Nucleation experiments on a titanium-carbon system imply nonclassical formation of presolar grains.

Just as the shapes of snowflakes provide us with information on the temperature and humidity of the upper atmosphere, the characteristics of presolar grains in meteorites place limits on their formation environment in a stellar outflow. However, even in the case of well-characterized presolar grains consisting of a titanium carbide core and a graphitic carbon mantle, it is not possible to delimit their formation environment. Here, we have demonstrated the formation of core-mantle grains in gravitational and microgravity environments and have found that core-mantle grains are formed by a nonclassical nucleation pathway involving the three steps: (i) primary nucleation of carbon at a substantially high supersaturation, (ii) heterogeneous condensation of titanium carbide on the carbon, and (iii) fusion of nuclei. We argue that the characteristics of not only core-mantle grains but also other presolar and solar grains might be accurately explained by considering a nonclassical nucleation pathway.

Theoretical analysis of crystallization by homogeneous nucleation of water droplets.

We propose a novel method for analyzing the crystallization process from supercooled water droplets. The method, which is based on nucleation theory, simultaneously evolves homogeneous ice nucleation and crystal growth in the cooling process and obtains the crystallization temperature and the number of crystal nuclei in the droplets. The model can reproduce not only the crystallization of water but also the vitrification process. The model well replicated the results of previous laboratory experiments, especially, the different responses of the crystallization temperatures of the micrometer- and nanometer-sized particles as a function of cooling rates. For particle sizes ranging from 1 to 1000 µm and cooling rates below 104 K s-1, the crystallization temperature was 230-240 K. At cooling rates above 104 K s-1, the crystallization temperature decreased rapidly. On the other hand, the crystallization temperature of 10 nm particles was 200-230 K at cooling rates below 104 K s-1. When describing the interfacial tension by σ = 29.1 + 0.1(T - 273.15) erg cm-2 (where T is the water droplet temperature in K), the analyses explained well the previously reported crystallization temperatures of droplets sized from a few nm to 100 µm under various cooling conditions. Our model also predicts the critical cooling rate for vitrification of the liquid water droplets. The critical cooling rate of vitrification is predicted to be 107-108 K s-1, consistent with the experimental rates. These analyses are useful not only for comprehensively understanding the ice nucleation process but also for predicting the crystallization processes in various environments such as cirrus clouds, which are difficult to reproduce in experiments.

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.

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.

Pure iron grains are rare in the universe.

The abundant forms in which the major elements in the universe exist have been determined from numerous astronomical observations and meteoritic analyses. Iron (Fe) is an exception, in that only depletion of gaseous Fe has been detected in the interstellar medium, suggesting that Fe is condensed into a solid, possibly the astronomically invisible metal. To determine the primary form of Fe, we replicated the formation of Fe grains in gaseous ejecta of evolved stars by means of microgravity experiments. We found that the sticking probability for the formation of Fe grains is extremely small; only a few atoms will stick per hundred thousand collisions so that homogeneous nucleation of metallic Fe grains is highly ineffective, even in the Fe-rich ejecta of type Ia supernovae. This implies that most Fe is locked up as grains of Fe compounds or as impurities accreted onto other grains in the interstellar medium.

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.

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.

Homogeneous SPC/E water nucleation in large molecular dynamics simulations.

We perform direct large molecular dynamics simulations of homogeneous SPC/E water nucleation, using up to ∼ 4 ⋅ 10(6) molecules. Our large system sizes allow us to measure extremely low and accurate nucleation rates, down to ∼ 10(19) cm(-3) s(-1), helping close the gap between experimentally measured rates ∼ 10(17) cm(-3) s(-1). We are also able to precisely measure size distributions, sticking efficiencies, cluster temperatures, and cluster internal densities. We introduce a new functional form to implement the Yasuoka-Matsumoto nucleation rate measurement technique (threshold method). Comparison to nucleation models shows that classical nucleation theory over-estimates nucleation rates by a few orders of magnitude. The semi-phenomenological nucleation model does better, under-predicting rates by at worst a factor of 24. Unlike what has been observed in Lennard-Jones simulations, post-critical clusters have temperatures consistent with the run average temperature. Also, we observe that post-critical clusters have densities very slightly higher, ∼ 5%, than bulk liquid. We re-calibrate a Hale-type J vs. S scaling relation using both experimental and simulation data, finding remarkable consistency in over 30 orders of magnitude in the nucleation rate range and 180 K in the temperature range.

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.

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

Recent very large molecular dynamics simulations of homogeneous nucleation with (1 - 8) × 10(9) Lennard-Jones atoms [J. Diemand, R. Angélil, K. K. Tanaka, and H. Tanaka, J. Chem. Phys. 139, 074309 (2013)] allow us to accurately determine the formation free energy of clusters over a wide range of cluster sizes. This is now possible because such large simulations allow for very precise measurements of the cluster size distribution in the steady state nucleation regime. The peaks of the free energy curves give critical cluster sizes, which agree well with independent estimates based on the nucleation theorem. Using these results, we derive an analytical formula and a new scaling relation for nucleation rates: ln J'/η is scaled by ln S/η, where the supersaturation ratio is S, η is the dimensionless surface energy, and J(') is a dimensionless nucleation rate. This relation can be derived using the free energy of cluster formation at equilibrium which corresponds to the surface energy required to form the vapor-liquid interface. At low temperatures (below the triple point), we find that the surface energy divided by that of the classical nucleation theory does not depend on temperature, which leads to the scaling relation and implies a constant, positive Tolman length equal to half of the mean inter-particle separation in the liquid phase.

Molecular dynamics simulations of the nucleation of water: determining the sticking probability and formation energy of a cluster.

We performed molecular dynamics simulations of the nucleation of water vapor in order to test nucleation theories. Simulations were performed for a wide range of supersaturation ratios (S = 3-25) and water temperatures (T(w) = 300-390 K). We obtained the nucleation rates and the formation free energies of a subcritical cluster from the cluster size distribution. The classical nucleation theory and the modified classical nucleation theory (MCNT) overestimate the nucleation rates in all cases. The semi-phenomenological model, which corrects the MCNT prediction using the second virial coefficient of a vapor, reproduces the formation free energy of a cluster with the size ≲20 to within 10% and the nucleation rate and cluster size distributions to within one order of magnitude. The sticking probability of the vapor molecules to the clusters was also determined from the growth rates of the clusters. The sticking probability rapidly increases with the supersaturation ratio S, which is similar to the Lennard-Jones system.

Properties of liquid clusters in large-scale molecular dynamics nucleation simulations.

We have performed large-scale Lennard-Jones molecular dynamics simulations of homogeneous vapor-to-liquid nucleation, with 10(9) atoms. This large number allows us to resolve extremely low nucleation rates, and also provides excellent statistics for cluster properties over a wide range of cluster sizes. The nucleation rates, cluster growth rates, and size distributions are presented in Diemand et al. [J. Chem. Phys. 139, 74309 (2013)], while this paper analyses the properties of the clusters. We explore the cluster temperatures, density profiles, potential energies, and shapes. A thorough understanding of the properties of the clusters is crucial to the formulation of nucleation models. Significant latent heat is retained by stable clusters, by as much as ΔkT = 0.1ε for clusters with size i = 100. We find that the clusters deviate remarkably from spherical-with ellipsoidal axis ratios for critical cluster sizes typically within b/c = 0.7 ± 0.05 and a/c = 0.5 ± 0.05. We examine cluster spin angular momentum, and find that it plays a negligible role in the cluster dynamics. The interfaces of large, stable clusters are thinner than planar equilibrium interfaces by 10%-30%. At the critical cluster size, the cluster central densities are between 5% and 30% lower than the bulk liquid expectations. These lower densities imply larger-than-expected surface areas, which increase the energy cost to form a surface, which lowers nucleation rates.

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.

Large scale molecular dynamics simulations of homogeneous nucleation.

We present results from large-scale molecular dynamics (MD) simulations of homogeneous vapor-to-liquid nucleation. The simulations contain between 1 × 10(9) and 8 × 10(9) Lennard-Jones (LJ) atoms, covering up to 1.2 µs (56 × 10(6) time-steps). They cover a wide range of supersaturation ratios, S ≃ 1.55-10(4), and temperatures from kT = 0.3 to 1.0ε (where ε is the depth of the LJ potential, and k is the Boltzmann constant). We have resolved nucleation rates as low as 10(17) cm(-3) s(-1) (in the argon system), and critical cluster sizes as large as 100 atoms. Recent argon nucleation experiments probe nucleation rates in an overlapping range, making the first direct comparison between laboratory experiments and molecular dynamics simulations possible: We find very good agreement within the uncertainties, which are mainly due to the extrapolations of argon and LJ saturation curves to very low temperatures. The self-consistent, modified classical nucleation model of Girshick and Chiu [J. Chem. Phys. 93, 1273 (1990)] underestimates the nucleation rates by up to 9 orders of magnitudes at low temperatures, and at kT = 1.0ε it overestimates them by up to 10(5). The predictions from a semi-phenomenological model by Laaksonen et al. [Phys. Rev. E 49, 5517 (1994)] are much closer to our MD results, but still differ by factors of up to 10(4) in some cases. At low temperatures, the classical theory predicts critical clusters sizes, which match the simulation results (using the first nucleation theorem) quite well, while the semi-phenomenological model slightly underestimates them. At kT = 1.0ε, the critical sizes from both models are clearly too small. In our simulations the growth rates per encounter, which are often taken to be unity in nucleation models, lie in a range from 0.05 to 0.24. We devise a new, empirical nucleation model based on free energy functions derived from subcritical cluster abundances, and find that it performs well in estimating nucleation rates.

Molecular dynamics simulations of nucleation from vapor to solid composed of Lennard-Jones molecules.

We performed molecular dynamics (MD) simulations of nucleation from vapor at temperatures below the triple point for systems consisting of 10(4)-10(5) Lennard-Jones (L-J) type molecules in order to test nucleation theories at relatively low temperatures. Simulations are performed for a wide range of initial supersaturation ratio (S(0) ≃ 10-10(8)) and temperature (kT = 0.2-0.6ε), where ε and k are the depth of the L-J potential and the Boltzmann constant, respectively. Clusters are nucleated as supercooled liquid droplets because of their small size. Crystallization of the supercooled liquid nuclei is observed after their growth slows. The classical nucleation theory (CNT) significantly underestimates the nucleation rates (or the number density of critical clusters) in the low-T region. The semi-phenomenological (SP) model, which corrects the CNT prediction of the formation energy of clusters using the second virial coefficient of a vapor, reproduces the nucleation rate and the cluster size distributions with good accuracy in the low-T region, as well as in the higher-T cases considered in our previous study. The sticking probability of vapor molecules onto the clusters is also obtained in the present MD simulations. Using the obtained values of sticking probability in the SP model, we can further refine the accuracy of the SP model.

Tests of the homogeneous nucleation theory with molecular-dynamics simulations. I. Lennard-Jones molecules.

Two kinds of the homogeneous nucleation theory exist at the present: the classical nucleation theory and the semiphenomenological model. To test them, we performed molecular-dynamics (MD) simulations of nucleation from vapor to liquid with 5000-20,000 Lennard-Jones-type molecules. Simulations were done for various values of supersaturation ratios (from 2 to 10) and temperatures (from 80 to 120 K). We compared the size distribution of clusters in MD simulations with those in the theoretical models because the number density of critical clusters governs the nucleation rate. We found that the semiphenomenological model achieves excellent agreements in size distributions of the clusters with all MD simulations we done. The classical theory underestimates the number density of the clusters in the temperature range of 80-100 K, but overestimates in 100-120 K. The semiphenomenological model also predicts well the nucleation rate in MD simulations, while the classical nucleation theory does not. Our results confirmed the validity of the semiphenomenological model for Lennard-Jones-type molecules.