Kinetic theory of binary nucleation based on a first passage time analysis

The binary classical nucleation theory (BCNT) is based on the Gibbsian thermodynamics and applies the macroscopic concept of surface tension to nanosize clusters. This leads to severe inconsistencies and large discrepancies between theoretical predictions and experimental results regarding the nucleation rate. We present an alternative approach to the kinetics of binary nucleation which avoids the use of classical thermodynamics for clusters. The new approach is an extension to binary mixtures of the kinetic theory previously developed by Narsimhan and Ruckenstein and Ruckenstein and Nowakowski [J. Colloid Interface Sci. 128, 549 (1989); 137, 583 (1990)] for unary nucleation which is based on molecular interactions and in which the rate of emission of molecules from a cluster is determined via a mean first passage time analysis. This time is calculated by solving the single-molecule master equation for the probability distribution of a "surface" molecule moving in a potential field created by the cluster. The starting master equation is a Fokker-Planck equation for the probability distribution of a surface molecule with respect to its phase coordinates. Owing to the hierarchy of characteristic time scales in the evolution of the molecule, this equation can be reduced to the Smoluchowski equation for the distribution function involving only the spatial coordinates. The new theory is combined with density functional theory methods to determine the density profiles. This is essential for nucleation in binary systems particularly when one of the components is surface active. Knowing these profiles, one can determine the potential fields created by the cluster, its rate of emission of molecules, and the nucleation rate more accurately than by using the uniform density approximation. The new theory is illustrated by numerical calculations for a model binary mixture of Lennard-Jones monomers and rigidly bonded dimers of Lennard-Jones atoms. The amphiphilic character of the dimer component (i.e., its surface activity) is induced by the asymmetry in the interaction between a monomer and the two different sites of a dimer. The inconsistencies of the BCNT are avoided in the new theory.     

Recent developments in the kinetic theory of nucleation

A review of recent progress in the kinetics of nucleation is presented. In the conventional approach to the kinetic theory of nucleation, it is necessary to know the free energy of formation of a new-phase particle as a function of its independent variables at least for near-critical particles. Thus the conventional kinetic theory of nucleation is based on the thermodynamics of the process. The thermodynamics of nucleation can be examined by using various approaches, such as the capillarity approximation, density functional theory, and molecular simulation, each of which has its own advantages and drawbacks. Relatively recently a new approach to the kinetics of nucleation was proposed [Ruckenstein E, Nowakowski B. J Colloid Interface Sci 1990;137:583; Nowakowski B, Ruckenstein E. J Chem Phys 1991;94:8487], which is based on molecular interactions and does not employ the traditional thermodynamics, thus avoiding such a controversial notion as the surface tension of tiny clusters involved in nucleation. In the new kinetic theory the rate of emission of molecules by a new-phase particle is determined with the help of a mean first passage time analysis. This time is calculated by solving the single-molecule master equation for the probability distribution function of a surface layer molecule moving in a potential field created by the rest of the cluster. The new theory was developed for both liquid-to-solid and vapor-to-liquid phase transitions. In the former case the single-molecule master equation is the Fokker-Planck equation in the phase space which can be reduced to the Smoluchowski equation owing to the hierarchy of characteristic time scales. In the latter case, the starting master equation is a Fokker-Planck equation for the probability distribution function of a surface layer molecule with respect to both its energy and phase coordinates. Unlike the case of liquid-to-solid nucleation, this Fokker-Planck equation cannot be reduced to the Smoluchowski equation, but the hierarchy of time scales does allow one to reduce it to the Fokker-Plank equation in the energy space. The new theory provides an equation for the critical radius of a new-phase particle which in the limit of large clusters (low supersaturations) yields the Kelvin equation and hence an expression for the macroscopic surface tension. The theory was illustrated with numerical calculations for a molecular pair interaction potential combining the dispersive attraction with the hard-sphere repulsion. The results for the liquid-to-solid nucleation clearly show that at given supersaturation the nucleation rate depends on the cluster structure (for three cluster structures considered-amorphous, fcc, and icosahedral). For both the liquid-to-solid and vapor-to-liquid nucleation, the predictions of the theory are consistent with the results of classical nucleation theory (CNT) in the limit of large critical clusters (low supersaturations). For small critical clusters the new theory provides higher nucleation rates than CNT. This can be accounted for by the fact that CNT uses the macroscopic interfacial tension which presumably overpredicts the surface tension of small clusters, and hence underpredicts nucleation rates.     

Mean-field kinetic nucleation theory

A new semiphenomenological model of homogeneous vapor-liquid nucleation is proposed in which the cluster kinetics follows the "kinetic approach to nucleation" and the thermodynamic part is based on the revised Fisher droplet model with the mean-field argument for the cluster configuration integral. The theory is nonperturbative in a cluster size and as such is valid for all clusters down to monomers. It contains two surface tensions: macroscopic (planar) and microscopic. The latter is a temperature dependent quantity related to the vapor compressibility factor at saturation. For Lennard-Jones fluids the microscopic surface tension possesses a universal behavior with the parameters found from the mean-field density functional calculations. The theory is verified against nucleation experiments for argon, nitrogen, water, and mercury, demonstrating very good agreement with experimental data. Classical nucleation theory fails to predict experimental results when a critical cluster becomes small.     

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.     

To the theory of homogeneous nucleation: Cluster energy

An attempt is made to critically analyze the modern state of the theory of homogeneous nucleation as concerns its ability to describe experiments with high accuracy. An analysis of the experimental data led us to conclude that the dependence of the nucleation rate on supersaturation and temperature T was not described by the theory, which underestimates the critical cluster size compared with the Gibbs-Thomson equation. The possibility of applying density functional theory (one of the latest achievements in the theory of homogeneous nucleation) was questioned. Within this theory, the Gibbs-Thomson equation remains valid even outside the classic capillary approximation. It is suggested that, to bring theory in consistency with experiment, certain fundamental propositions of the theory of nucleation should be revised. The inclusion of an additional contribution to the Gibbs energy of a cluster caused by the size dependence of the specific heat capacity of the cluster decreases the critical cluster size compared with the value calculated by the Gibbs-Thomson equation. The calculated dependence of nucleation rate on supersaturation was in agreement with the experimental results.     

Real-time time-dependent density functional theory using density fitting and the continuous fast multipole method

An implementation of real-time time-dependent density functional theory (RT-TDDFT) within the TURBOMOLE program package is reported using Gaussian-type orbitals as basis functions, second and fourth order Magnus propagator, and the self-consistent field as well as the predictor–corrector time integration schemes. The Coulomb contribution to the Kohn–Sham matrix is calculated combining density fitting approximation and the continuous fast multipole method. Performance of the implementation is benchmarked for molecular systems with different sizes and dimensionalities. For linear alkane chains, the wall time for density matrix time propagation step is comparable to the Kohn-Sham (KS) matrix construction. However, for larger two- and three-dimensional molecules, with up to about 5,000 basis functions, the computational effort of RT-TDDFT calculations is dominated by the KS matrix evaluation. In addition, the maximum time step is evaluated using a set of small molecules of different polarities. The photoabsorption spectra of several molecular systems calculated using RT-TDDFT are compared to those obtained using linear response time-dependent density functional theory and coupled cluster methods.     

Temperature effects on the nucleation mechanism of protein folding and on the barrierless thermal denaturation of a native protein

Recently, the authors proposed a kinetic model for the nucleation mechanism of protein folding where a protein was treated as a heteropolymer with all the bonds and bond angles equal and constant. As a crucial idea of the model, an overall potential around a cluster of native residues in which a protein residue performs a chaotic motion is considered to be a combination of three potentials: effective pairwise, average dihedral, and confining. The overall potential as a function of the distance from the cluster center has a double well shape which allows one to determine the rates with which the cluster emits and absorbs residues by using a first passage time analysis. One can then develop a kinetic theory for the nucleation mechanism of protein folding and evaluate the protein folding time. In the present paper we evaluate the optimal temperature at which the protein folding time is the shortest. A method is also proposed to determine the temperature dependence of the folding time without carrying out the time consuming calculations for a series of temperatures. Using Taylor series expansions in the formalism of the first passage time analysis, one can calculate the temperature dependence of the cluster emission and absorption rates in the vicinity of some temperature T(0) if they are known at T(0). Thus one can evaluate the protein folding time t(f) at any other temperature T in the vicinity of T(0) at which the folding time t(f) is known. We also present a model for the thermal denaturation of a protein occurring via the decay of the native structure of the protein. Due to a sufficiently large temperature increase or decrease, the rate with which a cluster of native residues within a protein emits residues becomes larger than the absorption rate in the whole range of cluster sizes up to the size of the whole protein. This leads to the unfolding of the protein in a barrierless way, i.e., as spinodal decomposition. Knowing the cluster emission and absorption rates as functions of temperature and cluster size, one can find the threshold temperatures of cold and hot barrierless denaturation as well as the corresponding unfolding times. Both proposed methods are illustrated by numerical calculations for two model proteins, one consisting of 124 amino acids, the other consisting of 2500 residues. The first one roughly mimicks a bovine pancreatic ribonuclease while the second one is a representative of the largest proteins which are extremely difficult to study by straightforward Monte Carlo or molecular dynamics simulations.     

Key thermodynamic characteristics of nucleation on charged and neutral cores of molecular sizes in terms of the gradient density functional theory

The gradient density functional theory and the Carnahan–Starling model formulated for describing the contribution of hard spheres have been used to calculate the profiles of condensate density in small critical droplets formed via homogeneous nucleation, as well as in stable and critical droplets formed via heterogeneous nucleation on solid charged and neutral condensation cores of molecular sizes. The calculations performed for water and argon at different values of condensate chemical potential have yielded the heights of the activation barriers for homoand heterogeneous nucleation as functions of vapor supersaturation at preset system temperatures. The interaction of condensate molecules with a solid core has been described by the resultant potential of molecular attractive forces. In the case of a charged core, the long-range Coulomb potential of electric forces has additionally been taken into account. Dielectric permittivities have been calculated as known functions of the local density of the fluid and temperature. The radius of the equimolecular droplet surface has been chosen as a variable describing the droplet size. Dependences of the chemical potential of condensate molecules in a droplet on its size have been plotted for water and argon with allowance for the action of capillary, electrostatic, and molecular forces. It has been shown that the role of the molecular force potential in heterogeneous nucleation increases with the size of condensation cores.     

A kinetic approach to the theory of heterogeneous nucleation on soluble particles during the deliquescence stage

Deliquescence is the dissolution of a solid nucleus in a liquid film formed on the nucleus due to vapor condensation. Previously, the kinetics of deliquescence was examined in the framework of the capillarity approximation which involves the thermodynamic interfacial tensions for a thin film and the approximation of uniform density therein. In the present paper we propose a kinetic approach to the theory of deliquescence which avoids the use of the above macroscopic quantities for thin films. The rates of emission of molecules from the liquid film into the vapor and from the solid core into the liquid film are determined through a first passage time analysis whereas the respective rates of absorption are calculated through the gas kinetic theory. The first passage time is obtained by solving the single-molecule master equation for the probability distribution of a "surface" molecule moving in a potential field created by the cluster. Furthermore, the time evolution of the liquid film around the solid core is described by means of two mass balance equations which involve the rates of absorption and emission of molecules by the film at its two interfaces. When the deliquescence of an ensemble of solid particles occurs by means of large fluctuations, the time evolution of the distribution of composite droplets (liquid film+solid core) with respect to the independent variables of state is governed by a Fokker-Planck kinetic equation. When both the vapor and the solid soluble particles are single component, this equation has the form of the kinetic equation of binary nucleation. A steady-state solution for this equation is obtained by the method of separation of variables. The theory is illustrated with numerical calculation regarding the deliquescence of spherical particles in a water vapor with intermolecular interactions of the Lennard-Jones kind. The new approach allows one to qualitatively explain an important feature of experimental data on deliquescence, namely the occurrence of nonsharp deliquescence, a feature that the previous deliquescence theory based on classical thermodynamics could not account for.     

Development of the cyclic cluster model formalism for Kohn-Sham auxiliary density functional theory methods

The development of the cyclic cluster model (CCM) formalism for Kohn-Sham auxiliary density functional theory (KS-ADFT) methods is presented. The CCM is a direct space approach for the calculation of perfect and defective systems under periodic boundary conditions. Translational symmetry is introduced in the CCM by integral weighting. A consistent weighting scheme for all two-center and three-center interactions appearing in the KS-ADFT method is presented. For the first time, an approach for the numerical integration of the exchange-correlation potential within the cyclic cluster formalism is derived. The presented KS-ADFT CCM implementation was applied to covalent periodic systems. The results of cyclic and molecular cluster model (MCM) calculations for trans-polyacetylene, graphene, and diamond are discussed as examples for systems periodic in one, two, and three dimensions, respectively. All structures were optimized. It is shown that the CCM results represent the results of MCM calculations in the limit of infinite molecular clusters. By analyzing the electronic structure, we demonstrate that the symmetry of the corresponding periodic systems is retained in CCM calculations. The obtained geometric and electronic structures are compared with available data from the literature.     

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.     

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.     

Correlation potentials for a multiconfigurational-based density functional theory with exact exchange

A density functional theory based on a complete active space self consistent field (CASSCF) reference function with exact exchange is discussed. It is first shown that such a theory may be formulated with a correlation potential dependent on the density function and on the active space used. Auxiliary functions, such as the on-top two-particle density, are used to define uniquely the potential for different active spaces. The paper also analyses the correlation functional for some atomic and molecular cases. Large ab initio calculations are performed to obtain accurate density functions. A correlation potential is then fitted such that the reference CASSCF function gives the same density. The correlation potential values are saved in a data base for future analysis.     

Model for the nucleation mechanism of protein folding

A nucleation-like pathway of protein folding involves the formation of a cluster containing native residues that grows by including residues from the unfolded part of the protein. This pathway is examined by using a heteropolymer as a protein model. The model heteropolymer consists of hydrophobic and hydrophilic beads with fixed bond lengths and bond angles. The total energy of the heteropolymer is determined by the pairwise repulsive/attractive interactions between nonlinked beads and by the contribution from the dihedral angles involved. The parameters of these interactions can be rigorously defined, unlike the ill-defined surface tension of a cluster of protein residues that constitutes the basis of a previous nucleation model. The main idea underlying the new model consists of averaging the dihedral potential of a selected residue over all possible configurations of all neighboring residues along the protein chain. The resulting average dihedral potential depends on the distance between the selected residue and the cluster center. Its combination with the average pairwise potential of the selected residue and with a confining potential caused by the bonds between the residues leads to an overall potential around the cluster that has a double-well shape. Residues in the inner (closer to the cluster) well are considered as belonging to the folded cluster, whereas those in the outer well are treated as belonging to the unfolded part of the protein. Transitions of residues from the inner well into the outer one and vice versa are considered as elementary emission and absorption events, respectively. The double-well character of the potential well around the cluster allows one to determine the rates of both emission and absorption of residues by the cluster using a first passage time analysis. Once these rates are found as functions of the cluster size, one can develop a self-consistent kinetic theory for the nucleation mechanism of folding of a protein. The model allows one to evaluate the size of the nucleus and the protein folding time. The latter is evaluated as the sum of the times necessary for the first nucleation event to occur and for the nucleus to grow to the maximum size (of the folded protein). Depending on the diffusion coefficients of the native residues in the range from 10(-6) to 10(-8) cm2/s, numerical calculations for a protein of 2500 residues suggest that the folding time ranges from several seconds to several hundreds of seconds.     

Molecular dynamics analysis of multiple site growth and coalescence effects on homogeneous and heterogeneous nucleations

Homogeneous and heterogeneous nucleations were simulated by molecular dynamics (MD). The behavior of Lennard-Jones molecules was studied inside a liquid-gas system where all dimensions of the wall were periodic and a soft core carrier gas within the system controlled the temperature. In this study, the classical nucleation theory was found to underestimate the homogeneous nucleation rate by five orders of magnitude, which complies with other MD studies. The discrepancy in the nucleation rate between theory and simulation was mainly caused by the fundamental assumption that there are no volumetric interactions in the growth process. In this particular case, however, growth was observed at multiple sites due to Ostwald ripening and coalescence between nuclei by Brownian motion. Furthermore, even though the supersaturation ratio is inadequate for homogeneous nucleation, once a seed is introduced to the system, a cluster can be created. The addition of seeds not only enhances nucleation but also renders coalescence as an important nucleation mechanism in the earlier stages compared to homogeneous nucleation.     

Numerical Modeling of Crystallization Nucleation in Glass with Temperature Dependent Viscosity

1 INTRODUCTION Knowing the nucleation rate at different tempe-ratures, especially the temperature when the nuclea-tion rate reaches its maximum for glass, is of bothscientific and technological importance. These dataare significant in understanding the stability of glassfor practical applications where the formation of nuc-lei and their subsequent growth to crystals must beavoided (such as for optical fibers, laser glass, opti-cal switching glass), as well as for preparing laserglass-ceram…     

Crossover model for the work of critical cluster formation in nucleation theory

We propose a relation for the work of critical cluster formation in nucleation theory W for the systems with long-range interparticle interactions. The method of bridge functions is used to combine the system behavior at sufficiently small quenches, adequately predicted by the classical nucleation theory, with nonclassical effects at deep quenches in the vicinity of the thermodynamic spinodal, described within the framework of the field theoretical approach with an appropriate Ginzburg-Landau functional. The crossover between the two types of nucleation behavior takes place in the vicinity of the kinetic spinodal where the lifetime of a metastable state is of the order of the relaxation time to local equilibrium. We argue that the kinetic spinodal corresponds to the minimum of the excess number of molecules in the critical cluster. This conjecture leads to the form of W containing no adjustable parameters. The barrier scaling function Gamma = W/W(cl), where W(cl) is the classical nucleation barrier, depends parametrically on temperature through the dimensionless combination of material properties. The results for argon nucleation are presented.     

Derivation of force field parameters for SnO2-H2O surface systems from plane-wave density functional theory calculations

Plane-wave density functional theory (DFT-PW) calculations were performed on bulk SnO2 (cassiterite) and the (100), (110), (001), and (101) surfaces with and without H2O present. A classical interatomic force field has been developed to describe bulk SnO2 and SnO2-H2O surface interactions. Periodic density functional theory calculations using the program VASP (Kresse et al., 1996) and molecular cluster calculations using Gaussian 03 (Frisch et al., 2003) were used to derive the parametrization of the force field. The program GULP (Gale, 1997) was used to optimize parameters to reproduce experimental and ab initio results. The experimental crystal structure and elastic constants of SnO2 are reproduced reasonably well with the force field. Furthermore, surface atom relaxations and structures of adsorbed H2O molecules agree well between the ab initio and force field predictions. H2O addition above that required to form a monolayer results in consistent structures between the DFT-PW and classical force field results as well.     

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.