Homogeneous nucleation of water between 200 and 240 K: New wave tube data and estimation of the Tolman length

Second inflection point of supercooled water surface tension induced by hydrogen bonds: A molecular-dynamics study

Surface tension of supercooled water is a fundamental property in various scientific processes. In this study, we perform molecular dynamics simulations with the TIP4P-2005 model to investigate the surface tension of supercooled water down to 220 K. Our results show a second inflection point (SIP) in the surface tension at temperature TSIP ≈ 267.5 ± 2.3 K. Using an extended IAPWS-E functional fit for the water surface tension, we calculate the surface excess internal-energy and entropy terms of the excess Helmholtz free energy. Similar to prior studies [Wang et al., Phys. Chem. Chem. Phys. 21, 3360 (2019); Gorfer et al., J. Chem. Phys. 158, 054503 (2023)], our results show that the surface tension is governed by two driving forces: a surface excess entropy change above the SIP and a surface excess internal-energy change below it. We study hydrogen-bonding near the SIP because it is the main cause of water's anomalous properties. With decreasing temperature, our results show that the entropy contribution to the surface tension reaches a maximum slightly below the SIP and then decreases. This is because the number of hydrogen bonds increases more slowly below the SIP. Moreover, the strengths and lifetimes of the hydrogen bonds also rise dramatically below the SIP, causing the internal-energy term to dominate the excess surface free energy. Thus, the SIP in the surface tension of supercooled TIP4P-2005 water is associated with an increase in the strengths and lifetimes of hydrogen bonds, along with a decrease in the formation rate (#/K) of new hydrogen bonds.

Acoustic Droplet Vaporization of Perfluorohexane Emulsions Induced by Heterogeneous Nucleation at an Ultrasonic Frequency of 1.1 MHz

Droplets made of liquid perfluorocarbon undergo a phase transition and transform into microbubbles when triggered by ultrasound of intensity beyond a critical threshold; this mechanism is called acoustic droplet vaporization (ADV). It has been shown that if the intensity of the signal coming from high ultrasonic harmonics are sufficiently high, superharmonic focusing is the mechanism leading to ADV for large droplets (>3 μm) and high frequencies (>1.5 MHz). In such a scenario, ADV is initiated due to a nucleus occurring at a specific location inside the droplet volume. But the question on what induces ADV in the case of nanometer-sized droplets and/or at low ultrasonic frequencies (<1.5 MHz) still remains. We investigated ADV of perfluorohexane (PFH) nano- and microdroplets at a frequency of 1.1 MHz and at conditions where there is no superharmonic focusing. Three types of droplets produced by microfluidics were studied: plain PFH droplets, PFH droplets containing many nanometer-sized water droplets, and droplets made of a PFH corona encapsulating a single micron-sized water droplet. The probability to observe a vaporization event was measured as a function of acoustic pressure. As our experiments were performed on droplet suspensions containing a population of monodisperse droplets, we developed a statistical model to extrapolate, from our experimental curves, the ADV pressure thresholds in the case where only one droplet would be insonified. We observed that the value of ADV pressure threshold decreases as the radius of a plain PFH droplet increases. This value was further reduced when a PFH droplet encapsulates a micron-sized water droplet, while the encapsulation of many nanometer-sized water droplets did not modify the threshold. These results cannot be explained by a model of homogeneous nucleation. However, we developed a heterogeneous nucleation model, where the nucleus appears at the surface in contact with PFH, that successfully predicts our experimental ADV results.

Hierarchical approximations to the nucleation work in the entire range of metastability

The work W to form a nucleus (also known as the critical nucleus) is a key quantity in the description of nucleation phenomena because of its exponentially strong effect on the nucleation rate. The present study provides a general approximate expression for W, which comprises a hierarchy of approximations to the dependence of W on the experimentally controlled overpressure Δp of a nucleating multicomponent phase. This general expression is used to derive explicit formulas for the lowest-order members of the W(Δp) hierarchy as well as for the respective lowest-order approximations to the Δp dependences of the nucleus surface tension, the nucleus radius, the Gibbs-Tolman length, and the stationary nucleation rate. The second-order and the third-order approximations to the W(Δp) dependence are confronted with available W(Δp) data, and the latter is found to agree very well with the data. The results obtained are applicable to homogeneous single-component or multicomponent nucleation from the binodal to the spinodal of the old phase, i.e., in the entire range of the old-phase metastability.

Probing the Free Energy of Small Water Clusters: Revisiting Classical Nucleation Theory

By addressing the defects in classical nucleation theory (CNT), we develop an approach for extracting the free energy of small water clusters from nucleation rate experiments without any assumptions about the form of the cluster free energy. For temperatures higher than ∼250 K, the extracted free energies from experimental data points indicate that their ratio to the free energies predicted by CNT exhibits nonmonotonic behavior as the cluster size changes. We show that this ratio increases from almost zero for monomers and passes through (at least) one maximum before approaching one for large clusters. For temperatures lower than ∼250 K, the behavior of the ratio between extracted energies and CNT's prediction changes; it increases with cluster size, but it remains below one for almost all of the experimental data points. We also applied a state-of-the-art quantum mechanics model to calculate free energies of water clusters (2-14 molecules); the results support the observed change in behavior based on temperature, albeit for temperatures above and below ∼298 K. We compared two different model chemistries, DLPNO-CCSD(T)/CBS//ωB97xD/6-31++G** and G3, against each other and the experimental value for formation of the water dimer.

Molecular dynamics study of wetting of alkanes on water: from high temperature to the supercooled region and the influence of second inflection points of interfacial tensions

To explore the wetting behavior of alkanes on bulk water interfaces, molecular dynamics (MD) simulations were carried out for united-atom PYS alkane models, and for SPC/E and TIP4P/2005 water models over a wide temperature range. The MD results at each temperature were used to find (1) the surface tension of the alkanes (octane, nonane) and water, and (2) the interfacial tensions of the alkane-water systems. These quantities were then used to calculate the spreading coefficient (S) and contact angle (θc) for each alkane on water. At higher temperatures, the contact angle of octane and nonane on water is found to behave in accord with conventional expectations, i.e., it decreases with increasing temperature for both water models as each system approaches the usual high-temperature transition to perfect wetting. At lower temperatures, we found an unusual temperature dependence of S and θc for each PYS alkane on SPC/E water. In contrast to conventional expectations, θc decreases with a decrease in the temperature. For octane-SPC/E water, this unusual behavior of θc occurs due to the presence of second inflection points (SIP) in the vapor-water and the octane-water interfacial tensions, whereas the SIP effect is much less important for the nonane-water system. The unusual temperature dependence of θc observed for nonane on SPC/E water is also found for nonane on TIP4P/2005 water. On the other hand, such unusual wetting behavior has not been observed in the PYS octane-TIP4P/2005 water system, except possibly for the two lowest temperatures studied.

Homogeneous water nucleation in carbon dioxide-nitrogen mixtures: Experimental study on pressure and carrier gas effects

New homogeneous nucleation experiments are presented at 240 K for water in carrier gas mixtures of nitrogen with carbon dioxide molar fractions of 5%, 15%, and 25%. The pulse expansion wave tube is used to test three different pressure conditions, namely, 0.1, 1, and 2 MPa. In addition, a restricted series of nucleation experiments is presented for 25% carbon dioxide mixtures at temperatures of 234 and 236 K at 0.1 MPa. As pressure and carbon dioxide content are increased, the nucleation rate increases accordingly. This behavior is attributed to the reduction in the water surface tension by the adsorption of carrier gas molecules. The new data are compared with theoretical predictions based on the classical nucleation theory and on extrapolations of empirical surface tension data to the supercooled conditions at 240 K. The extrapolation is carried out on the basis of a theoretical adsorption/surface tension model, extended to multi-component mixtures. The theoretical model appears to strongly overestimate the pressure and composition dependence. At relatively low pressures of 0.1 MPa, a reduction in the nucleation rates is found due to an incomplete thermalization of colliding clusters and carrier gas molecules. The observed decrease in the nucleation rate is supported by the theoretical model of Barrett, generalized here for water in multi-component carrier gas mixtures. The temperature dependence of the nucleation rate at 0.1 MPa follows the scaling model proposed by Hale [J. Chem. Phys. 122, 204509 (2005)].

Homogeneous water nucleation: Experimental study on pressure and carrier gas effects

Homogeneous nucleation of water is investigated in argon and in nitrogen at about 240 K and 0.1 MPa, 1 MPa, and 2 MPa by means of a pulse expansion wave tube. The surface tension reduction at high pressure qualitatively explains the observed enhancement of the nucleation rate of water in argon as well as in nitrogen. The differences in nucleation rates for the two mixtures at high pressure are consistent with the differences in adsorption behavior of the different carrier gas molecules. At low pressure, there is not enough carrier gas available to ensure the growing clusters are adequately thermalized by collisions with carrier gas molecules so that the nucleation rate is lower than under isothermal conditions. This reduction depends on the carrier gas, pressure, and temperature. A qualitative agreement between experiments and theory is found for argon and nitrogen as carrier gases. As expected, the reduction in the nucleation rates is more pronounced at higher temperatures. For helium as the carrier gas, non-isothermal effects appear to be substantially stronger than predicted by theory. The critical cluster sizes are determined experimentally and theoretically according to the Gibbs-Thomson equation, showing a reasonable agreement as documented in the literature. Finally, we propose an empirical correction of the classical nucleation theory for the nucleation rate calculation. The empirical expression is in agreement with the experimental data for the analyzed mixtures (water-helium, water-argon, and water-nitrogen) and thermodynamic conditions (0.06 MPa-2 MPa and 220 K-260 K).

Interfacial thermodynamics of spherical nanodroplets: molecular understanding of surface tension via a hydrogen bond network

Surface tension plays a ubiquitous role in phase transitions including condensation or evaporation of atmospheric liquid droplets. In particular, understanding of interfacial thermodynamics of the critical nucleus of 1 nm scale is important for molecular characterization of the activation energy barrier of nucleation. Here, we investigate surface tension of spherical nanodroplets with both molecular dynamics and density functional theory and find that surface tension decreases appreciably below 1 nm radius, whose analytical expression is consistently derived from the classic Tolman's equation. In particular, the free energy analysis of nanodroplets shows that the change of surface tension originates dominantly from the configurational energy of interfacial molecules, which is evidenced by the increasingly disrupted hydrogen bond network as the droplet size decreases. Our result can be applied to the interface-related phenomena associated with molecular fluctuations such as biomolecule adsorption at the sub-nm scale where macroscopic thermodynamic quantities are ill-defined.

Nucleation work, surface tension, and Gibbs-Tolman length for nucleus of any size

In the framework of the Gibbs approach to nucleation thermodynamics, expressions are derived for the nucleation work, nucleus size, surface tension, and Gibbs-Tolman length in homogeneous single-component nucleation at a fixed temperature. These expressions are in terms of the experimentally controlled overpressure of the nucleating phase and are valid for the entire overpressure range, i.e., for nucleus of any size. Analysis of available data for bubble and droplet nucleation in Lennard-Jones fluid shows that the theory describes well the data by means of a single free parameter, the Gibbs-Tolman length of the planar liquid/vapor interface. It is found that this length is about one-tenth of the Lennard-Jones molecular-diameter parameter and that it is positive for the bubble nucleus and negative for the droplet nucleus. In a sufficiently narrow temperature range, the nucleation work, nucleus radius, scaled surface tension, and Gibbs-Tolman length are apparently universal functions of scaled overpressure.

Ice-Crystal Nucleation in Water: Thermodynamic Driving Force and Surface Tension. Part I: Theoretical Foundation

A recently developed thermodynamic theory for the determination of the driving force of crystallization and the crystal–melt surface tension is applied to the ice-water system employing the new Thermodynamic Equation of Seawater TEOS-10. The deviations of approximative formulations of the driving force and the surface tension from the exact reference properties are quantified, showing that the proposed simplifications are applicable for low to moderate undercooling and pressure differences to the respective equilibrium state of water. The TEOS-10-based predictions of the ice crystallization rate revealed pressure-induced deceleration of ice nucleation with an increasing pressure, and acceleration of ice nucleation by pressure decrease. This result is in, at least, qualitative agreement with laboratory experiments and computer simulations. Both the temperature and pressure dependencies of the ice-water surface tension were found to be in line with the le Chatelier–Braun principle, in that the surface tension decreases upon increasing degree of metastability of water (by decreasing temperature and pressure), which favors nucleation to move the system back to a stable state. The reason for this behavior is discussed. Finally, the Kauzmann temperature of the ice-water system was found to amount TK=116K, which is far below the temperature of homogeneous freezing. The Kauzmann pressure was found to amount to pK=−212MPa, suggesting favor of homogeneous freezing on exerting a negative pressure on the liquid. In terms of thermodynamic properties entering the theory, the reason for the negative Kauzmann pressure is the higher mass density of water in comparison to ice at the melting point.

Extraction of monomer-cluster association rate constants from water nucleation data measured at extreme supersaturations

We utilize recently reported data for water nucleation in the uniform postnozzle flow of pulsed Laval expansions to derive water monomer association rates with clusters. The nucleation experiments are carried out at flow temperatures of 87.0 K and 47.5 K and supersaturations of lnS ∼ 41 and 104, respectively. The cluster size distributions are measured at different nucleation times by mass spectrometry coupled with soft single-photon ionization at 13.8 eV. The soft ionization method ensures that the original cluster size distributions are largely preserved upon ionization. We compare our experimental data with predictions by a kinetic model using rate coefficients from a previous ab initio calculation with a master equation approach. The prediction and our experimental data differ, in particular, at the temperature of 87.0 K. Assuming cluster evaporation to be negligible, we derive association rate coefficients between monomer and clusters purely based on our experimental data. The derived dimerization rate lies 2-3 orders of magnitude below the gas kinetic collision limit and agrees with the aforementioned ab initio calculation. Other than the dimerization rate, however, the derived rate coefficients between monomer and cluster j (j ≥ 3) are on the same order of magnitude as the kinetic collision limit. A kinetic model based on these results confirms that coagulation is indeed negligible in our experiments. We further present a detailed analysis of the uncertainties in our experiments and methodology for rate derivation and specify the dependency of the derived rates on uncertainties in monomer and cluster concentrations.

An improved model of homogeneous nucleation for high supersaturation conditions: aluminum vapor

A novel model of stationary nucleation, treating the thermodynamic functions of small clusters, has been built. The model is validated against the experimental data on the nucleation rate of water vapor obtained in a broad range of supersaturation values (S = 10-120), and, at high supersaturation values, it reproduces the experimental data much better than the traditional classical nucleation model. A comprehensive analysis of the nucleation of aluminum vapor with the usage of developed stationary and non-stationary nucleation models has been performed. It has been shown that, at some value of supersaturation, there exists a double potential nucleation barrier. It has been revealed that the existence of this barrier notably delayed the establishment of a stationary distribution of subcritical clusters. It has also been demonstrated that the non-stationary model of the present work and the model of liquid-droplet approximation predict different values of nucleation delay time, τ_s. In doing so, the liquid-droplet model can underestimate notably (by more than an order of magnitude) the value of τ_s.

Curvature Dependence of the Liquid-Vapor Surface Tension beyond the Tolman Approximation

Surface tension is a macroscopic manifestation of the cohesion of matter, and its value σ_{∞} is readily measured for a flat liquid-vapor interface. For interfaces with a small radius of curvature R, the surface tension might differ from σ_{∞}. The Tolman equation, σ(R)=σ_{∞}/(1+2δ/R), with δ a constant length, is commonly used to describe nanoscale phenomena such as nucleation. Here we report experiments on nucleation of bubbles in ethanol and n-heptane, and their analysis in combination with their counterparts for the nucleation of droplets in supersaturated vapors, and with water data. We show that neither a constant surface tension nor the Tolman equation can consistently describe the data. We also investigate a model including 1/R and 1/R^{2} terms in σ(R). We describe a general procedure to obtain the coefficients of these terms from detailed nucleation experiments. This work explains the conflicting values obtained for the Tolman length in previous analyses, and suggests directions for future work.

Size-Dependent Surface Free Energy and Tolman-Corrected Droplet Nucleation of TIP4P/2005 Water

Classical nucleation theory is notoriously inaccurate when using the macroscopic surface free energy for a planar interface. We examine the size dependence of the surface free energy for TIP4P/2005 water nanodroplets (radii ranging from 0.7 to 1.6 nm) at 300 K with the mitosis method, that is, by reversibly splitting the droplets into two subclusters. We calculate the Tolman length to be -0.56 ± 0.09 Å, which indicates that the surface free energy of water droplets that we investigated is 5-11 mJ/m(2) greater than the planar surface free energy. We incorporate the computed Tolman length into a modified classical nucleation theory (δ-CNT) and obtain modified expressions for the critical nucleus size and barrier height. δ-CNT leads to excellent agreement with independently measured nucleation kinetics.

Homogeneous water nucleation and droplet growth in methane and carbon dioxide mixtures at 235 K and 10 bar

Homogeneous nucleation rates and droplet growth rates of water in pure methane and mixtures of methane and carbon dioxide were measured in an expansion wave tube at 235 K and 10 bar. The nucleation rate in pure methane is three orders of magnitude higher than literature nucleation rates of water in low-pressure helium or argon. Addition of carbon dioxide to the carrier gas mixture increases the rates even more. Specifically, rates in a mixture of methane and 3% carbon dioxide are a factor of 10 higher than the rates in pure methane. With 25% carbon dioxide, the rates are four orders of magnitude higher than the rates in pure methane. An application of the nucleation theorem shows that the critical cluster consists of 22 water molecules and 5 methane molecules, for nucleation in pure methane. Growth rates of water droplets were measured in methane and in methane-carbon dioxide mixtures at 243 K and 11.5 bar. At equal temperature, pressure and water vapor fraction, the growth rate of the squared droplet radius is about 20% lower in the mixture with 25% carbon dioxide than in pure methane. The lower growth rate is caused by a smaller diffusion coefficient of water in the mixture with carbon dioxide; the difference of the diffusion coefficients is qualitatively reproduced by the empirical Fuller correlation combined with Blanc's law.

Quantum-mechanical solution to fundamental problems of classical theory of water vapor nucleation

The inconsistent temperature dependence of nucleation rates, disagreement of theoretical critical or onset supersaturations with experimental data, and insufficiently accurate predictions of nucleation rates are fundamental problems of the classical nucleation theory (CNT) of water vapors, which is a foundation of various multicomponent nucleation models widely used in the aerosol microphysics, physical chemistry, and chemical technology. In the present study, a correction to the CNT obtained from "first principles" has been derived and significant progress has been made in solving the fundamental problem of predicting nucleation rates of water vapors. The modified model with the quantum-mechanical correction incorporated is in very good agreement with experiments over the full range of temperatures (T=210-290K) , saturation ratios (S=2-100) , and nucleation rates (J= approximately 10{1}-10{17} cm-3).

Observation of a transition in the water-nanoparticle formation process at 167K

Rapid-scan Fourier transform infrared spectroscopy of the vapor/solid formation process of water nanoparticles in the 180-140 K temperature range at thermal-equilibrium conditions is reported. At 167 K a transition in the formation process was observed: the particle volume quintuples and the particle formation time triples within a temperature interval of +/-0.4 K caused by the temperature control. The authors interpret this behavior by an abrupt change in the nucleation rate of the H2O monomers in He buffer gas kept at 167 K and 200 mbar. A size and shape analysis of the particles during the formation process was carried out by application of the discrete dipole approximation method which delivers excellent accordance between experimental and calculated mid-IR spectra. Compared to other compact shapes (cube, prolate ellipsoid, and hexagonal prism) the ideal spherical shape fits the experimental spectra best. A distinct change in shape by particle conversion or agglomeration could be excluded to be involved in the formation process. As a possible explanation of the observed phenomenon, a transition from vapor/liquid/solid to vapor/solid nucleation with decreasing temperature is considered which was recently theoretically predicted by van Dongen and co-workers [J. Chem. Phys. 117, 5647 (2002); private communication; J. Chem. Phys. 120, 6314 (2004)].

Origin of the failure of classical nucleation theory: incorrect description of the smallest clusters

We carry out molecular Monte Carlo simulations of clusters in an imperfect vapor. We show that down to very small cluster sizes, classical nucleation theory built on the liquid drop model can be used very accurately to describe the work required to add a monomer to the cluster. However, the error made in modeling the smallest of clusters as liquid drops results in an erroneous absolute value for the cluster work of formation throughout the size range. We calculate factors needed to correct the cluster formation work given by the liquid drop model. The corrected work of formation results in nucleation rates in good agreement with recent nucleation experiments on argon and water.

Analysis of experimental data for the nucleation rate of water droplets

A formula for the stationary nucleation rate J is proposed and used for analysis of experimental data for the dependence of J on the supersaturation ratio S in isothermal homogeneous nucleation of water droplets in vapors. It is found that the experimental data are described quite successfully by the proposed formula which is based on (i) the Gibbs presentation of the nucleation work in terms of overpressure, (ii) the Girshick-Chiu [J. Chem. Phys. 93, 1273 (1990); 94, 826 (1991)] self-consistency correction to the equilibrium cluster size distribution, and (iii) the Reguera-Rubi [J. Chem. Phys. 115, 7100 (2001)] kinetic accounting of the nucleus translational-rotational motion. The formula, like that of Wolk and Strey [J. Phys. Chem. B 105, 11683 (2001)], could be used as a semiempirical relation describing the J(S) dependence for nucleation in vapors of single-component droplets or crystals of substances with insufficiently well known equations of state.