Energy dependent decay rates of Lennard-Jones clusters for use in nucleation theory

Decay rates of small clusters (containing between 10 and 40 Lennard-Jones atoms) are determined by molecular dynamics simulations. The cluster is defined by the condition that the atoms must lie within a specified distance of their center of mass, and initial isothermal states are generated using a Metropolis Monte Carlo method. Plots of the logarithm of the survival fraction against time are found to be nonlinear, indicating that the decay of constant temperature clusters is non-Markovian and depends on the collision rate with a thermalizing gas. However, when the clusters are banded according to their energies, exponential decay is seen. The energy dependent decay rates from simulations agree to within a factor of 2 with those estimated from equilibrium considerations (using free energies from thermodynamic integration and assuming a Gaussian energy distribution), indicating that clusters defined in this way can be used in Markovian rate equations. During nucleation, the cluster energy distribution is shifted from its equilibrium value, leading to a reduction in the nucleation rate by a temperature dependent factor of 100 or more, in the absence of a thermalizing carrier gas.     

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.     

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.     

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.     

Multicomponent dynamical nucleation theory and sensitivity analysis

Vapor to liquid multicomponent nucleation is a dynamical process governed by a delicate interplay between condensation and evaporation. Since the population of the vapor phase is dominated by monomers at reasonable supersaturations, the formation of clusters is governed by monomer association and dissociation reactions. Although there is no intrinsic barrier in the interaction potential along the minimum energy path for the association process, the formation of a cluster is impeded by a free energy barrier. Dynamical nucleation theory provides a framework in which equilibrium evaporation rate constants can be calculated and the corresponding condensation rate constants determined from detailed balance. The nucleation rate can then be obtained by solving the kinetic equations. The rate constants governing the multistep kinetics of multicomponent nucleation including sensitivity analysis and the potential influence of contaminants will be presented and discussed.     

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.     

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.     

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.     

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.     

Orientation of a benzene molecule inside a carbon nanotube

Benzene molecules confined in carbon nanotubes of varying radii are employed as semiconductors in electronic nanodevices, and their orientation determines the electrical properties of the system. In this paper, we investigate the interaction energy of all the possible configurations of a benzene molecule inside various carbon nanotubes and then we determine the equilibrium configuration. We adopt the continuous approach together with the semi-empirical Lennard-Jones potential function to model van der Waals interaction between a benzene molecule and a carbon nanotube. This approach results in an analytical expression, which accurately approximates the interaction energy and can be readily used to generate numerical data. We find that horizontal, tilted and perpendicular configurations on the axis of the carbon nanotube are all possible equilibrium configurations of the benzene molecule when the radius of the carbon nanotube is less than 5.580 Å. However, when the radius of the carbon nanotube is larger than 5.580 Å an offset horizontal orientation is the only possible equilibrium configuration of the benzene molecule. In the limiting case, the orientation of a benzene molecule on a graphene sheet can be derived simply by letting the radius of the carbon nanotube tend to infinity.     

Heterogeneous condensation of the Lennard-Jones vapor onto a nanoscale seed particle

The heterogeneous condensation of a Lennard-Jones vapor onto a nanoscale seed particle is studied using molecular dynamics simulations. Measuring the nucleation rate and the height of the free energy barrier using the mean first passage time method shows that the presence of a weakly interacting seed has little effect on the work of forming very small cluster embryos but accelerates the rate by lowering the barrier for larger clusters. We suggest that this results from a competition between the energetic and entropic features of cluster formation in the bulk and at the heterogeneity. As the interaction is increased, the free energy of formation is reduced for all cluster sizes. We also develop a simple phenomenological model of film formation on a small seed that captures the general features of the nucleation process for small heterogeneities. A comparison of our simulation results with the model shows that heterogeneous classical nucleation theory provides a good estimate of the critical size of the film but significantly overestimates the size of the barrier.     

A small-angle scattering study on equilibrium clusters in lysozyme solutions

We use small-angle scattering experiments to investigate the structural properties of aqueous lysozyme solutions under conditions where the existence of equilibrium clusters has recently been demonstrated (Nature 2004, 432, 492). We also discuss the possible emergence of a low angle scattering contribution, which recently attracted interest due to its appearance in solutions of various proteins. We demonstrate that in lysozyme solutions under our experimental conditions such rising low q intensities can only be observed under special circumstances and can thus not be attributed to the existence of a universal long-range attraction. We then focus on the structural properties of the equilibrium clusters as a function of protein concentration, temperature, and ionic strength. We show that the experimental structure factors obtained from the scattering measurements exhibit the typical cluster-cluster peak q(c) reflecting the mean distance between charged clusters as well as a monomer-monomer peak q(m), which represents the nearest neighbor shell of monomers within a single cluster. The underlying principle for the formation of these structures is the coexistence of two opposing forces, a short-range attraction and a long-range repulsion due to residual charges. We can quantitatively analyze our scattering data by applying a simple equilibrium cluster model and calculate an average cluster aggregation number, N(c). The thus obtained cluster aggregation number increases linearly with volume fraction. We also observe an increasing N(c) as temperature decreases and as the screening of residual charges increases. We point out the importance of the existence of equilibrium clusters and the universality of this phenomenon for self-assembling processes observed in nature. Finally, we discuss the limitations of our simple globular cluster model in view of recent findings from computer simulations.     

Computer simulations of nucleation of nanoparticle superclusters from solution

This paper presents simulation studies of nanoparticle supercluster (NPSC) nucleation from a temperature quenched system. The nanoparticles are represented as 5 nm, spherical gold nanoparticles ligated with alkane thiols. The pair potential accounts for the van der Waals interaction between the metallic cores and ligand-ligand and ligand-solvent interactions. Phenomena well-known for molecular systems are observed including a prenucleation induction period, fluctuating prenucleation clusters that predominately add monomers one at a time, a critical nucleus size, and growth of NPSCs from solution in the presence of an equilibrium supernatant, all consistent with classical nucleation theory. However, only the largest prenucleating clusters are dense, and the cluster size can occasionally range greater than the critical size in the prenucleation regime until a cluster with low enough energy occurs, then nucleation ensues. Late in the nucleation process, the clusters display a crystalline structure that is a random mix of face-centered cubic (fcc) and hexagonal close-packed (hcp) lattices and indistinguishable from a randomized icosahedra structure.     

氟化钾团簇的相变、成核和重结晶的分子动力学模拟

我们利用Born-Mayer-Huggins相互作用势函数对(KF)_N(N=108，256，500和864)团簇进行了分子动力学(MD)模拟。为了避免周期性边界条件对相变、成核和重结晶的干扰作用，对体系采用了自由边界。基于MD模拟结果，对团簇的熔化温度、熔化焓、扩散系数、成核速率、固液界面自由能、临界核大小等进行了计算和讨论。在对(KF)₈₆₄双晶团簇的热退火模拟中，观察到了固态的重结晶和晶粒的生长。经典的成核理论成功地解释了(KF)₈₆₄双晶团簇的重结晶MD模拟结果。     

Atomic‐Level Mechanisms of Nucleation of Pure Liquid Metals during Rapid Cooling

To obtain a material with the desired performance, the atomic‐level mechanisms of nucleation from the liquid to solid phase must be understood. Although this transition has been investigated experimentally and theoretically, its atomic‐level mechanisms remain debatable. In this work, the nucleation mechanisms of pure Fe under rapid cooling conditions are investigated. The local atomic packing stability and liquid‐to‐solid transition‐energy pathways of Fe are studied using molecular dynamics simulations and first‐principle calculations. The results are expressed as functions of cluster size in units of amorphous clusters (ACs) and body‐centered cubic crystalline clusters (BCC‐CCs). We found the prototypes of ACs in supercooled liquids and successfully divided these ACs to three categories according to their transition‐energy pathways. The information obtained in this study could contribute to our current understanding of the crystallization of metallic melts during rapid cooling.     

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.     

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

Molecular dynamics computer simulations have been carried out to study the effects of cluster size and temperature on the nucleation rate of rubidium chloride clusters in the temperature range of 500-650 K. Clusters with 256 and 500 RbCl molecules have been studied and the results are compared with those obtained from 108 molecule clusters. The melting point (MP) of the clusters was observed to increase with the size of the clusters and can be described by a linear equation MP=997-405 N^−1/3, where N is the number of molecules in the cluster. The nucleation rate is found to decrease with increasing cluster size or increasing nucleation temperature. Both classical nucleation theory and diffuse interface theory are used to interpret our observed results.     

Comparison between the classical theory predictions and molecular simulation results for heterogeneous nucleation of argon

We have performed Monte Carlo simulations of homogeneous and heterogeneous nucleations of Lennard-Jones argon clusters. The simulation results were interpreted using the major concept posing a difference between the homogeneous and heterogeneous classical nucleation theories-the contact parameter. Our results show that the multiplication concept of the classical heterogeneous nucleation theory describes the cluster-substrate interaction surprisingly well even for small molecular clusters. However, in the case of argon nucleating on a rigid monolayer of fcc(111) substrate at T=60 K, the argon-substrate atom interaction being approximately one-third as strong as the argon-argon interaction, the use of the classical theory concept results in an underestimation of the heterogeneous nucleation rate by two to three orders of magnitude even for large clusters. The main contribution to this discrepancy is induced by the failure of the classical theory of homogeneous nucleation to predict the energy involved in bringing one molecule from the vapor to the cluster for clusters containing less than approximately 15 molecules.     

Implementation of dynamical nucleation theory effective fragment potentials method for modeling aerosol chemistry

In this work, the dynamical nucleation theory (DNT) model using the ab initio based effective fragment potential (EFP) is implemented for evaluating the evaporation rate constant and molecular properties of molecular clusters. Predicting the nucleation rates of aerosol particles in different chemical environments is a key step toward understanding the dynamics of complex aerosol chemistry. Therefore, molecular scale models of nanoclusters are required to understand the macroscopic nucleation process. On the basis of variational transition state theory, DNT provides an efficient approach to predict nucleation kinetics. While most DNT Monte Carlo simulations use analytic potentials to model critical sized clusters, or use ab initio potentials to model very small clusters, the DNTEFP Monte Carlo method presented here can treat up to critical sized clusters using the effective fragment potential (EFP), a rigorous nonempirical intermolecular model potential based on ab initio electronic structure theory calculations, improvable in a systematic manner. The DNTEFP method is applied to study the evaporation rates, energetics, and structure factors of multicomponent clusters containing water and isoprene. The most probable topology of the transition state characterizing the evaporation of one water molecule from a water hexamer at 243 K is predicted to be a conformer that contains six hydrogen bonds, with a topology that corresponds to two water molecules stacked on top of a quadrangular (H(2)O)(4) cluster. For the water hexamer in the presence of isoprene, an increase in the cluster size and a 3-fold increase in the evaporation rate are predicted relative to the reaction in which one water molecule evaporates from a water hexamer cluster.     

Computer optimization of small atomic clusters

We have successfully identified stable configurations of both rare-gas and NaCl clusters with a new optimization procedure. An initial cluster configuration is prepared in a so-called shoot-and-stay process. Its total energy is then minimized with respect to the atomic coordinates. To prevent the system from being locked in local minima, the step size of each move is chosen as the width of the energy well at a higher level. As the system evolves, the global minimum is contained in the volume bounded by the decreasing value of step sizes. We have also carried out the optimization of NaCl clusters by the simulated annealing technique, for comparison. The results show that for such heterogeneous systems, the latter method cannot always find the global minimum, because of large energy gaps between different catchment regions in phase space.