A stochastic simulation of nonisothermal nucleation

The results of stochastic simulations of growth and evaporation of small clusters in vapor are reported. Energy dependent growth rates are determined from the monomer-cluster collision rate and decay rates are found from a detailed balance, with the equilibrium size and energy distribution of clusters calculated using the capillarity approximation and the equilibrium vapor pressure. These rates are used in simulations of two-dimensional random walks in size and energy space to determine the fraction of clusters in supersaturated vapor of size (i(min)+1) that reach a size i(max). By assuming that clusters of size i(min) are in equilibrium, this fraction can be related to the nonisothermal nucleation rate. The simulated rates show good agreement with the previously published analytical results. In the absence of an inert carrier gas, the nonisothermal nucleation rates are typically between 1% and 5% of the isothermal rates.     

Analysis of nucleation ability of cluster configurations with Monte Carlo simulations of argon

We determine the nucleation ability of argon clusters from Monte Carlo simulations. The nucleation rate appears to be defined by a sole characteristic of the clusters, namely, the stability. The stability is calculated as the ratio of grand canonical growth and decay rates and can be assigned to individual cluster configurations. We study the connection between the stability of the cluster configurations and their volume and total potential energy. Neither the potential energy nor the volume of a cluster configuration has a clear relation to its stability, and thus to the nucleation ability. On the other hand, we show that it is possible to use a specific volume for each cluster size to calculate the work of the cluster formation. These clusters with a unique volume have the same average stability as the full set of clusters. Our simulation method allows us to study the effect of possible deviations from equilibrium in the cluster configuration distributions. We argue that the nucleation process itself can produce a source for such a deviation. We show that even a small deviation from equilibrium in the cluster configuration distribution can lead to a dramatic deceleration of the nucleation rate. Although our simulations may overestimate the magnitude of the effect, they give qualitative estimates for its importance.     

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.     

Monte Carlo simulations of critical cluster sizes and nucleation rates of water

We have calculated the critical cluster sizes and homogeneous nucleation rates of water at temperatures and vapor densities corresponding to experiments by Wolk and Strey [J. Phys. Chem B 105, 11683 (2001)]. The calculations have been done with an expanded version of a Monte Carlo method originally developed by Vehkamaki and Ford [J. Chem. Phys. 112, 4193 (2000)]. Their method calculates the statistical growth and decay probabilities of molecular clusters. We have derived a connection between these probabilities and kinetic condensation and evaporation rates, and introduce a new way for the calculation of the work of formation of clusters. Three different interaction potential models of water have been used in the simulations. These include the unpolarizable SPC/E [J. Phys. Chem. 91, 6269 (1987)] and TIP4P [J. Chem. Phys. 79, 926 (1983)] models and a polarizable model by Guillot and Guissani [J. Chem. Phys. 114, 6720 (2001)]. We show that TIP4P produces critical cluster sizes and a temperature and vapor density dependence for the nucleation rate that agree well with the experimental data, although the magnitude of nucleation rate is constantly overestimated by a factor of 2 x 10(4). Guissani and Guillot's model is somewhat less successful, but both the TIP4P and Guillot and Guissani models are able to reproduce a much better experimental temperature dependency of the nucleation rate than the classical nucleation theory. Using SPC/E results in dramatically too small critical clusters and high nucleation rates. The water models give different average binding energies for clusters. We show that stronger binding between cluster molecules suppresses the decay probability of a cluster, while the growth probability is not affected. This explains the differences in results from different water models.     

Evaporation rate of nucleating clusters

The Becker-D?ring kinetic scheme is the most frequently used approach to vapor liquid nucleation. In the present study it has been extended so that master equations for all cluster configurations are included into consideration. In the Becker-D?ring kinetic scheme the nucleation rate is calculated through comparison of the balanced steady state and unbalanced steady state solutions of the set of kinetic equations. It is usually assumed that the balanced steady state produces equilibrium cluster distribution, and the evaporation rates are identical in the balanced and unbalanced steady state cases. In the present study we have shown that the evaporation rates are not identical in the equilibrium and unbalanced steady state cases. The evaporation rate depends on the number of clusters at the limit of the cluster definition. We have shown that the ratio of the number of n-clusters at the limit of the cluster definition to the total number of n-clusters is different in equilibrium and unbalanced steady state cases. This causes difference in evaporation rates for these cases and results in a correction factor to the nucleation rate. According to rough estimation it is 10(-1) by the order of magnitude and can be lower if carrier gas effectively equilibrates the clusters. The developed approach allows one to refine the correction factor with Monte Carlo and molecular dynamic simulations.     

The molecular approach to heterogeneous nucleation

A molecular approach to heterogeneous nucleation has been developed. The expressions for the equilibrium cluster distribution, the reversible work of the cluster formation, and the nucleation rate have been derived. Two separate statements for the work of formation were formulated. If the equilibrium cluster distribution is normalized on the monomer concentration near the substrate surface, the reversible work of formation is expressed by DeltaG(het) (I) = (F(n) (het)-F(n) (hom))-(F(1) (het)-F(1) (hom)) + DeltaG(hom) where F(n) (het) and F(n) (hom) are the Helmholtz free energies of a cluster interacting with a substrate and a cluster not interacting with the substrate, respectively. If the equilibrium cluster distribution is normalized on the monomer concentration far from the substrate surface, the work of cluster formation is given by DeltaG(het) (II) = (F(n) (het)-F(n) (hom)) + DeltaG(hom). The former expression corresponds to the approach of the classical heterogeneous nucleation theory. The cluster partition function appears to be dependent on the location of a virtual plane, which separates the volume, where the interaction of the clusters with the substrate is effective from the one where interaction is negligible. Our Monte Carlo simulations have shown that the dependence is rather weak and thus the location of the plane is not very important. According to the simulations the variation of the plane position in the range from 20 to 50 Angstroms does not lead to a considerable change of the heterogeneous nucleation rate.     

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.     

Molecular dynamics calculation of half-lives for thermal decay of Lennard-Jones clusters

Molecular dynamics has been used with a Lennard-Jones (6–12) potential in order to study the decay behavior of neutral Argon clusters containing between 12 and 14 atoms. The clusters were heated to temperatures well above their melting points and then tracked in time via molecular dynamics until evaporation of one or more atoms was observed. In each simulation, the mode of evaporation, energy released during evaporation, and cluster lifetime were recorded. Results from roughly 2000 simulation histories were combined in order to compute statistically significant values of cluster half-lives and decay energies. It was found that cluster half-life decreases with increasing energy and that for a given value of excess energy (defined asE=(E _tot ?E _gnd)/n), the 13 atom cluster is more stable against decay than clusters containing either 12 or 14 atoms. The dominant decay mechanism for all clusters was determined to be single atom emission.     

Simulation of cluster impact fusion

We report molecular dynamics simulations of the impact of TiD clusters on TiD targets. In each cluster collision the total fusion probability seems to be due to a single deuterium deuterium collision. The kinetic energies of incident deuterium atoms gradually level off around the initial cluster energy, but do not reach the high energy tail of a corresponding Maxwell-Boltzmann distribution. Neither any other support for a thermonuclear fusion mechanism was observed. On the contrary, our simulations imply that the enhanced fusion rate is rather due to channeled many atom collision cascade type mechanism.     

Microscopic simulations of molecular cluster decay: Does the carrier gas affect evaporation?

We develop a kinetic theory of cluster decay by considering the stochastic motion of molecules within an effective potential of mean force (PMF) due to the cluster. We perform molecular dynamics simulations on a 50-atom argon cluster to determine the mean radial force on a component atom and hence the confining potential of mean force. Comparisons between isolated clusters and clusters thermostatted through the presence of a 100-atom helium carrier gas show that the heat bath has only a slight effect upon the PMF. This confirms the validity of calculations of cluster properties using isolated cluster simulations. The PMF is used to calculate the atomic evaporation rate from these clusters, and results are compared with the predictions of the capillarity approximation together with detailed balance, both components of the classical theory of aerosol nucleation.     

Influence of thermostats and carrier gas on simulations of nucleation

We investigate the influence of carrier gas and thermostat on molecular dynamics (MD) simulations of nucleation. The task of keeping the temperature constant in MD simulations is not trivial and an inefficient thermalization may have a strong influence on the results. Different thermostating mechanisms have been proposed and used in the past. In particular, we analyze the efficiency of velocity rescaling, Nose-Hoover, and a carrier gas (mimicking the experimental situation) by extensive MD simulations. Since nucleation is highly sensitive to temperature, one would expect that small variations in temperature might lead to differences in nucleation rates of up to several orders of magnitude. Surprisingly, the results indicate that the choice of the thermostating method in a simulation does not have--at least in the case of Lennard-Jones argon--a very significant influence on the nucleation rate. These findings are interpreted in the context of the classical theory of Feder et al. [Adv. Phys. 15, 111 (1966)] by analyzing the temperature distribution of the nucleating clusters. We find that the distribution of cluster temperatures is non-Gaussian and that subcritically sized clusters are colder while postcritically sized clusters are warmer than the bath temperature. However, the average temperature of all clusters is found to be always higher than the bath temperature.     

Kinetic stability of complex molecular clusters

This investigation is concerned with modeling the evaporation, or decay, of n-nonane molecular clusters. We use a unique cluster decay model that was first developed to estimate the decay time scale of argon clusters using molecular-dynamics simulations. In this study we seek to enhance the model so that it represents a more complex cluster decay dynamic, suitable for n-nonane clusters. Experimental measurements of nucleation rates of n-nonane droplets have been used to deduce the rate at which a molecule escapes from the cluster. Typically for an n-nonane cluster containing 40 molecules, at an experimental temperature of 225 K, the empirical decay time, which is the inverse of the decay rate, is estimated to be 50 ns. For this time scale, the direct observation of n-nonane cluster decay from a molecular-dynamics trajectory is not feasible, since decay events are so rare. However, the cluster decay model uses a combination of molecular dynamics and stochastic dynamics in order to resolve the problem associated with long decay time scales. The model is based on a Langevin treatment that views cluster decay as single-particle escape from a confining potential of mean force. It is used to predict kinetic decay times of n-nonane clusters. We discover this result differs significantly from a classically derived decay time scale determined from a continuum thermodynamic treatment of the population balance equations of clusters. However, the dynamically generated results obtained from the kinetic decay model compare more favorably than the classical results with the empirical decay times that are deduced from experimental measurements of n-nonane clusters.     

A kinetic model for nonisothermic homogeneous nucleation

A model for isothermal homogeneous nucleation is proposed that improves the classical model. A quasiequilibrium distribution of clusters was calculated on a basis of the Frenkel’-Lothe-Pound theory. The dependence of the free energy of clusters on their size was represented by an interpolation formula relating the free energy of dimers and large clusters to which a notion of macroscopic surface tension is applicable. The nucleation rate and the dependence of the cluster temperature on their size were calculated by balance equations describing the heating of from a cluster due to the condensation of monomers and its cooling due to collisions with an ambient gas. It is shown that the nucleation rate in excess buffer gas is higher than for the pure condensing gas by approximately two orders of magnitude. The model adequately describes the experimental data for the nucleation of methanol supersaturated vapor.     

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.     

Fluorescence of rare gas clusters excited with synchrotron radiation

A cluster beam experiment for fluorescence measurements on rare gas clusters has been built up. First results on the evolution of energy levels of neutral krypton clusters with a cluster size between 200–8500 atoms/cluster are reported. The well known bulk excitons of solid krypton do not merge into the first atomic lines and appear only in clusters larger than 300 atoms/cluster. Smaller clusters absorb only at energies which fit well with surface excitons of the solid.     

On the statistical nature of collision and surface-induced dissociation: a theoretical investigation of aluminum clusters

The unimolecular dissociation dynamics of aluminum clusters following collision with either a rare gas atom or a surface is investigated by classical trajectory simulations with model potentials. Two conformers of Al(6) with very distinct shapes, i.e., the spherical O(h) and planar C(2)(h) clusters, are considered in this work. The initial vibrational energy and angular momentum distributions resulting from collision, as well as the energy and angular momentum resolved lifetime distributions, of excited clusters were determined for both collision-induced dissociation (CID) and surface-induced dissociation (SID) processes. The partitioning of excitation energy acquired upon collision was found to depend on the excitation mechanism (CID or SID), as well as on the cluster molecular shape, especially in the case of CID. For both types of processes, the energy and angular momentum resolved excited cluster lifetime distributions were found to decay exponentially, in agreement with statistical theories of chemical reactions, suggesting intrinsic Rice-Ramsperger-Kassel-Marcus (RRKM) behavior. Moreover, the simulated microcanonical rate constants determined from the cluster lifetime distributions are in good agreement with the predictions of the orbiting transition state model of phase space theory (OTS/PST), which further supports the statistical character of cluster CID and SID. Thus, in the CID and SID of highly fluxional systems such as aluminum clusters, the rate of intramolecular vibrational energy redistribution (IVR) is much faster than the dissociation rate, which validates one of the key assumptions, i.e., post-collision statistical behavior, underlying the models that are routinely used to determine cluster binding energies from experimental CID/SID cross sections.     

Molecular Dynamics Studies of the Kinetics of Phase Changes in Clusters IV： Crystal Nucleation from Molten （NaCl）256 and （NaCl）500 Clusters

Molecular dynamics computer simulation based on the Born-Mayer-Huggins potential function has been carried out to study the effects of duster size and temperature on the nucleation rate of sodium chloride dusters in the temperature range of 580 K to 630 K. Clusters with 256 and 500 NaCl molecules have been studied and the results have been compared with those obtained from 108 molecule dusters. The melting point (MP) of the clusters were observed to increase with the size of the clusters and can be well described by a linear equation MP =1107(37)-1229(23)N^-1/3(N is the number of molecules in the duster).The nucleation rate was found to decrease with increasing the duster size or temperature. Various nucleation theories have been used to interpret the nucleation rates obtained from this molecular dynamics simulation. It is possible to use a constant diffuse interface thickness to interpret the nucleation rate from the diffuse interface theory in the temperature range of this study. However, the interfacinl free energy estimated from classical nucleation theory and diffuse interface theory increases too fast with increasing the temperature while that from Gran-Gunton theory does not change with changing temperatures.The sizes of critical nuclei estimated from all the theories are smaller than those estimated from our simulations.     

Magic numbers, excitation levels, and other properties of small neutral 4He clusters (N < or = 50)

The ground-state energies and the radial and pair distribution functions of neutral 4He clusters are systematically calculated by the diffusion Monte Carlo method in steps of one 4He atom from 3 to 50 atoms. In addition the chemical potential and the low-lying excitation levels of each cluster are determined with high precision. These calculations reveal that the "magic numbers" observed in experimental 4He cluster size distributions, measured for free jet gas expansions by nondestructive matter-wave diffraction, are not caused by enhanced stabilities. Instead they are explained in terms of an enhanced growth due to sharp peaks in the equilibrium concentrations in the early part of the expansion. These peaks appear at cluster sizes which can just accommodate one more additional stable excitation. The good agreement with experiment provides not only experimental confirmation of the energy level and the chemical potential calculations, but also evidence for a new mechanism which can lead to magic numbers in cluster size distributions. By accounting for the falloff of the radial density distributions at the surface and a size-dependent surface tension, the energy levels are demonstrated to be consistent with a modified Rayleigh model of surface excitations. The compressibility coefficient of these small clusters is found to be one order of magnitude smaller than the bulk compressibility.     

Quasi-unary homogeneous nucleation of H2SO4-H2O

We show that the binary homogeneous nucleation (BHN) of H2SO4-H2O can be treated as quasi-unary nucleation of H2SO4 in equilibrium with H2O vapor. A scheme to calculate the evaporation coefficient of H2SO4 molecules from H2SO4-H2O clusters is presented and a kinetic model to simulate the quasi-unary nucleation of H2SO4-H2O is developed. In the kinetic model, the growth and evaporation of sulfuric acid clusters of various sizes are explicitly simulated. The kinetic quasi-unary nucleation model does not have two well-recognized problems associated with the classical BHN theory (violation of the mass action law and mismatch of the cluster distribution for monomers) and is appropriate for the situations where the assumption of equilibrium cluster distribution is invalid. The nucleation rates predicted with our quasi-unary kinetic model are consistent with recent experimental nucleation experiments in all the cases studied, while the most recent version of the classical BHN model systematically overpredicts the nucleation rates. The hydration of sulfuric acid clusters, which is not considered in the classical model but is accounted for implicitly in our kinetic quasi-unary model, is likely to be one of physical mechanisms that lead to lower nucleation rates. Further investigation is needed to understand exactly what cause the difference between the kinetic quasi-unary model and the classical BHN model.     

Evaporation of hot mesoscopic atomic metal clusters

The evaporation times of hot mesoscopic sodium clusters are estimated using the microscopic and microcanonical statistical model of Weisskopf and the macroscopic canonical kinetic gas theory. The level density of liquid sodium clusters as function of the internal excitation energy is computed from the experimentally known thermodynamic quantities of the bulk matter. In using the known temperature dependent surface tension and standard thermodynamics, both theories are corrected to take into account finite size effects. By this method it is possible to obtain decay rates and internal level densities of mesoscopic clusters at excitations where any harmonic approximation breaks down. For clusters having more than about 200 atoms, the evaporation times obtained from the two models are practically equal.