Free Energy Of Clusters Research Articles

Quantum Effects on the Free Energy of Ionic Aqueous Clusters Evaluated by Nonequilibrium Computational Methods

Nonequilibrium simulation methods and rigid-body path-integral techniques are combined to estimate the relevance of protonic quantum effects in the free energy of ion-water clusters. The Crooks' fluctuation relation is used to quantitatively characterize the impact of quantum effects on the dissociation free energy of the paradigm I(-)(H(2)O)(5). By use of a rigorous smoothing procedure in the calculation of the work distributions, the effects are found to be about 11% and therefore non-negligible. Quantum effects on the potential of mean force of Na(+)(H(2)O)(12) were also evaluated using Jarzynski's work theorem for a reaction coordinate, and they were also found to be significant. The results suggest that quantization should play a significant role in the kinetics of ionic transport in aqueous environments.

Application of cluster dynamics modeling to the precipitation in aluminum alloys

Abstract The cluster dynamics method is used to model at the atomic scale the kinetics of first order phase transformations. Clusters are embryos of the growing phase. Their formation kinetics from the solid solution are obtained by solving a set of master differential equations. This set is the continuity equation of the cluster size distribution function and is based on inter-cluster solute exchanges. These exchanges, absorption and emission of solute atoms, are controlled by the solute diffusivities and the cluster free energies. The model is applied here to the precipitation of Al3(Zr, Sc) dispersoids with the L12 structure in Al–rich Zr–Sc solid solutions. Several characteristic features are obtained and discussed: 1) In the ternary alloy, the fast (Sc) diffusing species always controls the nucleation, in contrast to classical thermodynamical descriptions. 2) A Zr–Sc thermodynamic coupling induces heterogeneous nucleation on Zr atoms. 3) A segregation on the dispersoids outer shell of the slow diffusing solute (Zr) occurs during the coarsening stage, slowing down their coarsening rate. Finally, some extensions and prospects of the method are considered.

Structure and thermodynamics of colloidal protein cluster formation: Comparison of square-well and simple dipolar models

Reversible formation of weakly associated protein oligomers or clusters is a key early step in processes such as protein aggregation and colloidal phase separation. A previously developed cell-based, quasichemical model for lattice fluids [T. M. Young and C. J. Roberts, J. Chem. Phys. 127, 165101 (2007)] is extended here to treat continuous-space systems. It is illustrated using two simplified limiting cases for globular proteins at the isoelectric point: spherical square-well (SW) particles with an isotropic short-ranged attraction and screened dipolar particles with SW attractions and square-shoulder repulsions. Cluster free energies (DeltaA(i)) and structures are analyzed as a function of the reduced second virial coefficient b(2)(*). DeltaA(i) values and the average structures of clusters up to pentamers have distinct differences due to the anisotropic nature of the dipolar interactions. However, DeltaA(i) values can be mapped semiquantitatively between the two cases if compared at common values of b(2)(*). Free energy landscapes of oligomerization are constructed, illustrating significant differences in landscape ruggedness for small clusters of dipolar versus SW fluids, and suggesting a possible molecular interpretation for empirical models of nucleation-dependent aggregation of proteins.

Theoretical study of the structure, energetics and vibrational frequencies of water–acetone and water–2-butanone complexes

The complexes of acetone and 2-butanone with up to two water molecules hydrogen bonded to the carbonyl oxygen were studied computationally using Density Functional Theory at the B3LYP/6-311++G(d, p) level. Gas phase calculations were followed by optimizations in the solvent field, simulated with the Polarizable Continuum Model (PCM). Vibrational frequencies obtained with the help of empirical correlation agreed well with experimental spectra of HDO in aqueous solutions of the studied ketones. Accurate clustering enthalpies and free energies were obtained at various levels of theory, up to Coupled Cluster. Hydration free energies of the solutes were also calculated and compared with experiment.

Mesoscale model parameters from molecular cluster calculations

We present an efficient, systematic, and universal method to estimate the interaction parameters used in mesoscale simulation methods such as dissipative particle dynamics and self-consistent field methods from molecular cluster calculations. The method is based on a generalized Flory-Huggins model in which molecules, or fragments thereof, are in contact with their van der Waals surface. We sample the density of states of molecular clusters in the space spanned by the coarse-grained degrees of freedom. From here, we calculate the sum over states and free energy of the cluster at a temperature of interest by histogram reweighting. The method allows to calculate the energy and entropy contributions to the cluster free energy explicitly. For two components, we then obtain the excess free energy of mixing and the Flory-Huggins chi-parameter, and their energetic and entropic contributions. We present two applications of the method: a simple liquid mixture of hexane and nitrobenzene, and a series of polymer blends. In the case of hexane/nitrobenzene, we compare to alternative simulation methods; here we find that the energy of mixing alone is too high to explain the critical point. By including the excess entropy of mixing, however, the predicted phase behavior is in reasonable agreement with experiment. The tendency of calculations based on average energy alone to overestimate the chi-parameter is also apparent in the polymer blend calculations.

A quasichemical approach for protein-cluster free energies in dilute solution

Reversible formation of protein oligomers or small clusters is a key step in processes such as protein polymerization, fibril formation, and protein phase separation from dilute solution. A straightforward, statistical mechanical approach to accurately calculate cluster free energies in solution is presented using a cell-based, quasichemical (QC) approximation for the partition function of proteins in an implicit solvent. The inputs to the model are the protein potential of mean force (PMF) and the corresponding subcell degeneracies up to relatively low particle densities. The approach is tested using simple two and three dimensional lattice models in which proteins interact with either isotropic or anisotropic nearest-neighbor attractions. Comparison with direct Monte Carlo simulation shows that cluster probabilities and free energies of oligomer formation (DeltaG(i) (0)) are quantitatively predicted by the QC approach for protein volume fractions approximately 10(-2) (weight/volume concentration approximately 10 g l(-1)) and below. For small clusters, DeltaG(i) (0) depends weakly on the strength of short-ranged attractive interactions for most experimentally relevant values of the normalized osmotic second virial coefficient (b(2) (*)). For larger clusters (i"2), there is a small but non-negligible b(2) (*) dependence. The results suggest that nonspecific, hydrophobic attractions may not significantly stabilize prenuclei in processes such as non-native aggregation. Biased Monte Carlo methods are shown to accurately provide subcell degeneracies that are intractable to obtain analytically or by direct enumeration, and so offer a means to generalize the approach to mixtures and proteins with more complex PMFs.

The Critical Role of Anharmonicity in Aqueous Ionic Clusters Relevant to Nucleation

Calculating accurate free energies of aqueous ionic clusters is important in many areas of chemical physics. A common practice used in ab initio and other methods to determine the cluster free energy is to use a single energetically favorable configuration and then assume the harmonic approximation to the potential energy surface. In the present study, we compare and contrast harmonic free energies to those determined from anharmonic configurational sampling and how both compare to experiment. It is found that anharmonicity plays a critical role in determining accurate free energies for the clusters underlying ion-induced nucleation. Consequently, energies, geometries, and frequencies obtained by using a single or multiple configurations within the harmonic approximation lead to inaccurate aqueous cluster thermodynamics.

Single-ion solvation free energies and the normal hydrogen electrode potential in methanol, acetonitrile, and dimethyl sulfoxide.

The division of thermodynamic solvation free energies of electrolytes into contributions from individual ionic constituents is conventionally accomplished by using the single-ion solvation free energy of one reference ion, conventionally the proton, to set the single-ion scales. Thus, the determination of the free energy of solvation of the proton in various solvents is a fundamental issue of central importance in solution chemistry. In the present article, relative solvation free energies of ions and ion-solvent clusters in methanol, acetonitrile, and dimethyl sulfoxide (DMSO) have been determined using a combination of experimental and theoretical gas-phase free energies of formation, solution-phase reduction potentials and acid dissociation constants, and gas-phase clustering free energies. Applying the cluster pair approximation to differences between these relative solvation free energies leads to values of -263.5, -260.2, and -273.3 kcal/mol for the absolute solvation free energy of the proton in methanol, acetonitrile, and DMSO, respectively. The final absolute proton solvation free energies are used to assign absolute values for the normal hydrogen electrode potential and the solvation free energies of other single ions in the solvents mentioned above.

Ion–water cluster free energy computer simulation using some of most popular ion–water and water–water pair interaction models

The bicanonical Monte-Carlo method allows effective computer simulation of free energy, entropy as well as internal energy of ion molecular clusters. The advantages of this method have been exploited to test how much adequately frequently applied pair interaction models reproduce not only energetics, but also statistical properties of the investigated ion–water clusters. In present paper, the series of reproductions of mass spectroscopy data on stepwise binding Gibbs energy, enthalpy and entropy of Ion(H 2O) n , n = 1 ÷ 6 clusters for a set of monovalent ions has been carried out with the help of bicanonical Monte-Carlo technique. Each reproduction from the series has been performed using one of pair interaction ion–water plus water–water models, the experimental data on free energy, entropy as well as internal energy of ion molecular clusters. Based on the result of the simulations, the quantitative analysis of the investigated pair interaction models accuracy has been accomplished. The accuracy of the pair interaction ion–water plus water–water models has been compared with the accuracy of the one of polarizable models taking into account polarization of ions and water molecules explicitly.

Aqueous Solvation Free Energies of Ions and Ion−Water Clusters Based on an Accurate Value for the Absolute Aqueous Solvation Free Energy of the Proton

Thermochemical cycles that involve pKa, gas-phase acidities, aqueous solvation free energies of neutral species, and gas-phase clustering free energies have been used with the cluster pair approximation to determine the absolute aqueous solvation free energy of the proton. The best value obtained in this work is in good agreement with the value reported by Tissandier et al. (Tissandier, M. D.; Cowen, K. A.; Feng, W. Y.; Gundlach, E.; Cohen, M. J.; Earhart, A. D.; Coe, J. V. J. Phys. Chem. A 1998, 102, 7787), who applied the cluster pair approximation to a less diverse and smaller data set of ions. We agree with previous workers who advocated the value of -265.9 kcal/mol for the absolute aqueous solvation free energy of the proton. Considering the uncertainties associated with the experimental gas-phase free energies of ions that are required to use the cluster pair approximation as well as analyses of various subsets of data, we estimate an uncertainty for the absolute aqueous solvation free energy of the proton of no less than 2 kcal/mol. Using a value of -265.9 kcal/mol for the absolute aqueous solvation free energy of the proton, we expand and update our previous compilation of absolute aqueous solvation free energies; this new data set contains conventional and absolute aqueous solvation free energies for 121 unclustered ions (not including the proton) and 147 conventional and absolute aqueous solvation free energies for 51 clustered ions containing from 1 to 6 water molecules. When tested against the same set of ions that was recently used to develop the SM6 continuum solvation model, SM6 retains its previously determined high accuracy; indeed, in most cases the mean unsigned error improves when it is tested against the more accurate reference data.

Theoretical Analysis of Non-Catalytic Growth of Nanorods on a Substrate

A theoretical analysis explaining the whole process of the growth of nanorods on a substrate without a catalyst is presented. Prior to the growth of the nanorods, the reaction precursors form nuclei on the substrate. The nuclei undergo cluster migration caused by the surface diffusion of adatoms on the substrate, and this migration continues until the mean free time of the adatoms is larger than surface diffusion time. The most probable mechanism by which cluster migration takes place is the one that leads to the minimization of the cluster free-energy, namely the migration of six adatoms into one fixed adatom. This cluster migration continues during several (typically smaller than 6) consecutive nuclei growth steps. After the process of cluster migration comes to an end, the nuclei grow in an isotropic manner by collection of the adatoms, until the nucleus reaches the thermodynamic size limit. The one-dimensional growth of nanorods on the nuclei, which is associated with the critical radius, begins when the reactant dose is smaller than a certain value, which is determined by the thermodynamic size limit and the mass transport parameter. The mass transport of the reaction precursors leads to the expansion of the radius and elongation of the height of the nanorods, and the growth rate of the height is greater than that of the radius. This difference in the growth rate causes the aspect ratio to increase with increasing growth time. By comparing the experimental data in the literature (ZnO nanorods), the presented analysis explains well the noncatalytic growth of nanorods on a substrate.

B3LYP/6-311++G** geometry-optimization study of pentahydrates of α- and β- d-glucopyranose

Five water molecules were placed in 37 different configurations around α- and β- d-glucopyranose in the gt, gg, and tg conformational states, and the glucose–water complexes were geometry optimized using density functionals at the B3LYP/6-311++G** level of theory. The five water molecules were organized in space and energy minimized using an empirical potential, AMB02C, and then further geometry optimized using DFT algorithms to minimum energy positions. Electronic energy, zero point vibrational energy, enthalpy, entropy, stress energy on glucose and the water cluster, hydrogen-bond energy, and relative free energy were obtained for each configuration using thermodynamic procedures and an analytical Hessian program. The lowest energy complex was that of a clustering of water molecules around the 1- and 6-hydroxyl positions of the β- gt anomer. Configurations in which the water molecules created a favorable network completely around and under glucose were found to have low energy for both α and β anomers. Calculation of the α/β anomeric ratio using the zero point corrected energy gave, ∼32/68%, highly favoring the β anomer in agreement with the experimental ∼36/64% value. This ratio is better than the ∼50/50% ratio found in our previous monohydrate study. An approximate hydroxymethyl population was obtained by noting average relative energies among the three conformational states, gt, gg, and tg. In the β anomer complexes the gt conformation was favored over the gg state, while in the α anomer complexes the gg state was favored over the gt conformation, with the tg conformations all being of higher energy making little or no contribution to the rotamer population. Some geometry variances, found between glucose in vacuo and glucose after interaction with water molecules, are described and account for some observed C-5–C-6 bond length anomalies reported by us previously for the vacuum glucose structures.

Monte Carlo simulation study of droplet nucleation

A new rigorous Monte Carlo simulation approach is employed to study nucleation barriers for droplets in Lennard-Jones fluid. Using the gauge cell method we generate the excess isotherm of critical clusters in the size range from two to six molecular diameters. The ghost field method is employed to compute the cluster free energy and the nucleation barrier with desired precision of (1-2)kT. Based on quantitative results obtained by Monte Carlo simulations, we access the limits of applicability of the capillarity approximation of the classical nucleation theory and the Tolman equation. We show that the capillarity approximation corrected for vapor nonideality and liquid compressibility provides a reasonable assessment for the size of critical clusters in Lennard-Jones fluid; however, its accuracy is not sufficient to predict the nucleation barriers for making practical estimates of the rate of nucleation. The established dependence of the droplet surface tension on the droplet size cannot be approximated by the Tolman equation for small droplets of radius less than four molecular diameters. We confirm the conclusion of ten Wolde and Frenkel [J. Chem. Phys. 109, 9901 (1998)] that integration of the normal component of the Irving-Kirkwood pressure tensor severely underestimates the nucleation barriers for small clusters.

Scaling of the Nucleation Rate and a Monte Carlo Discrete Sum Approach to Water Cluster Free Energies of Formation

The intent of this work is to examine small cluster discrete size effects and their effect on the free energy of cluster formation. There is evidence that such terms can cancel in part the temperature dependence of the monomer flux factor of the classical nucleation rate and result in a scaled form for the nucleation rate. In this work, Monte Carlo configurational free energy differences between neighboring sized n molecule TIP4P water clusters are calculated and used in a Monte Carlo discrete summation (MCDS) technique to generate steady-state nucleation rates. The free energy differences, when plotted versus n-1/3, show evidence of a bulklike effective surface tension for n ≥ 10, and for the range of T examined the free energy differences appear to scale in temperature like (Tc/T − 1). This scaling can provide estimates of nucleation rates for arbitrary temperatures within the range of T simulated. Nucleation rates generated from the model TIP4P free energy differences are compared with the experimental water nucleation rate data of Wölk and Strey (J. Chem. Phys. 2001, 105, 11683) and with the data of Miller et al. (J. Chem. Phys. 1983, 78, 3204). The TIP4P MCDS results provide some evidence of the cancellation effect and generate the scaling of the nucleation rate data at higher temperatures. The magnitudes of the nucleation rates are, however, too large by a factor of 104. Other discrete sum models are also presented and give similar results.

First-principles calculation on free energy of precipitate nucleation

A slight modification of the nucleation theory enables reliable calculations of cluster free energies of precipitate nucleation. This method divides the free energy into the cluster energy and entropy terms. The former, consisting of the enthalpy of driving force and the interface energy, is precisely calculated by first-principles calculations. The latter, which is entropy loss from scattered atoms condensing into a cluster, is estimated by the ideal solution model. Model calculations have been performed for an Fe–Cu alloy, Ni added Fe–Cu alloy, and vacancy behavior around Cu clusters.

Nucleation and growth in cluster dynamics: A quantitative test of the classical kinetic approach

Nucleation and size dependent growth of nanometer sized crystalline particles in glassy media have been studied by numerically solving the Turnbull–Fisher master equations that describe the time evolution of cluster population. Time dependencies of the formation rate and number density are determined for large clusters (built of up to 2×105 formula units, containing 1.8×106 atoms). We demonstrate that the formation rate and number density of such clusters are well approximated by Shneidman’s asymptotically exact analytical solution. A quantitative test of the kinetic Turnbull–Fisher model has been performed: Evaluating the kinetic coefficients and interfacial parameters from the transient time and steady-state nucleation rates measured on six stoichiometric oxide glass compositions (lithium–disilicate, barium–disilicate, lithium–diborate, wollastonite, 1:2:3 and 2:1:3 soda–lime–silica glass compositions), we calculated the macroscopic growth rates and compared with experiments. For wollastonite, lithium–diborate and the 1:2:3 soda–lime–silica glass, differences of 2 to 4 orders of magnitude have been observed between theory and experiment. This inadequacy of the microscopic kinetic parameters in describing macroscopic growth cannot be explained by either the curvature effect on the interfacial free energy or the self-consistency correction for the cluster free energy. The origin of the discrepancy is discussed.

Efficient free energy calculations by variationally optimized metric scaling: Concepts and applications to the volume dependence of cluster free energies and to solid–solid phase transitions

Finite-time variational switching is an efficient method for obtaining converging upper and lower bounds to free energy changes by computer simulation. Over the course of the simulation, the Hamiltonian is changed continuously between the system of interest and a reference system for which the partition function has an analytic form. The bounds converge most rapidly when the system is kept close to equilibrium throughout the switching. In this paper we introduce the technique of metric scaling to improve adherence to equilibrium and thereby obtain more rapid convergence of the free energy bounds. The method involves scaling the coordinates of the particles, perhaps in a nonuniform way, so as to assist their natural characteristic evolution over the course of the switching. The scaling schedule can be variationally optimized to produce the best convergence of the bounds for a given Hamiltonian switching path. A correction due to the intrinsic work of scaling is made at the end of the calculation. The method is illustrated in a pedagogical one-dimensional example, and is then applied to the volume dependence of cluster free energies, a property of direct relevance to vapor–liquid nucleation theory. Order-of-magnitude improvements in efficiency are obtained in these simple examples. As a contrasting application, we use metric scaling to calculate directly the free energy difference between face-centered-cubic and body-centered-cubic Yukawa crystals. A continuous distortion is applied to the lattice, avoiding the need for separate comparison of the two phases with an independent reference system.

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.

Ionic charging free energies: Spherical versus periodic boundary conditions

Ionic charging free energies calculated by Ewald summation differ substantially from those calculated in spherical cluster calculations, with or without the inclusion of a Born correction in the latter. Using Gauss’ law, we derive an electrostatic potential for ions in spherical clusters that involves contributions only from the interior solvent. This “interior” potential agrees with the “P-summation” approach proposed by Hummer et al. [J. Phys. Chem. B 101, 3017 (1997)], and leads to charging free energies which agree, within simulation error, with those given by Ewald summation with finite-size corrections. The difference in charging free energies between this approach and the conventional cluster free energies including the Born correction is traced to the surface potential of water.

Nucleation theorems, the statistical mechanics of molecular clusters, and a revision of classical nucleation theory

The nucleation theorems relate the temperature and supersaturation dependence of the rate of nucleation of droplets from a metastable vapor phase to properties of the critical molecular cluster, the size that is approximately equally likely to grow or decay. They are derived here using a combination of statistical mechanics and cluster population dynamics, using an arbitrary model cluster definition. The theorems are employed to test the validity of the classical theory of homogeneous nucleation and its ``internally consistent'' form. It is found that the properties of the critical cluster for these models are incorrect, and it emerges that this occurs because the classical theory employs the free energy of a fixed droplet, rather than one free to take any position in space. Thus a term representing positional, or mixing, entropy is missing from the cluster free energy. A revised model is proposed, based on the capillarity approximation but with such a term included, and it is shown that it is fully consistent with the nucleation theorems. The model increases classical rates by factors of approximately ${10}^{4}$--${10}^{6}$. Other nucleation models should be tested for internal consistency using the same methods. Finally, the nucleation theorems are used to extract the excess internal energies of molecular clusters from experimental data for several substances.