of the Nucleation Theorem, the proper incorporation of translational and rotational degrees of freedom, and the influence of inhomogeneities and gradients in the process, is discussed. The theory sketched here provides a unified framework in which nucleation can be realistically described.

Correlations for Vapor Nucleating Critical Embryo Parameters

Condensation nucleation light scattering detection in principle works by converting the effluent of the chromatographic separation into an aerosol and then selectively evaporating the mobile phase, leaving less volatile analytes and nonvolatile impurities as dry aerosol particles. The dry particles produced are then exposed to an environment that is saturated with the vapors of an organic solvent (commonly n-butanol). The blend of aerosol particles and organic vapor is then cooled so that conditions of vapor supersaturation are achieved. In principle, the vapor then condenses onto the dry particles, growing each particle (ideally) from as small as a few nanometers in diameter into a droplet with a diameter up to about 10 μm. The grown droplets are then passed through a beam of light, and the light scattered by the droplets is detected and used as the detector response. This growth and detection step is generally carried out using commercial continuous-flow condensation nucleus counters. In the present research, the possibility of using other fluids than the commonly used n-butanol is investigated. The Kelvin equation and the Nucleation theorem [Anisimov et al. (1978)] are used to evaluate a range of fluids for efficacy of growing small particles by condensation nucleation. Using the available experimental data on vapor nucleation, the correlations of Kelvin diameters (the critical embryo sizes) and the bulk surface tension with dielectric constants of working liquids are found. A simple method for choosing the most efficient fluid, within a class of fluids, for growth of small particles is suggested.

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.

Formal study of nucleation as described by fluctuation theory

A formal description of the nucleation process with out appeal to any geometrical cluster definition is attempted. This need is avoided by simply specifying the system size and monitoring a global order parameter. Consistent with this assumption, systematic comparison between fluctuation theory and classical thermodynamics leads to the equal-area and equal-height rules for phase coexistence and to a general relation between the surface free energy and fluctuations. An alternative definition for the surface free energy which may be measured by means of a simple thermodynamic integration path arises naturally from the formalism. A proof of the so-called “nucleation theorem” is given. A general expression for the nucleation barrier is obtained, which when applied to a capillary drop model explains a recently observed scaling law. The nucleation barrier as obtained from simulations is seen to be systematically smaller than that predicted by classical nucleation theory. It is argued that a microscopic phase transition, where the homogeneous phase is transformed into an inhomogeneous phase with no free energy cost, is responsible for this behavior.

Kinetic extensions of the nucleation theorem

Kinetic extensions of the nucleation theorem (KNT) are derived using the law of mass action and detailed balance. Results are obtained for the first- and higher-order derivatives of the nucleation rate, J, with change in supersaturation, S, in terms of the cumulants, κn, of a molecular distribution of reciprocal equilibrium cluster growth rates. At constant temperature we find d ln J/d ln S=κ1+1, an exact formulation of the nucleation theorem in terms of nucleation rate, and the extension dn ln J/d(ln S)n=(−1)n+1κn for the higher-order derivatives (n⩾2). The case n=2 is related to the Kelvin relation. Analysis of recent water vapor nucleation rates [Wölk and Strey, J. Phys. Chem. B 105, 11683 (2001)] provides molecular-based estimates for κ1 and κ2 suitable for comparison with the predictions of classical nucleation theory. The KNT is applied to ion-induced nucleation from the gas phase, by a sequence of reversible chemical reactions, and extensions to multistep kinetics and multicomponent nucleation are presented. Nucleation theorems enable one to deduce molecular-level properties directly from macroscopic rate measurements. Here we show these properties are not those of a single cluster, the critical nucleus, as approximate forms of the theorems would suggest, but instead are averages over a weighted distribution of clusters near critical size.

Homogeneous bubble nucleation in liquids: The classical theory revisited

The classical theory of homogeneous bubble nucleation is reconsidered by employing a phenomenological nucleation barrier in the capillarity approximation that utilizes the superheat threshold achieved in experiments. Consequently, an algorithm is constructed for the evaluation of the superheat temperatures in homogeneous boiling (tensile strengths in cavitation), the critical radii and steady-state nucleation rates. The nucleation theorem is written in this framework and is applied to the classical theory of homogeneous bubble nucleation for the phenomenological nucleation barrier employed. The superheat temperatures calculated show excellent agreement over a wide range of liquid pressures for most of the substances investigated. The steady-state nucleation rates are also altered by many orders of magnitude, in agreement with the results of previous investigators using different approaches.

Verification of the nucleation theorem for the internally constrained one-dimensional hard rod fluid

The generalized nucleation theorem as derived by Bowles et al. [J. Chem. Phys. 115, 1853 (2001)] had been verified for nonuniformities generated by a cavity and a cluster within the one-dimensional hard rod fluid. We further verify the nucleation theorem for nonuniformities created by a cavity and by a cluster containing m particles within the internally constrained hard rod fluid. The given results provide additional evidence of the validity of the generalized nucleation theorem even down to the molecular level.

Erratum: “A theorem for inhomogeneous systems: The generalization of the nucleation theorem” [J. Chem. Phys. 115, 1853 (2001)

Erratum: “A theorem for inhomogeneous systems: The generalization of the nucleation theorem” [J. Chem. Phys. <b>115</b>, 1853 (2001)

Properties of Ice Clusters from an Analysis of Freezing Nucleation

Properties of the critical clusters that are instrumental in the nucleation of a first-order phase transition can be obtained through the analysis of nucleation rate data. The theoretical tools for doing so are known as nucleation theorems. We consider solid-phase critical clusters forming from a liquid phase, and it is shown that their size and binding energies can be extracted from data gathered at a range of temperatures and pressures. Ice crystal nucleation rates described recently are used to study ice clusters containing several hundred water molecules.

High Pressure Nucleation Experiments in Binary and Ternary Mixtures

New experimental high-pressure nucleation results of n-nonane in methane and first nucleation rate data of water in methane are presented. The nucleation behavior of the ternary system water and n-nonane in methane is investigated experimentally. For this purpose, a new dedicated mixture preparation device was designed, rendering more accurate and controllable vapor fractions. This has eliminated the large scatter in the nucleation rates of n-nonane in methane at 40 bar and 240 K, previously measured. These experiments also show that nucleation theories are unable to predict the rates (even qualitatively) at these extreme conditions. The composition of the critical cluster is determined for all of the binary systems investigated, by applying the nucleation theorem. Furthermore, a criterion for the application of one component theories to binary gas-vapor systems is applied. The nucleation data for the ternary mixtures are analyzed using the experimental results for the binary mixtures. From this, it can b...