Homogeneous nucleation rates for water

Thermodynamic Theory of Curvature-Dependent Surface Tension: Tolman's Theory Revisited

An exact equation for determining the Tolman length (TL) as a function of radius is obtained within the framework of classical thermodynamics and a computational procedure for solving it is proposed. As a result of implementing this procedure, the dependences of the TL and surface tension on radius are obtained for the drop and bubble cases and various equations of state. As one of the results of the thermodynamic study, a new equation for the dependence of surface tension on radius (curvature effect) alternative to the corresponding Tolman equation and associated with the spinodal point is obtained. The fundamental impossibility to determine the curvature effect analytically from the binodal point, i.e., using the Tolman equation, is established; it is calculated only from the spinodal point and is determined by the characteristics of the system at this point. The sign of the TL asymptotic value debated in the literature in recent decades is uniquely determined in the theory: it is negative for drops and positive for bubbles.

Electric Field-Induced Water Condensation Visualized by Vapor-Phase Transmission Electron Microscopy

Understanding the nanoscale water condensation dynamics in strong electric fields is important for improving the atmospheric modeling of cloud dynamics and emerging technologies utilizing electric fields for direct air moisture capture. Here, we use vapor-phase transmission electron microscopy (VPTEM) to directly image nanoscale condensation dynamics of sessile water droplets in electric fields. VPTEM imaging of saturated water vapor stimulated condensation of sessile water nanodroplets that grew to a size of ∼500 nm before evaporating over a time scale of a minute. Simulations showed that electron beam charging of the silicon nitride microfluidic channel windows generated electric fields of ∼10⁸ V/m, which depressed the water vapor pressure and effected rapid nucleation of nanosized liquid water droplets. A mass balance model showed that droplet growth was consistent with electric field-induced condensation, while droplet evaporation was consistent with radiolysis-induced evaporation via conversion of water to hydrogen gas. The model quantified several electron beam-sample interactions and vapor transport properties, showed that electron beam heating was insignificant, and demonstrated that literature values significantly underestimated radiolytic hydrogen production and overestimated water vapor diffusivity. This work demonstrates a method for investigating water condensation in strong electric fields and under supersaturated conditions, which is relevant to vapor-liquid equilibrium in the troposphere. While this work identifies several electron beam-sample interactions that impact condensation dynamics, quantification of these phenomena here is expected to enable delineating these artifacts from the physics of interest and accounting for them when imaging more complex vapor-liquid equilibrium phenomena with VPTEM.

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.

Experimental investigation of homogeneous nucleation of water in various gases using an expansion chamber

Nucleation is an integral part of phase transitions and plays an important role in technology. The control of nucleation in devices such as turbines, rocket and jet engines, wind tunnels, and combustion processes is key to achieving efficient, ecologically sound operation. Our paper presents information about a modernized experimental setup used for homogeneous water nucleation research. The main goal of the study is an analysis of the influence of carrier gas on homogeneous nucleation. The experimental setup based on an expansion chamber has been used for research of homogeneous nucleation over several decades and has undergone various technical improvements. In the Institute of Thermomechanics, the setup was been completely disassembled and cleaned. Pressure transducers, laser, and the data acquisition system have been replaced with modern parts. After that, the chamber was assembled, and thoroughly tested and calibrated. Our investigations were carried with argon and nitrogen as carrier gases in a range of nucleation temperatures 220 – 260 K, pressure range 80-150 kPa, at several concentrations of water vapour. The results thus obtained are consistent with literature data. The findings suggest that this experimental approach is useful for homogeneous water nucleation research.

Possible Anomaly in the Surface Tension of Supercooled Water: New Experiments at Extreme Supercooling down to -31.4 °C

The surface tension of water is suspected to show a substantial increase at low temperatures, which is considered to be one of the many anomalies of water. The second inflection point (SIP) anomaly, originally claimed to be at around -8 °C, was experimentally refuted down to -25 °C by Hrubý et al. (J. Phys. Chem. Lett. 2014, 5, 425-428). Recent molecular simulations predict the SIP anomaly near or even below the homogeneous freezing limit of around -38 °C. To contribute to an ongoing discussion about the SIP anomaly, new experiments focused on extreme levels of supercooling were carried out in this study. Unique experimental data down to -31.4 °C were collected using two measuring techniques based on the capillary rise method. A significant deviation from the extrapolated IAPWS formulation R1-76(2014) for surface tension of ordinary water was detected below -20 °C. Contrary to previous data, new experiments provide room for an anomaly in the course of surface tension in the deeply supercooled region.

Microcanonical Nucleation Theory for Anisotropic Materials Validated on Alumina Clusters

Nucleation kinetics in gas phase remains an open issue with no general model. The derivation of the reaction constants assuming a canonical ensemble fails to describe anisotropic materials such as oxides. We have developed a general and versatile model using activated complex kinetics with a microcanonical approach. This approach handles the kinetics issue in cluster growth when the transient nature of the processes hinders the use of the canonical ensemble. The model efficiently reproduces experimental size distributions of alumina clusters formed by laser ablation with different buffer gas densities, including magic numbers. We show that the thermodynamic equilibrium is not reached during the growth. The bounding energy measured is 10 times lower than the one deduced from DFT calculation, but also the one expected from the bulk cohesive energy.

Curvature Corrections Remove the Inconsistencies of Binary Classical Nucleation Theory

The study of nucleation in fluid mixtures exposes challenges beyond those of pure systems. A striking example is homogeneous condensation in highly surface-active water-alcohol mixtures, where classical nucleation theory yields an unphysical, negative number of water molecules in the critical embryo. This flaw has rendered multicomponent nucleation theory useless for many industrial and scientific applications. Here, we show that this inconsistency is removed by properly incorporating the curvature dependence of the surface tension of the mixture into classical nucleation theory for multicomponent systems. The Gibbs adsorption equation is used to explain the origin of the inconsistency by linking the molecules adsorbed at the interface to the curvature corrections of the surface tension. The Tolman length and rigidity constant are determined for several water-alcohol mixtures and used to show that the corrected theory is free of physical inconsistencies and provides accurate predictions of the nucleation rates. In particular, for the ethanol-water and propanol-water mixtures, the average error in the predicted nucleation rates is reduced from 11-15 orders of magnitude to below 1.5. The curvature-corrected nucleation theory opens the door to reliable predictions of nucleation rates in multicomponent systems, which are crucial for applications ranging from atmospheric science to research on volcanos.

Application of the Nucleation Theorem to Crystallization of Liquids: Some General Theoretical Results

Different aspects in applying the nucleation theorem to the description of crystallization of liquids are analyzed. It is shown that, by employing the classical Gibbs’ approach in the thermodynamic description of heterogeneous systems, a general form of the nucleation theorem can be formulated that is valid not only for one-component but generally for multi-component systems. In this analysis, one basic assumption of classical nucleation theory is utilized. In addition, commonly employed in application to crystallization, it is supposed that the bulk properties of the critical clusters are widely identical to the properties of the newly evolving crystal phase. It is shown that the formulation of the nucleation theorem as proposed by Kashchiev [J. Chem. Phys. 76, 5098-5102 (1982)], also relying widely on the standard classical approach in the description of crystal nucleation, holds for multi-component systems as well. The general form of the nucleation theorem derived by us is taken then as the starting point for the derivation of particular forms of this theorem for the cases that the deviation from equilibrium is caused by variations of either composition of the liquid phase, temperature, or pressure. In this procedure, expressions recently developed by us for the curvature dependence of the surface tension, respectively, its dependence on pressure and/or temperature are employed. The basic assumption of classical nucleation theory mentioned above is, however, in general, not true. The bulk and surface properties of the critical crystal clusters may differ considerably from the properties of the evolving macroscopic phases. Such effects can be incorporated into the theoretical description by the application of the generalized Gibbs approach for the specification of the dependence of the properties of critical crystal clusters on the degree of metastability of the liquid phase. Applying this method, it is demonstrated that a similar formulation of the nucleation theorem, as derived based on classical nucleation theory, holds true also in cases when a dependence of the state parameters of the critical clusters on the degree of deviation from equilibrium is appropriately accounted for.

Derivation of an exact, nonequilibrium framework for nucleation: Nucleation is a priori neither diffusive nor Markovian

We discuss the structure of the equation of motion that governs nucleation processes at first order phase transitions. From the underlying microscopic dynamics of a nucleating system, we derive by means of a nonequilibrium projection operator formalism the equation of motion for the size distribution of the nuclei. The equation is exact, i.e., the derivation does not contain approximations. To assess the impact of memory, we express the equation of motion in a form that allows for direct comparison to the Markovian limit. As a numerical test, we have simulated crystal nucleation from a supersaturated melt of particles interacting via a Lennard-Jones potential. The simulation data show effects of non-Markovian dynamics.

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.

Water nucleation in helium, methane, and argon: A molecular dynamics study

Nucleation of highly supersaturated water vapor in helium, methane, and argon carrier gases at 350 K was investigated using molecular dynamics simulations. Nucleation rates obtained from the mean first passage time (MFPT) method are typically one order of magnitude lower than those from the Yasuoka and Matsumoto method, which can be attributed to the overestimation of the critical cluster size in the MFPT method. It was found that faster nucleation will occur in carrier gases that have better thermalization properties such that latent heat is removed more efficiently. These thermalization properties are shown to be strongly dependent on the molecular mass and Lennard-Jones (LJ) parameters. By varying the molecular mass, for unaltered LJ parameters, it was found that a heavier carrier gas removes less heat although it has a higher collision rate with water than a lighter carrier. Thus, it was shown that a clear distinction between water vapor-carrier gas collisions and water cluster-carrier gas collisions is indispensable for understanding the effect of collision rates on thermalization. It was also found that higher concentration of carrier gas leads to higher nucleation rate. The nucleation rates increased by a factor of 1.3 for a doubled concentration and by almost a factor of two for a tripled concentration.

Surface tension of aqueous binary mixtures under the supercooled conditions – Development of the measuring technique and preliminary data for water + lower alcohols

An experimental apparatus originally developed for the measurement of surface tension of supercooled water was modified such that it allows for measurement of binary aqueous mixtures. The measuring principle based on the capillary rise technique is similar to that employed in the previous measurements with pure water [J. Hrubý et al., J. Phys. Chem. Lett. 5 (2014) 425 and V. Vinš et al., J. Phys. Chem. B 119 (2015) 5567]. The temperature dependence of the surface tension is determined from the measured height of the liquid column elevated in a vertical capillary tube with an inner diameter of 0.32 mm. The aqueous liquid rises to a height of 60 mm to 95 mm depending on the solute concentration and the temperature. Preliminary tests performed with the binary mixtures of water with methanol, ethanol, and n-propanol at various concentrations and temperatures down to –28 °C approved functionality of the measuring technique. Some difficulties, such as influence of impurities on the liquid column elevation or formation of bubbles in the liquid column, were observed. The experimental apparatus is further being modified in order to obtain more accurate data for various aqueous mixtures.

Targeted Nanoparticle Thermometry: A Method to Measure Local Temperature at the Nanoscale Point Where Water Vapor Nucleation Occurs

An optical nanothermometer technique based on laser trapping, moving and targeted attaching an erbium oxide nanoparticle cluster is developed to measure the local temperature. The authors apply this new nanoscale temperature measuring technique (limited by the size of the nanoparticles) to measure the temperature of vapor nucleation in water. Vapor nucleation is observed after superheating water above the boiling point for degassed and nondegassed water. The average nucleation temperature for water without gas is 560 K but this temperature is lowered by 100 K when gas is introduced into the water. The authors are able to measure the temperature inside the bubble during bubble formation and find that the temperature inside the bubble spikes to over 1000 K because the heat source (optically-heated nanorods) is no longer connected to liquid water and heat dissipation is greatly reduced.

Cavitation in a metallic liquid: homogeneous nucleation and growth of nanovoids

Large-scale molecular dynamics (MD) simulations are performed to investigate homogeneous nucleation and growth of nanovoids during cavitation in liquid Cu. We characterize in detail the atomistic cavitation processes by following the temporal evolution of cavities or voids, analyze the nucleation behavior with the mean first-passage time (MFPT) and survival probability (SP) methods, and discuss the results against classical nucleation theory (CNT), the Tolman equation for surface energy, independent calculation of surface tension via integrating the stress profiles, the Johnson-Mehl-Avrami (JMA) growth law, and the power law for nucleus size distributions. Cavitation in this representative metallic liquid is a high energy barrier Poisson processes, and the steady-state nucleation rates obtained from statistical runs with the MFPT and SP methods are in agreement. The MFPT method also yields the critical nucleus size and the Zeldovich factor. Fitting with the Tolman's equation to the MD simulations yields the surface energy of a planar interface (~0.9 J m⁻²) and the Tolman length (0.4-0.5 Å), and those values are in accord with those from integrating the stress profiles of a planar interface. Independent CNT predictions of the nucleation rate (10(33 - 34) s(-1) m(-3)) and critical size (3-4 Å in radius) are in agreement with the MFPT and SP results. The JMA law can reasonably describe the nucleation and growth process. The size distribution of subcritical nuclei appears to follow a power law with an exponent decreasing with increasing tension owing to coupled nucleation and growth, and that of the supercritical nuclei becomes flattened during further stress relaxation due to void coalescence.

Mesoscopic nucleation theory for confined systems: a one-parameter model

Classical nucleation theory has been recently reformulated based on fluctuating hydrodynamics [J. F. Lutsko and M. A. Durán-Olivencia, Classical nucleation theory from a dynamical approach to nucleation, J. Chem. Phys. 138, 244908 (2013). The present work extends this effort to the case of nucleation in confined systems such as small pores and vesicles. The finite available mass imposes a maximal supercritical cluster size and prohibits nucleation altogether if the system is too small. We quantity the effect of system size on the nucleation rate. We also discuss the effect of relaxing the capillary-model assumption of zero interfacial width resulting in significant changes in the nucleation barrier and nucleation rate.

Limit of metastability for liquid and vapor phases of water

We report the limits of superheating of water and supercooling of vapor from Monte Carlo simulations using microscopic models with configurational enthalpy as the order parameter. The superheating limit is well reproduced. The vapor is predicted to undergo spinodal decomposition at a temperature of Tspvap=46±10 °C (0 °C≪Tspvap≪100 °C) under 1 atm. The water-water network begins to form at the supercooling limit of the vapor. Three-dimensional water-water and cavity-cavity unbroken networks are interwoven at critically superheated liquid water; if either network breaks, the metastable state changes to liquid or vapor.

Monte Carlo tests of nucleation concepts in the lattice gas model

The conventional theory of homogeneous and heterogeneous nucleation in a supersaturated vapor is tested by Monte Carlo simulations of the lattice gas (Ising) model with nearest-neighbor attractive interactions on the simple cubic lattice. The theory considers the nucleation process as a slow (quasistatic) cluster (droplet) growth over a free energy barrier ΔF(*), constructed in terms of a balance of surface and bulk term of a critical droplet of radius R(*), implying that the rates of droplet growth and shrinking essentially balance each other for droplet radius R=R(*). For heterogeneous nucleation at surfaces, the barrier is reduced by a factor depending on the contact angle. Using the definition of physical clusters based on the Fortuin-Kasteleyn mapping, the time dependence of the cluster size distribution is studied for quenching experiments in the kinetic Ising model and the cluster size ℓ(*) where the cluster growth rate changes sign is estimated. These studies of nucleation kinetics are compared to studies where the relation between cluster size and supersaturation is estimated from equilibrium simulations of phase coexistence between droplet and vapor in the canonical ensemble. The chemical potential is estimated from a lattice version of the Widom particle insertion method. For large droplets it is shown that the physical clusters have a volume consistent with the estimates from the lever rule. Geometrical clusters (defined such that each site belonging to the cluster is occupied and has at least one occupied neighbor site) yield valid results only for temperatures less than 60% of the critical temperature, where the cluster shape is nonspherical. We show how the chemical potential can be used to numerically estimate ΔF(*) also for nonspherical cluster shapes.

On the interfacial thermodynamics of nanoscale droplets and bubbles

We present a new self-consistent thermodynamic formalism for the interfacial properties of nanoscale embryos whose interiors do not exhibit bulklike behavior and are in complete equilibrium with the surrounding mother phase. In contrast to the standard Gibbsian analysis, whereby a bulk reference pressure based on the same temperature and chemical potentials of the mother phase is introduced, our approach naturally incorporates the normal pressure at the center of the embryo as an appropriate reference pressure. While the interfacial properties of small embryos that follow from the use of these two reference pressures are different, both methods yield by construction the same reversible work of embryo formation as well as consistency between their respective thermodynamic and mechanical routes to the surface tension. Hence, there is no a priori reason to select one method over another. Nevertheless, we argue, and demonstrate via a density-functional theory (with the local density approximation) analysis of embryo formation in the pure component Lennard-Jones fluid, that our new method generates more physically appealing trends. For example, within the new approach the surface tension at all locations of the dividing surface vanishes at the spinodal where the density profile spanning the embryo and mother phase becomes completely uniform (only the surface tension at the Gibbs surface of tension vanishes in the Gibbsian method at this same limit). Also, for bubbles, the location of the surface of tension now diverges at the spinodal, similar to the divergent behavior exhibited by the equimolar dividing surface (in the Gibbsian method, the location of the surface of tension vanishes instead). For droplets, the new method allows for the appearance of negative surface tensions (the Gibbsian method always yields positive tensions) when the normal pressures within the interior of the embryo become less than the bulk pressure of the surrounding vapor phase. Such a prediction, which is allowed by thermodynamics, is consistent with the interpretation that the mother phase's attempted compression of the droplet is counterbalanced by the negative surface tension, or free energy cost to decrease the interfacial area. Furthermore, for these same droplets, the surface of tension can no longer be meaningfully defined (the surface of tension always remains well defined in the Gibbsian method). Within the new method, the dividing surface at which the surface tension equals zero emerges as a new lengthscale, which has various thermodynamic analogs to and similar behavior as the surface of tension.

Atomistic Characterization of Stochastic Cavitation of a Binary Metallic Liquid under Negative Pressure

We demonstrate the stochastic nature of cavitation in a binary metallic liquid Cu46Zr54 during hydrostatic expansion by employing molecular dynamics (MD) simulations using a quantum mechanics (QM)-derived potential. The activation volume is obtained from MD simulations and transition-state theory. Extrapolation of the pressure dependence of the activation volume from our MD simulations to low tensile pressure agrees remarkably with macroscale cavitation experiments. We find that classical nucleation theory can predict the cavitation rate if we incorporate the Tolman length derived from the MD simulations.