Heterogeneous multicomponent nucleation theorems for the analysis of nanoclusters

In this paper we present a new form of the nucleation theorems applicable to heterogeneous nucleation. These heterogeneous nucleation theorems allow, for the first time, direct determination of properties of nanoclusters formed on pre-existing particles from measured heterogeneous nucleation probabilities. The theorems can be used to analyze the size (first theorem) and the energetics (second theorem) of heterogeneous clusters independent of any specific nucleation model. We apply the first theorem to the study of small water and n-propanol clusters formed at the surface of 8 nm silver particles. According to the experiments the size of the two-component critical clusters is found to be below 90 molecules, and only less than 20 molecules for pure water, less than 300 molecules for pure n-propanol. These values are drastically smaller than the ones predicted by the classical nucleation theory, which clearly indicates that the nucleating clusters are too small to be quantitatively described using a macroscopic theory.     

A study of nucleation in supersaturated ibuprofen vapor in a flow diffusion chamber

Isothermal nucleation of supersaturated ibuprofen racemate vapor has been experimentally studied in a flow diffusion chamber at 293.3 and 301.2 K. Nucleation rates have been measured in the range of 10⁴?10⁴ cm^?3 s^?1 as functions of supersaturation. According to the first nucleation theorem, the numbers of molecules in critical nuclei have been found and used to determine the nucleation rate and supersaturation values as depending on the sizes of critical nuclei. The comparison of the experimental data with theoretical predictions has shown that the nucleation rates measured as functions of the numbers of molecules in critical nuclei are higher than the rates predicted by the classical theory by six to seven decimal orders of magnitude but, within one order of magnitude, coincide with the rates predicted by a theory previously proposed in a work by one of the authors, in which nucleation clusters were considered to be microscopic objects.     

微电极上单核沉积的现场电化学原子力显微镜研究

用现场电化学原子力显微镜（ＥＣＡＦＭ）观察玻碳微电极上银单核电化学成核与生长的不同阶段的行为,结果表明银单核在玻碳边缘生成以（１００）面择优的晶粒,证实了成核过程强烈地信赖于玻碳棋底的状态,有成核历史的基底沉积多核,经特殊方法“清洗”的基底沉积单核,本工作首次把ＥＣＡＦＭ应用于微电极系统的研究,对微是极上的单核生长进行实时观察;同非现场测量进行比较,演示一种轩单晶微电极的可行方法。     

Rethinking the application of the first nucleation theorem to particle formation

The critical cluster is the threshold size above which a cluster will be more likely to grow than to evaporate. In field and laboratory measurements of new particle formation, the number of molecules of a given species in the critical cluster is commonly taken to be the slope of the log-log plot of the formation rate versus the concentration of the species. This analysis is based on an approximate form of the first nucleation theorem, which is derived with the assumption that there are no minima in the free energy surface prior to the maximum at the critical size. However, many atmospherically relevant systems are likely to exhibit such minima, for example, ions surrounded by condensable vapour molecules or certain combinations of acids and bases. We have solved numerically the birth-death equations for both an electrically neutral one-component model system with a local minimum at pre-critical sizes and an ion-induced case. For the ion-induced case, it is verified that the log-log slope of the nucleation rate versus particle concentration plot gives accurately the difference between the cluster sizes at the free energy maximum and minimum, as is expected from the classical form of the ion-induced nucleation rate. However, the results show that applying the nucleation theorem to neutral systems with stable pre-nucleation clusters may lead to erroneous interpretations about the nature of the critical cluster.     

Volumes of critical bubbles from the nucleation theorem

A corollary of the nucleation theorem due to Kashchiev [Nucleation: Basic Theory with Applications (Butterworth-Heinemann, Oxford, 2000)] allows the volume V(*) of a critical bubble to be determined from nucleation rate measurements. The original derivation was limited to one-component, ideal gas bubbles with a vapor density much smaller than that of the ambient liquid. Here, an exact result is found for multicomponent, nonideal gas bubbles. Provided a weak density inequality holds, this result reduces to Kashchiev's simple form which thus has a much broader range of applicability than originally expected. Limited applications to droplets are also mentioned, and the utility of the p(T,x) form of the nucleation theorem as a sum rule is noted.     

Comparison of theories of ion-induced nucleation

A theory of nucleation on ions and a theory of nucleation in the presence of ions are presented. The calculated data obtained within these theories are compared to the experimental data for the nucleation of vapors of different compounds. It is shown that the theory of nucleation in the presence of ions is in better agreement with the experimental data than the theory of nucleation on ions. It is assumed within the theory of nucleation in the presence of ions that each ion in the vapor phase forms a stable ion associate with vapor molecules.     

Multivariate analysis of homogeneous nucleation rate measurements. Nucleation in the p-toluic acid/sulfuric acid/water system

Recent kinetic extensions of the nucleation theorem suggest that the logarithm of the steady-state nucleation rate has strong multilinear dependence on the log concentrations of condensable species present in the vapor phase. A further remarkable result is that the coefficients of this linear dependency provide a direct determination of the molecular content of the critical nucleus itself. Building on these results, the powerful utility of multivariate statistical methods is demonstrated here for physically based parametrization and interpretation of nucleation rate measurements. The new approach is applied to recent measurements by Zhang et al. [Science 304, 1487 (2004)] on the p-toluic acid/sulfuric acid/water ternary vapor system. A linear minimum variance parametrization for nucleation rate dependence on vapor composition, accurate over the range of the measurements, is obtained. Estimates of critical nucleus molecular composition are also presented. These suggest that a single molecule of p-toluic acid present in the critical nucleus is sufficient to trigger a ternary nucleation event. Efforts under way to apply the new methods to analysis of new particle formation in the atmosphere are discussed.     

Multidimensional view of amyloid fibril nucleation in atomistic detail

Starting from a disordered aggregate, we have simulated the formation of ordered amyloid-like beta structures in a system formed by 18 polyvaline chains in explicit solvent, employing molecular dynamics accelerated by bias-exchange metadynamics. We exploited 8 different collective variables to compute the free energy of hundreds of putative aggregate structures, with variable content of parallel and antiparallel β-sheets and different packing among the sheets. This allowed characterizing in detail a possible nucleation pathway for the formation of amyloid fibrils: first the system forms a relatively large ordered nucleus of antiparallel β-sheets, and then a few parallel sheets start appearing. The relevant nucleation process culminates at this point: when a sufficient number of parallel sheets is formed, the free energy starts to decrease toward a new minimum in which this structure is predominant. The complex nucleation pathway we found cannot be described within classical nucleation theory, namely employing a unique simple reaction coordinate like the total content of β-sheets.     

Atomistic theory of amyloid fibril nucleation

We consider the nucleation of amyloid fibrils at the molecular level when the process takes place by a direct polymerization of peptides or protein segments into β-sheets. Employing the atomistic nucleation theory (ANT), we derive a general expression for the work to form a nanosized amyloid fibril (protofilament) composed of successively layered β-sheets. The application of this expression to a recently studied peptide system allows us to determine the size of the fibril nucleus, the fibril nucleation work, and the fibril nucleation rate as functions of the supersaturation of the protein solution. Our analysis illustrates the unique feature of ANT that the size of the fibril nucleus is a constant integer in a given supersaturation range. We obtain the ANT nucleation rate and compare it with the rates determined previously in the scope of the classical nucleation theory (CNT) and the corrected classical nucleation theory (CCNT). We find that while the CNT nucleation rate is orders of magnitude greater than the ANT one, the CCNT and ANT nucleation rates are in very good quantitative agreement. The results obtained are applicable to homogeneous nucleation, which occurs when the protein solution is sufficiently pure and/or strongly supersaturated.     

Multicomponent nucleation: thermodynamically consistent description of the nucleation work

A thermodynamically consistent formula is derived for the nucleation work in multicomponent homogeneous nucleation. The derivation relies on the conservative dividing surface which defines the nucleus as having specific surface energy equal to the specific surface energy sigma0 of the interface between the macroscopically large new and old phases at coexistence. Expressions are given for the radius of the nucleus defined by the conservative dividing surface and by the surface of tension. As a side result, the curvature dependence of the surface tension sigmaT of the nucleus defined by the surface of tension is also determined. The analysis is valid for nuclei of any size, i.e., for nucleation in the whole range of conditions between the binodal and the spinodal of the metastable old phase provided the inequality sigmaT < or = sigma0 is satisfied. It is found that under the conditions of validity of the analysis the nucleation rate is higher than the nucleation rate given by the classical nucleation theory. The general results are applied to nucleation of unary liquids or solids in binary gaseous, liquid or solid mixtures.     

Determining the nucleation rate from the dimer growth probability

A new method is proposed for the determination of the stationary one-component nucleation rate J with the help of data for the growth probability P2 of a dimer which is the smallest cluster of the nucleating phase. The method is based on an exact formula relating J and P2, and is readily applicable to computer simulations of nucleation. Using the method, the dependence of J on the supersaturation s is determined by kinetic Monte Carlo simulations of two-dimensional (2D) nucleation of monolayers on the (100) face of Kossel crystal. The change of J over nearly 11 orders of magnitude is followed and it is found that the classical nucleation theory overestimates the simulation J values by an s-dependent factor. The 2D nucleus size evaluated via the nucleation theorem is described satisfactorily by the classical Gibbs-Thomson equation and its corrected version accounting for the spinodal limit of 2D nucleation.     

Correlation problems in atomic and molecular systems III. Rederivation of the coupled-pair many-electron theory using the traditional quantum chemical methodst

The equations of the coupled-pair many-electron theory (CPMET ) for the closed shell systems are rederived both in the spin-orbital and orbital forms without the use of second quantization, Wick's theorem or the technique of Feynman-like diagrams. Only the Slater rules are used for the calculation of necessary matrix elements. A comparison with earlier papers shows clearly the usefulness and conceptual simplicity of the mathematical methods of quantum field theory both in the derivation of the CPMET , in spin-orbital form, and in the process of excluding spin variables.     

Multivariable kinetic theory of the first order phase transitions

The problem of calculation of the steady state homogeneous nucleation rate in the multidimensional space of the variables describing a nucleus is considered. Within the framework of the theory proposed, expressions for the nucleation rate and the steady state distribution function of nuclei are derived. The expression for the nucleation rate is invariant with respect to the space dimensionality and, in particular, involves the result of the one-dimensional theory. The distribution function is obtained in the initial, physical variables. In connection with the analysis of restrictions on the current direction, the question of symmetry of the matrix of diffusivities is considered; on the basis of the detailed balance principle it is shown that this matrix is symmetric. The question of normalizing the equilibrium distribution functions is investigated and the physical picture of the equilibrium state is described. The procedure of reducing the multidimensional theory to the one-dimensional one is described.     

Obtaining reaction coordinates by likelihood maximization

We present a new approach for calculating reaction coordinates in complex systems. The new method is based on transition path sampling and likelihood maximization. It requires fewer trajectories than a single iteration of existing procedures, and it applies to both low and high friction dynamics. The new method screens a set of candidate collective variables for a good reaction coordinate that depends on a few relevant variables. The Bayesian information criterion determines whether additional variables significantly improve the reaction coordinate. Additionally, we present an advantageous transition path sampling algorithm and an algorithm to generate the most likely transition path in the space of collective variables. The method is demonstrated on two systems: a bistable model potential energy surface and nucleation in the Ising model. For the Ising model of nucleation, we quantify for the first time the role of nuclei surface area in the nucleation reaction coordinate. Surprisingly, increased surface area increases the stability of nuclei in two dimensions but decreases nuclei stability in three dimensions.     

Metadynamics simulations of ice nucleation and growth

The metadynamics method for accelerating rate events in molecular simulations is applied to the problem of ice freezing. We demonstrate homogeneous nucleation and growth of ice at 180 K in the isothermal-isobaric ensemble without the presence of external fields or surfaces. This result represents the first report of continuous and dynamic ice nucleation in a system of freely evolving density. Simulations are conducted using a variety of periodic simulation domains. In all cases the cubic polymorph ice I(c) is grown. The influence of boundary effects on estimates of the nucleation free energy barrier are discussed in relation to differences between this and earlier work.     

Koopmans' theorem in the ROHF method: canonical form for the Hartree-Fock Hamiltonian

Since the classic work of Roothaan [Rev. Mod. Phys. 32, 179 (1960)], the one-electron energies of a ROHF method are known as ambiguous quantities having no physical meaning. Together with this, it is often assumed in present-day computational studies that Koopmans' theorem is valid in a ROHF method. In this work we analyze the specific dependence of orbital energies on the choice of the basic equations in a ROHF method which are the Euler equations and different forms of the generalized Hartree-Fock equation. We first prove that the one-electron open-shell energies epsilon(m) derived by the Euler equations can be related to the respective ionization potentials I(m) via the modified Koopmans' formula I(m)= -epsilon(m)f(m) where f(m) is an occupation number. As compared to this, neither the closed-shell orbital energies nor the virtual ones derived by the Euler equations can be related to the respective ionization potentials and electron affinities via Koopmans' theorem. Based on this analysis, we derive the new (canonical) form for the Hamiltonian of the Hartree-Fock equation, the eigenvalues of which obey Koopmans' theorem for the whole energy spectrum. A discussion of new orbital energies is presented on the examples of a free N atom and an endohedral N@C(60) (I(h)). The vertical ionization potentials and electron affinities estimated via Koopmans' theorem are compared with the respective observed data and, for completeness, with the respective estimates derived via a DeltaSCF method. The agreement between observed data and their estimates via Koopmans' theorem is qualitative and, in general, appears to possess the same accuracy level as in the closed-shell SCF.     

Homogeneous nucleation and growth in supersaturated zinc vapor investigated by molecular dynamics simulation

Homogeneous nucleation and growth of zinc from supersaturated vapor are investigated by nonequilibrium molecular dynamics simulations in the temperature range from 400 to 800 K and for a supersaturation ranging from log S=2 to 11. Argon is added to the vapor phase as carrier gas to remove the latent heat from the forming zinc clusters. A new parametrization of the embedded atom method for zinc is employed for the interaction potential model. The simulation data are analyzed with respect to the nucleation rates and the critical cluster sizes by two different methods, namely, the threshold method of Yasuoka and Matsumoto [J. Chem. Phys. 109, 8451 (1998)] and the mean first passage time method for nucleation by Wedekind et al. [J. Chem. Phys. 126, 134103 (2007)]. The nucleation rates obtained by these methods differ approximately by one order of magnitude. Classical nucleation theory fails to describe the simulation data as well as the experimental data. The size of the critical cluster obtained by the mean first passage time method is significantly larger than that obtained from the nucleation theorem.     

Probability of fluctuations in the number of the nearest neighbours in a hard-sphere liquid: Probabilities of large deviations

The paper analyzes the physical implications of the result obtained in the work [1] that a random variable — the number of the nearest neighbors M _λ in a fluid consisting of N hard spheres in the Gibbs canonical ensemble — can be expressed as a sum of N independent and similarly distributed random variables. The mean value of a functional of this random variable determines the free Helmholtz energy of a fluid with the SW interaction potential (hard sphere plus square well). The paper shows that the said property of the random variable M _λ allows one to average this functional using the general methods of the probability theory (the Cramer theorem or the method of large deviations) without expanding it in a perturbation theory series. All the mathematical variables introduced in the proof of the Cramer theorem have a simple physical meaning and are determined by the thermodynamic characteristics of the SW fluid. The conditions of the Cramer theorem are shown to hold in the entire region of the fluid existence except the critical point area.     

Nucleation near the spinodal: limitations of mean field density functional theory

We investigate the diverging size of the critical nucleus near the spinodal using the gradient theory (GT) of van der Waals and Cahn and Hilliard and mean field density functional theory (MFDFT). As is well known, GT predicts that at the spinodal the free energy barrier to nucleation vanishes while the radius of the critical fluctuation diverges. We show numerically that the scaling behavior found by Cahn and Hilliard for these quantities holds quantitatively for both GT and MFDFT. We also show that the excess number of molecules Deltag satisfies Cahn-Hilliard scaling near the spinodal and is consistent with the nucleation theorem. From the latter result, it is clear that the divergence of Deltag is due to the divergence of the mean field isothermal compressibility of the fluid at the spinodal. Finally, we develop a Ginzburg criterion for the validity of the mean field scaling relations. For real fluids with short-range attractive interactions, the near-spinodal scaling behavior occurs in a fluctuation dominated regime for which the mean field theory is invalid. Based on the nucleation theorem and on Wang's treatment of fluctuations near the spinodal in polymer blends, we infer a finite size for the critical nucleus at the pseudospinodal identified by Wang.     

Numerical verification of the generalized Crooks nonequilibrium work theorem for non-Hamiltonian molecular dynamics simulations

The generalized Crooks theorem (GCT) for deterministic non-Hamiltonian molecular dynamics simulations [Phys. Rev. E 75, 050101 (2007)] connects the probabilities of nonequilibrium realizations switching the system between two thermodynamic states, to the partition functions of these states. In comparison to the "classical" Crooks nonequilibrium work theorem [J. Stat. Phys. 90, 1481 (1998)], which deals with realizations involving only mechanical work, the GCT also accounts for additional work resulting from changes of the intensive and extensive thermodynamic variables of the system. In this article we present a numerical verification of the GCT using a Lennard-Jones fluid model where two particles are subject to a time-dependent external potential. Moreover, in order to switch the system between different thermodynamic states, the temperature and the pressure (or volume), which are controlled through the Martyna-Tobias-Klein equations of motion [J. Chem. Phys. 101, 4177 (1994)], are also varied externally. The free energy difference between states characterized by different distances of the target particles is evaluated using both a standard methodology (pair radial distribution functions) and the GCT. In order to exploit the various options provided by the GCT approach, i.e., the possibility of temperature/pressure/volume changes during the realizations, the free energy difference is recovered via arbitrary thermodynamic cycles. In all tests, the GCT is quantitatively verified.