On the similarity of equilibrium and critical clusters in atomic vapors.

In a previous paper [I. Napari, J. Julin, and H. Vehkamäki, J. Chem. Phys. 131, 244511 (2009)] we compared sizes of critical and equilibrium clusters in Lennard-Jones vapors using various geometrical clusters definitions. In particular, we found that the critical and equilibrium clusters were of the same size if each constituent atom in the cluster was required to have at least five neighboring atoms (ten Wolde-Frenkel cluster) but the critical clusters were much larger if only one neighboring atom was sufficient to fulfill the cluster definition (Stillinger cluster). The conclusion was that the critical clusters at high vapor densities have more ramified structure than the corresponding equilibrium clusters. In this study we have performed new molecular dynamics simulations to enlighten this matter. It is found that the surprising conclusion of the earlier work can be traced to the mean first passage time method which was used to obtain critical cluster sizes from simulations. When a certain sized Stillinger cluster first appears in the simulation, the cluster tends to have a more ramified structure than Stillinger clusters of that size observed later in the simulation. However, for the latter clusters the ratio of Stillinger and ten Wolde-Frenkel sizes in the vapor is the same as in the equilibrium simulations, implying similar structure of critical and equilibrium clusters.

Performance of some nucleation theories with a nonsharp droplet-vapor interface.

Nucleation theories involving the concept of nonsharp boundary between the droplet and vapor are compared to recent molecular dynamics (MD) simulation data of Lennard-Jones vapors at temperatures above the triple point. The theories are diffuse interface theory (DIT), extended modified liquid drop-dynamical nucleation theory (EMLD-DNT), square gradient theory (SGT), and density functional theory (DFT). Particular attention is paid to thermodynamic consistency in the comparison: the applied theories either use or, with a proper parameter adjustment, result in the same values of equilibrium vapor pressure, bulk liquid density, and surface tension as the MD simulations. Realistic pressure-density correlations are also used. The best agreement between the simulated nucleation rates and calculations is obtained from DFT, SGT, and EMLD-DNT, all of which, in the studied temperature range, show deviations of less than one order of magnitude in the nucleation rate. DIT underestimates the nucleation rate by up to two orders of magnitude. DFT and SGT give the best estimate of the molecular content of the critical nuclei. Overall, at the vapor conditions of this study, all the investigated theories perform better than classical nucleation theory in predicting nucleation rates.

Vapor-liquid nucleation of argon: exploration of various intermolecular potentials.

The homogeneous vapor-liquid nucleation of argon has been explored at T=70 and 90 K using classical nucleation theory, semiempirical density functional theory, and Monte Carlo simulations using the aggregation-volume-bias algorithm with umbrella sampling and histogram-reweighting. In contrast with previous simulation studies, which employed only the Lennard-Jones intermolecular potential, the current studies were carried out using various pair potentials including the Lennard-Jones potential, a modified Buckingham exponential-six potential, the Barker-Fisher-Watts pair potential, and a recent ab initio potential developed using the method of effective diameters. It was found that the differences in the free energy of formation of the critical nuclei between the potentials cannot be explained solely in terms of the difference in macroscopic properties of the potentials, which gives a possible reason for the failure of classical nucleation theory.

A thermodynamically consistent determination of surface tension of small Lennard-Jones clusters from simulation and theory.

We have determined the surface tension of small Lennard-Jones clusters using molecular dynamics and Monte Carlo simulation methods as well as density functional theory calculations. For the two simulation methods the surface tension is calculated via a rigorous thermodynamic route using simulation data as input. The capillary approximation of the classical nucleation theory, where the surface tension of a planar surface is used for cluster surface, is found to be quite reasonable even when the cluster size is as small as 100-150 atoms. For smaller cluster sizes the cluster surface tension is considerably lower than the planar value. We have also obtained an approximative value for the Tolman length by extrapolating to the planar limit the difference between the equimolar radius and the radius of the surface of tension. A negative Tolman length is suggested by all the methods used.

Cluster sizes in direct and indirect molecular dynamics simulations of nucleation.

We performed molecular dynamics simulations of a Lennard-Jones fluid, and compared the sizes of critical clusters in direct simulations of a nucleation event in vapor phase with the sizes of clusters in stable equilibrium with the surrounding vapor. By applying different cluster criteria it is shown that both the critical clusters and the equilibrium clusters have dense cores of similar size but the critical clusters have more outlying cluster atoms surrounding this core. The cluster definition introduced by ten Wolde and Frenkel [J. Chem. Phys. 109, 9901 (1998)], where each cluster atom must have at least five neighboring atoms within the distance of 1.5 times the Lennard-Jones length parameter, agrees well with the cluster size obtained from classical nucleation theory, and we find this agreement to be independent of temperature. The cluster size obtained from the observed nucleation rates by the first nucleation theorem is larger than the classical estimate and much smaller than the size given by the density profile of the equilibrium cluster.

Equilibrium vapor pressure and surface tension from cluster data: density functional results.

Density functional theory is applied to investigate the possibility of using the data from atomic and molecular clusters for the prediction of equilibrium vapor pressure and surface tension. For this purpose free energies of center of mass clusters constrained to a spherical volume are calculated at various temperatures. Clusters composed of Lennard-Jones atoms and molecules of two Lennard-Jones sites are considered. The desired bulk values are extracted from cluster data using the method by Merikanto and et al. [Phys. Rev. Lett. 98, 145702 (2007)] and a consistent comparison to the exact values obtained from the density functional theory is made. At temperatures not much above the triple point the estimates of both the equilibrium vapor pressure and surface tension are within 4% of the exact values for all the molecular models, including those with a structured liquid-vapor interface, if the clusters used for the estimates have more than about hundred molecules. The dependence on the constraining volume is found weak.

Equilibrium sizes and formation energies of small and large Lennard-Jones clusters from molecular dynamics: a consistent comparison to Monte Carlo simulations and density functional theories.

We have performed molecular dynamics simulations of Lennard-Jones argon clusters in equilibrium with a surrounding vapor and combined them with simulations of nucleation events in supersaturated vapor to investigate the dependence of critical cluster size on the vapor density in the cluster size range of 20-300 atoms. The simulations are performed at reduced temperature T(') = 0.662, which with the parameter values of Lennard-Jones argon corresponds to 80 K. We obtain bulk equilibrium values by simulating a planar liquid-vapor interface. In the studied cluster size range, we find a linear relation between critical size Delta N(*) and Delta mu(-3), where Delta mu is the chemical potential difference between supersaturated vapor and saturated vapor, but the slope of the line is not given by the Kelvin relation of classical nucleation theory. With this relation, along with the known formation energy of the small critical cluster of the nucleation simulations, we proceed to calculate the formation energies for larger critical sizes by integrating the nucleation theorem. We compare the molecular dynamics results to results from Monte Carlo simulations and both perturbative density functional theory and square gradient theory calculations. We find that the molecular dynamics results are in excellent agreement with the density functional and square gradient values. However, the Monte Carlo critical sizes and formation energies are somewhat lower than the molecular dynamics ones.

Two-component heterogeneous nucleation kinetics and an application to Mars.

We develop a two-component heterogeneous nucleation model that includes exact calculation of the Stauffer-type [D. Stauffer, J. Aerosol Sci. 7, 319 (1976)] steady-state kinetic prefactor using the correct heterogeneous Zeldovich factor for a heterogeneous two-component system. The model, and a simplified version of it, is tested by comparing its predictions to experimental data for water-n-propanol nucleating on silver particles. The model is then applied to water-carbon dioxide system in Martian conditions, which has not been modeled before. Using the ideal mixture assumption, the model shows theoretical possibilities for two-component nucleation adjacent to the initial stages of one-component water nucleation, especially with small water vapor amounts. The numbers of carbon dioxide molecules in the critical cluster are small in the case of large water amounts (up to 300 ppm) in the gas phase, but larger when there is very little water vapor (1 ppm).

Connection between the virial equation of state and physical clusters in a low density vapor.

We carry out Monte Carlo simulations of physical Lennard-Jones and water clusters and show that the number of physical clusters in vapor is directly related to the virial equation of state. This relation holds at temperatures clearly below the critical temperatures, in other words, as long as the cluster-cluster interactions can be neglected--a typical assumption used in theories of nucleation. Above a certain threshold cluster size depending on temperature and interaction potential, the change in cluster work of formation can be calculated analytically with the recently proposed scaling law. The breakdown of the scaling law below the threshold sizes is accurately modeled with the low order virial coefficients. Our results indicate that high order virial coefficients can be analytically calculated from the lower order coefficients when the scaling law for cluster work of formation is valid. The scaling law also allows the calculation of the surface tension and equilibrium vapor density with computationally efficient simulations of physical clusters. Our calculated values are in good agreement with those obtained with other methods. We also present our results for the curvature dependent surface tension of water clusters.

Comparative study on methodology in molecular dynamics simulation of nucleation.

Gas-liquid nucleation of 1000 Lennard-Jones atoms is simulated to evaluate temperature regulation methods and methods to obtain nucleation rate. The Berendsen and the Andersen thermostats are compared. The Berendsen thermostat is unable to control the temperature of clusters larger than the critical size. Independent of the thermostating method the velocities of individual atoms and the translational velocities of clusters up to at least six atoms are accurately described by the Maxwell velocity distribution. Simulations with the Andersen thermostat yield about two times higher nucleation rates than those with the Berendsen thermostat. Nucleation rate is extracted from the simulations by direct observation of times of nucleation onset and by the method of Yasuoka and Matsumoto [J. Chem. Phys. 109, 8451 (1998)]. Compared to the direct observation, the nucleation rates obtained from the method of Yasuoka and Matsumoto are higher by a factor of 3.

Surface tension and scaling of critical nuclei in diatomic and triatomic fluids.

Density functional theory has been used to investigate surface tension and scaling of critical clusters in fluids consisting of diatomic and rigid triatomic molecules. The atomic sites are hard spheres with attractive interactions obtained from the tail part of the Lennard-Jones potential. Asymmetry in attractive interactions between the atomic sites has been introduced to cause molecular orientation and oscillatory density profiles at liquid-vapor interfaces. The radial dependence of cluster surface tension in fluids showing modest orientation in unimolecular layer at the interface or no orientation at all resembles the surface tension behavior of clusters in simple monoatomic fluids, although the surface tension maximum becomes more pronounced with increasing chain length of the molecule. Surface tension of clusters having multiple oscillatory layers at the interface shows a prominent maximum at small cluster sizes; however, the surface tension of large clusters is lower than the planar value. The scaling relation for the number of molecules in the critical cluster and the nucleation barrier height developed by McGraw and Laaksonen [Phys. Rev. Lett. 76, 2754 (1996)] are well obeyed for fluids with little structure at liquid-vapor interface. However, fluids having enhanced interfacial structure show some deviation from the particle number scaling, and the barrier height scaling breaks up seriously.

A comparison of rigid and flexible water models in collisions of monomers and small clusters.

In this study we have investigated the dynamics of small water clusters using microcanonical molecular dynamics simulations. The clusters are formed by colliding vapor monomers with target clusters of two and five molecules. The monomers are sampled from a thermal ensemble at T=300 K and target clusters with several total energies are considered. We compare rigid extended simple point charge water with flexible counterparts having intramolecular harmonic bonds with force constants 10(3) and 10(5) kcal(mol A2). We show that the lifetimes of the clusters formed via collision process are similar for the rigid model and the flexible model with the bigger force constant, if the translational temperatures of the target cluster molecules are equal. The model with the smaller force constant results in much longer lifetimes due to the stabilizing effect caused by the kinetic energy transfer into internal vibration of the molecules. This process may take several hundreds of picoseconds, giving rise to time-dependent decay rates of constant-energy clusters. A study of binary collisions of water molecules shows that the introduction of flexibility to the molecules increases the possibility of dimer formation and thus offers an alternative route for dimer production in vapors. Our results imply that allowing for internal degrees of freedom is likely to enhance gas-liquid nucleation rates in water simulations.

Postcollision relaxation of small atomic clusters.

Molecular-dynamics simulations are performed to investigate the effects caused by the lack of internal equilibration on the dynamics and properties of atomic clusters. The studied systems consist of Lennard-Jones clusters of five to ten atoms and a colliding vapor monomer. Cluster radius and potential energy are shown to reach a time-independent value within 30 ps after a collision with a vapor monomer. The relaxation in terms of rotational energy takes at least 200 ps. During the first couple of picoseconds after the collision time-dependent cluster decay rates are observed. The unrelaxed cluster states are expected to have minimal effect on gas-liquid nucleation rates.

Evaluation of surface composition of surface active water-alcohol type mixtures: a comparison of semiempirical models.

We study adsorption at planar liquid-vapor interface of surface active binary mixtures and test three well-known models for the composition of surface phase. The models were originally presented by Guggenheim. These are compared to results for model fluids from density functional theory (DFT). The model of Laaksonen and Kulmala is in best agreement with DFT calculations. Surface mole fraction of the solute component from the Guggenheim model exceeds one for a mixture with high surface activity. The failure of the Guggenheim model is also evident in our calculations for water-methanol, water-ethanol, and water-n-propanol mixtures.

Stable ammonium bisulfate clusters in the atmosphere.

Liquid drop model based equilibrium thermodynamics predicts that in the presence of even small ammonia concentrations practically all the atmospheric sulfuric acid molecules are bound to tiny, stable ammonium-bisulfate clusters. Hitherto sulfuric acid has been believed to form hydrates with water. Thermodynamic theory predicts correctly the hydrate formation observed experimentally. Results from ab initio computer simulations contradict the thermodynamic results and also the experimental findings for cluster formation in both sulfuric acid-water and ammonia-sulfuric acid-water mixtures.

Molecular dynamic simulations of atom-cluster collision processes.

Monomer-cluster collisions of Lennard-Jones argon atoms have been studied using molecular dynamics simulation for target cluster sizes of 2, 3, 4, 5, 10, and 20 atoms. Capture probability of monomers by clusters and the lifetimes of the resulting clusters have been calculated as a function of impact parameter and the total energy of the target cluster. Cluster lifetime is further integrated over all impact parameters to obtain the average lifetime for each cluster size and energy. The average lifetime of the smallest aggregates is shown to be short compared to the collision time between monomers and clusters unless the vapor is highly supersaturated. The formation probability of a new cluster decreases steeply if a minimum lifetime is required for the cluster.

Binary homogeneous nucleation in water-succinic acid and water-glutaric acid systems.

Binary homogeneous nucleation of water-succinic acid and water-glutaric acid systems have been investigated. The numerical approach was based on the classical nucleation theory. Usually, nucleation is discussed in terms of kinetics, but the thermodynamics involved is undoubtedly equally important. In this paper we studied the above mentioned binary systems giving a quantitative insight into the nucleation process and a detailed consideration of the thermodynamics involved. Both diacids in study are in solid state at room temperature. They behave in environment according to their liquid state properties because of the absence of crystalline lattice energies, and therefore their subcooled liquid state thermodynamics have to be considered. The lack of consistent thermodynamic data for pure organic components and their aqueous solutions represent a high source of uncertainty. However, the present simulations indicate that in atmospheric conditions these binary systems will not form new particles.

On the closure conjectures for the Gibbsian approximation model of a binary droplet.

Within the framework of Gibbsian thermodynamics, a binary droplet is regarded to consist of a uniform interior and dividing surface. The properties of the droplet interior are those of the bulk liquid solution, but the dividing surface is a fictitious phase whose chemical potentials cannot be rigorously determined. The state of the nucleus interior and free energy of nucleus formation can be found without knowing the surface chemical potentials, but the latter are still needed to determine the state of the whole nucleus (including the dividing surface) and develop the kinetics of nucleation. Thus it is necessary to recur to additional conjectures in order to build a complete, thermodynamic, and kinetic theory of nucleation within the framework of the Gibbsian approximation. Here we consider and analyze the problem of closing the Gibbsian approximation droplet model. We identify micro- and Gamma-closure conjectures concerning the surface chemical potentials and excess surface coverages, respectively, for the droplet surface of tension. With these two closure conjectures, the Gibbsian approximation model of a binary droplet becomes complete so that one can determine both the surface and internal characteristics of the whole nucleus and develop the kinetic theory, based on this model. Theoretical results are illustrated by numerical evaluations for binary nucleation in a water-methanol vapor mixture at T=298.15 K. Numerical results show a striking increase in the droplet surface tension with decreasing droplet size at constant overall droplet composition. A comparison of the Gibbsian approximation with density functional calculations for a model surfactant system indicate that the excess surface coverages from the Gibbsian approximation are accurate enough for large droplets and droplets that are not too concentrated with respect to the solute.

The role of dimers in evaporation of small argon clusters.

Evaporation of small Lennard-Jones argon clusters has been studied using molecular dynamic simulations. An extensive library of clusters with 4, 5, 6, 11, and 21 atoms has been obtained from an earlier study. Analysis of the evaporation properties of the clusters indicate, that the fraction of dimer evaporations of all evaporation events increases with the total energy of the cluster. The fraction of evaporated dimers from clusters with a constant lifetime is independent of the cluster size for short-lived clusters and increases with cluster size for long-lived clusters. Only a few percent of the clusters which are long lived enough to participate in vapor-liquid nucleation decay by emitting dimers. The mean cluster lifetime as a function of total energy shows the same exponentially decreasing trend for monomer and dimer evaporation channels. The fraction of trimer evaporations is found to be vanishingly small.