Free energy of cluster formation and a new scaling relation for the nucleation rate

Effects of Hydrogen Bond Networks on Viscosity in Aqueous Solutions

In aqueous solutions, the impact of ions on hydrogen bond networks plays a crucial role in transport properties. We used molecular dynamics simulations to explain how ions affect viscosity through structural changes. We developed a quantitative model to describe the effect of ions on viscosity. The model comprises two parts: the addition of ions alters hydrogen bond networks, and changes in hydrogen bond networks exponentially lead to changes in viscosity. The influence of ions on hydrogen bond networks involves the following mechanisms: first, ions can disrupt the tetrahedral structures within the first solvation shell into three-coordinated structures through substitution; second, structural changes within the first shells affect the global hydrogen bond network through electrostatic forces and the hindrance of ionic volumes. By analyzing the mechanisms of how hydrogen bond networks determine viscosity through the decomposition of viscosity, we found that the proportion of potential viscosity in aqueous solutions primarily increases due to the enhancement of non-hydrogen bonding interactions, and the proportion of hydrogen bonding viscosity decreases accordingly. Our results demonstrate that hydrogen bond networks are crucial for describing the changes in transport phenomena affected by external factors.

Multiscale approach to the theory of nonisothermal homogeneous nucleation

Molecular dynamics (MD) of the Lennard-Jones cluster in the environment of supersaturated vapor at fixed temperature and density is used for the investigation of nonisothermal nucleation. The results allow one to single out different processes occurring at different time scales, the Ornstein-Uhlenbeck fluctuations at the short time scale and a combination of slow diffusion and drift of the fluctuation packet that represents a cluster, at the long time scale. The multiscale approach is developed, in which a separate treatment of different time scales makes it possible to consider strongly correlated cluster size and temperature. This reduces the nonisothermal cluster evolution to a one-dimensional problem. The fluctuation packet drift velocity and diffusivity are calculated based on the cluster microscopic thermophysical parameters determined in this work from MD data for isothermal clusters. The proposed approach is consistent with the results of our MD simulation.

Heterogeneous Ion-Induced Nucleation of Water and Butanol Vapors Studied via Computational Quantum Chemistry beyond Prenucleation and Critical Cluster Sizes

Water and butanol are used as working fluids in condensation particle counters, and condensation of a single vapor onto an ion can be used as a simple model system for the study of ion-induced nucleation in the atmosphere. Motivated by this, we examine heterogeneous nucleation of water (H2O) and n-butanol (BuOH) vapors onto three positively (Li+, Na+, K+) and three negatively charged (F-, Cl-, Br-) ions using classical nucleation theory and computational quantum chemistry methods. We study phenomena that cannot be captured by Kelvin-Thomson equation for small nucleation ion cores. Our quantum chemistry calculations reveal the molecular mechanism behind ion-induced nucleation for each studied system. Typically, ions become solvated from all sides after several vapor molecules condense onto the ion. However, we show that the clusters of water and large negatively charged ions (Cl- and Br-) thermodynamically prefer the ion being migrated to the cluster surface. Although our methods generally do not show clear sign-preference for ion-water nucleation, we identified positive sign-preference for ion-butanol nucleation caused by the possibility to form stabilizing hydrogen bonds between butanol molecules condensed onto a positively charged ion. These bonds cannot form when butanol condenses onto a negatively charged ion. Therefore, we show that ion charge, its sign, as well as vapor properties have effects on the prenucleation and critical cluster/droplet sizes and also on the molecular mechanism of ion-induced nucleation.

Homogeneous nucleation of carbon dioxide in supersonic nozzles II: molecular dynamics simulations and properties of nucleating clusters

Large scale molecular dynamics simulations of the homogeneous nucleation of carbon dioxide in an argon atmosphere were carried out at temperatures between 75 and 105 K. Extensive analyses of the nucleating clusters' structural and energetic properties were performed to quantify these details for the supersonic nozzle experiments described in the first part of this series [Dingilian et al., Phys. Chem. Chem. Phys., 2020, 22, 19282-19298]. We studied ten different combinations of temperature and vapour pressure, leading to nucleation rates of 10²³-10²⁵ cm^-3 s^-1. Nucleating clusters possess significant excess energy from monomer capture, and the observed cluster temperatures during nucleation - on both sides of the critical cluster size - are higher than that of the carrier gas. Despite strong undercooling with respect to the triple point, most clusters are clearly liquid-like during the nucleation stage. Only at the lowest simulation temperatures and vapour densities, clusters containing over 100 molecules are able to undergo a second phase transition to a crystalline solid. The formation free energies retrieved from the molecular dynamics simulations were used to improve the classical nucleation theory by introducing a Tolman-like term into the classical liquid-drop model expression for the formation free energy. This simulation-based theory predicts the simulated nucleation rates perfectly, and improves the prediction of the experimental rates compared to self-consistent classical nucleation theory.

Homogeneous nucleation of carbon dioxide in supersonic nozzles I: experiments and classical theories

We studied the homogeneous nucleation of carbon dioxide in the carrier gas argon for concentrations of CO2 ranging from 2 to 39 mole percent using three experimental methods. Position-resolved pressure trace measurements (PTM) determined that the onset of nucleation occurred at temperatures between 75 and 92 K with corresponding CO2 partial pressures of 39 to 793 Pa. Small angle X-ray scattering (SAXS) measurements provided particle size distributions and aerosol number densities. Number densities of approximately 1012 cm-3, and characteristic times ranging from 6 to 13 μs, resulted in measured nucleation rates on the order of 5 × 1017 cm-3 s-1, values that are consistent with other nucleation rate measurements in supersonic nozzles. Finally, we used Fourier transform infrared (FTIR) spectroscopy to identify that the condensed CO2 particles were crystalline cubic solids with either sharp or rounded corners. Molecular dynamics simulations, however, suggest that CO2 forms liquid-like critical clusters before transitioning to the solid phase. Furthermore, the critical clusters are not in thermal equilibrium with the carrier gas. Comparisons with nucleation theories were therefore made assuming liquid-like critical clusters and incorporating non-isothermal correction factors.

Vapor-phase nucleation of n-pentane, n-hexane, and n-heptane: Critical cluster properties

The first and second nucleation theorems provide a way to determine the molecular content and excess internal energies of critical clusters, which rely solely on experimental nucleation rates measured at constant temperatures and supersaturations, respectively. Here, we report the size n^* and excess internal energy E_x(n^*) of n-pentane, n-hexane, and n-heptane critical clusters when particles form under the highly supersaturated conditions present in supersonic expansions. In summary, critical clusters contain from ∼2 to ∼11 molecules and exhibit the expected increase in the critical cluster size with increasing temperature and decreasing supersaturation. Surprisingly, the n^* values for all three alkanes appear to lie along a single line when plotted as a function of supersaturation. Within the framework of the capillarity approximation, the excess internal energies determined for the n-heptane critical clusters formed under the low temperature (∼150 K) conditions in our supersonic nozzle are reasonably consistent with those determined under higher temperature (∼250 K) conditions in the thermal diffusion cloud chamber by Rudek et al. [J. Chem. Phys. 105, 4707 (1996)].

Water nucleation at extreme supersaturation

We report water cluster formation in the uniform postnozzle flow of a Laval nozzle at low temperatures of 87.0 and 47.5 K and high supersaturations of lnS ∼ 41 and 104, respectively. Cluster size distributions were measured after soft single-photon ionization at 13.8 eV with mass spectrometry. Critical cluster sizes were determined from cluster size distributions recorded as a function of increasing supersaturation, resulting in critical sizes of 6-15 and 1, respectively. Comparison with previous data for propane and toluene reveals a systematic trend in the nucleation behavior, i.e., a change from a steplike increase to a gradual increase of the maximum cluster size with increasing supersaturation. Experimental nucleation rates of 5 · 10¹⁵ cm^-3 s^-1 and 2 · 10¹⁵ cm^-3 s^-1 for lnS ∼ 41 and 104, respectively, were retrieved from cluster size distributions recorded as a function of nucleation time. These lie 2-3 orders of magnitude below the gas kinetic collision limit assuming unit sticking probability, but they agree very well with a recent prediction by a master equation model based on ab initio transition state theory. The experimental observations are consistent with barrierless growth at 47.5 K, but they hint at a more complex nucleation behavior for the measurement at 87.0 K.

Condensation and dissociation rates for gas phase metal clusters from molecular dynamics trajectory calculations

In gas phase synthesis systems, clusters form and grow via condensation, in which a monomer binds to an existing cluster. While a hard-sphere equation is frequently used to predict the condensation rate coefficient, this equation neglects the influences of potential interactions and cluster internal energy on the condensation process. Here, we present a collision rate theory-molecular dynamics simulation approach to calculate condensation probabilities and condensation rate coefficients. We use this approach to examine atomic condensation onto 6-56-atom Au and Mg clusters. The probability of condensation depends upon the initial relative velocity (v) between atom and cluster and the initial impact parameter (b). In all cases, there is a well-defined region of b-v space where condensation is highly probable, and outside of which the condensation probability drops to zero. For Au clusters with more than 10 atoms, we find that at gas temperatures in the 300-1200 K range, the condensation rate coefficient exceeds the hard-sphere rate coefficient by a factor of 1.5-2.0. Conversely, for Au clusters with 10 or fewer atoms and for 14- and 28-atom Mg clusters, as cluster equilibration temperature increases, the condensation rate coefficient drops to values below the hard-sphere rate coefficient. Calculations also yield the self-dissociation rate coefficient, which is found to vary considerably with gas temperature. Finally, calculations results reveal that grazing (high b) atom-cluster collisions at elevated velocity (>1000 m s^-1) can result in the colliding atom rebounding (bounce) from the cluster surface or binding while another atom dissociates (replacement). The presented method can be applied in developing rate equations to predict material formation and growth rates in vapor phase systems.

Analyzing multistep homogeneous nucleation in vapor-to-solid transitions using molecular dynamics simulations

In this paper, we present multistep homogeneous nucleations in vapor-to-solid transitions as revealed by molecular dynamics simulations on Lennard-Jones molecules, where liquidlike clusters are created and crystallized. During a long, direct NVE (constant volume, energy, and number of molecules) involving the integration of (1.9-15)×10^{6} molecules in up to 200 million steps (=4.3 μs), crystallization in many large, supercooled nanoclusters is observed once the liquid clusters grow to a certain size (∼800 molecules for the case of T≃0.5ɛ/k). In the simulations, we discovered an interesting process associated with crystallization: the solid clusters lost 2-5 % of their mass during crystallization at low temperatures below their melting temperatures. Although the crystallized clusters were heated by latent heat, they were stabilized by cooling due to evaporation. The clusters crystallized quickly and completely except at surface layers. However, they did not have stable crystal structures, rather they had metastable structures such as icosahedral, decahedral, face-centered-cubic-rich (fcc-rich), and hexagonal-close-packed-rich (hcp-rich). Several kinds of cluster structures coexisted in the same size range of ∼1000-5000 molecules. Our results imply that multistep nucleation is a common first stage of condensation from vapor to solid.

Canonical free-energy barrier of particle and polymer cluster formation

A common approach to study nucleation rates is the estimation of free-energy barriers. This usually requires knowledge about the shape of the forming droplet, a task that becomes notoriously difficult in macromolecular setups starting with a proper definition of the cluster boundary. Here we demonstrate a shape-free determination of the free energy for temperature-driven cluster formation in particle as well as polymer systems. Combined with rigorous results on equilibrium droplet formation, this allows for a well-defined finite-size scaling analysis of the effective interfacial free energy at a fixed density. We first verify the theoretical predictions for the formation of a liquid droplet in a supersaturated particle gas by generalized-ensemble Monte Carlo simulations of a Lennard-Jones system. Going one step further, we then generalize this approach to cluster formation in a dilute polymer solution. Our results suggest an analogy with particle condensation, when the macromolecules are interpreted as extended particles.

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.

Comment on "Simple improvements to classical bubble nucleation models"

A critical analysis of several statements concerning experimental studies, molecular dynamics simulations, and the theoretical interpretation of bubble nucleation processes is performed. In particular, it is shown that the Tolman equation does not supply us, in general, with a satisfactory theoretically founded description of the curvature dependence of the surface tension and the dependence of the steady-state nucleation rate of bubbles and droplets on supersaturation in the framework of classical nucleation theory.

Reply to "Comment on 'Simple improvements to classical bubble nucleation models' "

We reply to the Comment by Schmelzer and Baidakov [Phys. Rev. E 94, 026801 (2016)].10.1103/PhysRevE.94.026801 They suggest that a more modern approach than the classic description by Tolman is necessary to model the surface tension of curved interfaces. Therefore we now consider the higher-order Helfrich correction, rather than the simpler first-order Tolman correction. Using a recent parametrization of the Helfrich correction provided by Wilhelmsen et al. [J. Chem. Phys. 142, 064706 (2015)]JCPSA60021-960610.1063/1.4907588, we test this description against measurements from our simulations, and find an agreement stronger than what the pure Tolman description offers. Our analyses suggest a necessary correction of order higher than the second for small bubbles with radius ≲1 nm. In addition, we respond to other minor criticism about our results.

Droplet formation and growth inside a polymer network: A molecular dynamics simulation study

We present a molecular dynamics simulation study that focuses on the formation and growth of nanoscale droplets inside polymer networks. Droplet formation and growth are investigated by the liquid-vapor phase separation of a dilute Lennard-Jones (LJ) fluid inside regularly crosslinked, polymer networks with varying mesh sizes. In a polymer network with small mesh sizes, droplet formation can be suppressed, the extent of which is dependent on the attraction strength between the LJ particles. When droplets form in a polymer network with intermediate mesh sizes, subsequent growth is significantly slower when compared with that in bulk without a polymer network. Interestingly, droplet growth beyond the initial nucleation stage occurs by different mechanisms depending on the mesh size: droplets grow mainly by diffusion and coalescence inside polymer networks with large mesh sizes (as observed in bulk), whereas Ostwald ripening becomes a more dominant mechanism for droplet growth for small mesh sizes. The analysis of droplet trajectories clearly reveals the obstruction effect of the polymer network on the movement of growing droplets, which leads to Ostwald ripening of droplets. This study suggests how polymer networks can be used to control the growth of nanoscale droplets.

Spontaneous cavitation in a Lennard-Jones liquid: Molecular dynamics simulation and the van der Waals-Cahn-Hilliard gradient theory

The process of bubble nucleation in a Lennard-Jones (LJ) liquid is studied by molecular dynamics (MD) simulation. The bubble nucleation rate J is determined by the mean life-time method at temperatures above that of the triple point in the region of negative pressures. The results of simulation are compared with classical nucleation theory (CNT) and modified classical nucleation theory (MCNT), in which the work of formation of a critical bubble is determined in the framework of the van der Waals-Cahn-Hilliard gradient theory (GT). It has been found that the values of J obtained in MD simulation systematically exceed the data of CNT, and this excess in the nucleation rate reaches 8-10 orders of magnitude close to the triple point temperature. The results of MCNT are in satisfactory agreement with the data of MD simulation. To describe the properties of vapor-phase nuclei in the framework of GT, an equation of state has been built up which describes stable, metastable and labile regions of LJ fluids. The surface tension of critical bubbles γ has been found from CNT and data of MD simulation as a function of the radius of curvature of the surface of tension R*. The dependence γ(R*) has also been calculated from GT. The Tolman length has been determined, which is negative and in modulus equal to ≈(0.1 - 0.2) σ. The paper discusses the applicability of the Tolman formula to the description of the properties of critical nuclei in nucleation.

Simple improvements to classical bubble nucleation models

We revisit classical nucleation theory (CNT) for the homogeneous bubble nucleation rate and improve the classical formula using a correct prefactor in the nucleation rate. Most of the previous theoretical studies have used the constant prefactor determined by the bubble growth due to the evaporation process from the bubble surface. However, the growth of bubbles is also regulated by the thermal conduction, the viscosity, and the inertia of liquid motion. These effects can decrease the prefactor significantly, especially when the liquid pressure is much smaller than the equilibrium one. The deviation in the nucleation rate between the improved formula and the CNT can be as large as several orders of magnitude. Our improved, accurate prefactor and recent advances in molecular dynamics simulations and laboratory experiments for argon bubble nucleation enable us to precisely constrain the free energy barrier for bubble nucleation. Assuming the correction to the CNT free energy is of the functional form suggested by Tolman, the precise evaluations of the free energy barriers suggest the Tolman length is ≃0.3σ independently of the temperature for argon bubble nucleation, where σ is the unit length of the Lennard-Jones potential. With this Tolman correction and our prefactor one gets accurate bubble nucleation rate predictions in the parameter range probed by current experiments and molecular dynamics simulations.

A two-parameter extension of classical nucleation theory

A two-variable stochastic model for diffusion-limited nucleation is developed using a formalism derived from fluctuating hydrodynamics. The model is a direct generalization of the standard classical nucleation theory (CNT). The nucleation rate and pathway are calculated in the weak-noise approximation and are shown to be in good agreement with direct numerical simulations for the weak-solution/strong-solution transition in globular proteins. We find that CNT underestimates the time needed for the formation of a critical cluster by two orders of magnitude and that this discrepancy is due to the more complex dynamics of the two variable model and not, as often is assumed, a result of errors in the estimation of the free energy barrier.

Scaling law for crystal nucleation time in glasses

Due to high viscosity, glassy systems evolve slowly to the ordered state. Results of molecular dynamics simulation reveal that the structural ordering in glasses becomes observable over "experimental" (finite) time-scale for the range of phase diagram with high values of pressure. We show that the structural ordering in glasses at such conditions is initiated through the nucleation mechanism, and the mechanism spreads to the states at extremely deep levels of supercooling. We find that the scaled values of the nucleation time, τ1 (average waiting time of the first nucleus with the critical size), in glassy systems as a function of the reduced temperature, T˜, are collapsed onto a single line reproducible by the power-law dependence. This scaling is supported by the simulation results for the model glassy systems for a wide range of temperatures as well as by the experimental data for the stoichiometric glasses at the temperatures near the glass transition.

Bubble evolution and properties in homogeneous nucleation simulations

We analyze the properties of naturally formed nanobubbles in Lennard-Jones molecular dynamics simulations of liquid-to-vapor nucleation in the boiling and the cavitation regimes. The large computational volumes provide a realistic environment at unchanging average temperature and liquid pressure, which allows us to accurately measure properties of bubbles from their inception as stable, critically sized bubbles, to their continued growth into the constant speed regime. Bubble gas densities are up to 50% lower than the equilibrium vapor densities at the liquid temperature, yet quite close to the gas equilibrium density at the lower gas temperatures measured in the simulations: The latent heat of transformation results in bubble gas temperatures up to 25% below those of the surrounding bulk liquid. In the case of rapid bubble growth-typical for the cavitation regime-compression of the liquid outside the bubble leads to local temperature increases of up to 5%, likely significant enough to alter the surface tension as well as the local viscosity. The liquid-vapor bubble interface is thinner than expected from planar coexistence simulations by up to 50%. Bubbles near the critical size are extremely nonspherical, yet they quickly become spherical as they grow. The Rayleigh-Plesset description of bubble-growth gives good agreement in the cavitation regime.

Direct simulations of homogeneous bubble nucleation: Agreement with classical nucleation theory and no local hot spots

We present results from direct, large-scale molecular dynamics simulations of homogeneous bubble (liquid-to-vapor) nucleation. The simulations contain half a billion Lennard-Jones atoms and cover up to 56 million time steps. The unprecedented size of the simulated volumes allows us to resolve the nucleation and growth of many bubbles per run in simple direct micro-canonical simulations while the ambient pressure and temperature remain almost perfectly constant. We find bubble nucleation rates which are lower than in most of the previous, smaller simulations. It is widely believed that classical nucleation theory (CNT) generally underestimates bubble nucleation rates by very large factors. However, our measured rates are within two orders of magnitude of CNT predictions; only at very low temperatures does CNT underestimate the nucleation rate significantly. Introducing a small, positive Tolman length leads to very good agreement at all temperatures, as found in our recent vapor-to-liquid nucleation simulations. The critical bubbles sizes derived with the nucleation theorem agree well with the CNT predictions at all temperatures. Local hot spots reported in the literature are not seen: Regions where a bubble nucleation event will occur are not above the average temperature, and no correlation of temperature fluctuations with subsequent bubble formation is seen.