Free Energy Of Clusters Research Articles

Helmholtz free energy of a cluster on the coherent substrate: Monte Carlo calculations

Monte Carlo simulations of clusters on underlying substrate have been performed. A cluster free energy has been calculated for different substrate lattice parameters. Computing efforts have been focused on the case when a substrate lattice was approximately the same as a bulk crystal lattice consisting of the cluster molecules. The present calculations qualitatively confirm the dependence of the cluster free energy on a misfit between the lattices found in previous studies [D. Turnbull and B. Vonnegut, Ind. Eng. Chem. 44, 1292 (1952) and N. Cabrera, Mem. Sci. Rev. Metall. 62, 205 (1965)]. The effect of the misfit turns out to grow when the number of molecules in the cluster increases. The aforementioned studies of Turnbull and Vonnegut, and Cabrera predicted less strong dependence of the effect of the misfit on the number of molecules in the cluster.

Cluster free energies and nucleation: a density functional approach

Density functional methods of statistical mechanics are applied to calculate the average density profiles and free energies of small (50–500 particles) liquid clusters in unstable or metastable equilibrium with respect to the vapor. The results are compared with those obtained from experiments on homogeneous nucleation of liquids from the vapor.

Cluster-size distributions of ionic and colloidal systems

We develop a method for calculating the physical and geometric properties of atomic clusters in a very general aggregation model. Our method accommodates an arbitrary interparticle potential. It also allows the use of a very wide range of pairing criteria for specifying which configuations are considered to form clusters. We develop an exact differential equation satisfied by the generating function for the quantities {${\mathit{n}}_{\mathit{c}}$(k)}, where ${\mathit{n}}_{\mathit{c}}$(k) is the mean number per unit volume of clusters containing exactly k atoms. We discuss economic methods of solving this equation; a second paper is planned to discuss numerical solutions in detail. We show that the analog, for continuum percolation, of the virial theorem is an exact rate equation of Smoluchowski type for the cluster densities, in which the time is replaced by the total particle density. In this equation, the rate constants are given by contact values of connectedness functions. Applications of this rate equation are discussed. We also give an exact equation for the generating function for cluster volumes. Extensions of the method to calculate cluster free energies economically are described.

On exact bounds for the cluster free energy in the three-dimensional lattice-gas model

The exact bounds by Lebowitz and Penrose for the cluster distribution of the simple cubic lattice-gas model with attractive interactions lead here to exact upper and lower bounds for the limiting cluster free energy or free energy per particle of a cluster of macroscopic size and, consequently, for the renormalized fugacity relating cluster distribution and partition functions. We also give practical realizations of the bounds for the cluster distribution. Our upper and lower bound expressions are always close enough to each other for 4J/kBT>1.5. The nature of the bounds at higher temperatures is also analyzed.

Proof of the existence of the cluster free energy

LetZ p be the partition function over internal configurational degrees of freedom of ap-particle cluster. The existence of the limit, asp→∞, of (βp)−1 logZ p is demonstrated for the square, triangle, hexagon, simple cubic, face-centered cubic, and body-centered cubic lattice gas models, and for continuous space models with potentials satisfying stability and tempering conditions. Pairwise, finite-range bonding is assumed throughout.

Interfacial contribution to cluster free energy

Interfacial contribution to cluster free energy

Interfacial contribution to cluster free energy

The Monte Carlo technique of “overlapping distributions” is used for computing directly the free energy difference F N + 1 − F N between two clusters containing respectively N + 1 and N solute atoms, in the square and the simple cubic Ising model with nearest neighbour interactions. High accuracy results are obtained within reasonable computer times. The capillarity approximation gives a good fit to the data, provided the following is taken into account: (i) the specific bulk and surface energies are given their macroscopic equilibrium values; (ii) a curvature correction to the surface specific energy has to be introduced: it is positive for two dimensions and negative for three dimensions; (iii) a size independent term is to be added in three dimensions; it may be viewed as a corner contribution; (iv) the coefficient of the logarithmic term is given its more recent value. When introduced in the master equation which describes the kinetics of the cluster population, within the simplifying assumptions of the classical nucleation theory, good agreement is found, for the shapes of the cluster size distributions, with the numerical experiments on the kinetic Ising model in two dimensions. However, the time scales of both computations do not match linearly. Possible reasons for this discrepancy are discussed.

Measurements of homogeneous nucleation rates for n-nonane vapor using a two-piston expansion chamber

The homogeneous nucleation rate J in supersaturated n-nonane vapor was measured as a function of supersaturation S and temperature T using a newly developed expansion chamber. Homogeneous nucleation is observed only during a short supercritical nucleation pulse so that nucleation and droplet growth are decoupled under the considered experimental conditions. Reproducible nucleation pulses with a duration of about 1 ms are achieved by means of two spring-operated pistons moving in a well-defined time sequence. The number concentration of the droplets growing in the expansion chamber is determined immediately after the end of the nucleation pulse by means of constant angle Mie scattering. Taking into account the duration of the nucleation pulse we measured homogeneous nucleation rates ranging from about 5×105 to 5×109 cm−3 s−1. Homogeneous nucleation in n-nonane vapor was observed in the temperature range from 199 to 241 K. Argon was used as the carrier gas. The measured nucleation rates were compared with the classical theory of nucleation. The slopes of the J–S curves were found to be in good agreement and thus for n-nonane the dependence of the nucleation rate on the supersaturation appears to be correctly described by the classical nucleation theory. However, the actual values of the experimental nucleation rates were found to be several orders of magnitude higher than the theoretical prediction and show a significantly different temperature dependence. A qualitative explanation of the obtained experimental results can probably be given by means of Burton’s recent microscopic calculations of the cluster free energy.

Cluster free energy in the simple-cubic Ising model

The Monte Carlo technique previously developed to compute directly the free-energy difference between two clusters containing $N+1$ and $N$ particles is applied to the three-dimensional cubic Ising model with nearest-neighbor interactions. For cluster sizes $1\ensuremath{\le}N\ensuremath{\le}19$, at two reduced temperatures ($\frac{T}{{T}_{c}}=0.4 \mathrm{and} 0.59$) it is found that down to surprisingly small sizes ($N\ensuremath{\ge}8$) the free energy ${F}_{N}$ can be written as ${F}_{N}=Na+{N}^{\frac{2}{3}}b+{N}^{\frac{1}{3}}c+d+\ensuremath{\tau}{\ensuremath{\beta}}^{\ensuremath{-}1}\mathrm{ln}N$, where $a$ and $\ensuremath{\tau}$ are given their theoretical value for bulk phases, and $b$, $c$, and $d$ can be understood, respectively, as the surface contribution, a step contribution, and a vertex contribution. A Fisher-type model (${F}_{N}=Na+{N}^{\ensuremath{\sigma}}\ensuremath{\Sigma}+\ensuremath{\tau}{\ensuremath{\beta}}^{\ensuremath{-}1}\mathrm{ln}N$) when fitted to the data yields a size- and temperature-dependent value of $\ensuremath{\sigma}$. Other proposed models are shown to depart systematically from our data. Typical values of nucleation rates for this model, as predicted from the present data, range from ${10}^{\ensuremath{-}3}$ to ${10}^{+2}$\char22{}those expected from the classical capillarity approximation of ${F}_{N}$.

Monte Carlo computation of cluster free energies in the Ising model: a test for the validity of the capillarity approximation

A new Monte Carlo technique is proposed to compute directly the free energy difference FN+1-FN of two clusters with size N+1 and N. The algorithm is applied to the two-dimensional square Ising model with nearest-neighbour interactions. Results are given for 1 9, but for smaller cluster sizes the obtained energies are larger than predicted by the capillarity model. The deviation exhibits no systematic size dependence, but a strong temperature dependence. The existence of a size independent term in FN is confirmed by the values of the internal energy UN obtained as a by-product of the calculation.

A monte carlo method for approximating critical cluster size in the nucleation of model systems

A formalism is presented for estimating critical cluster size as defined in classical models for nucleation phenomena. The method combines Bennett's Monte Carlo technique for determining free-energy differences for clusters containingn andn- 1 atoms with the steady state nucleation rate formalism. A simple form for the free energy of formation of then cluster [including a termA (n)n2/3] is used to predict critical cluster size and critical supersaturation ratio, S*. This approach is applied to Lennard-Jones vapor clusters at 60 K. Results for free-energy differences for the 13, 18, 24, and 43 clusters predict a critical cluster size of 70 ± 5 atoms at a critical supersaturation ratio given bylnS*=2,45 0.15. This method is intended to provide estimates of critical cluster size for more ambitious attempts to calculate cluster free energies or for initializing conditions in microscopic simulations of nucleating systems.

?Critical clusters? in a supersaturated vapor: Theory and Monte Carlo simulation

A new thermodynamic analysis is given for the equilibrium between a liquid cluster and the surrounding supersaturated gas phase in afinite constant volume. It is shown that for constant total density and intermediate volume this equilibrium is stable, although it is unstable for very large volume. We show that observation of the critical cluster sizel* then yields information on the surface free energy of the liquid cluster. The accuracy of previous approximate prescriptions for obtaining the free energy of physical clusters is investigated. As an application, the theory is used to analyze Monte Carlo simulations of the two-dimensional lattice gas model at low temperatures. We obtain cluster surface area, diffusivity, and free energy for clusters with 26≥l≥500. It is found that the capillarity approximation is inaccurate forl≥100, but the free energy of small clusters ishigher than the result of classical nucleation theory, in contrast to what one expects from Tolman-like corrections. We interpret these results, deriving low-temperature series expansions for very small clusters, thus showing that the capillarity approximation both underestimates the surface energy and overestimates the surface entropy of very small clusters. Finally, we use our results to give a speculative explanation of recent nucleation experiments. The dependence of the cluster diffusivity on cluster size is tentatively explained in terms of a crossover between two mechanisms yielding different power laws.

Microscopic capillarity approximation: Free energies of small clusters

Free energies of crystalline microclusters containing 2⩽i⩽100 atoms as well as the bulk chemical potential and surface free energy of a microscopic fcc crystal are calculated for a model substance exhibiting a pairwise additive potential energy and only nearest neighbor interactions. These results are used to test various existing approximations for cluster free energy. An alternative treatment of small clusters is suggested. This microscopic capillarity approximation gives an improved description of the free energy of small clusters, at the expense of requiring knowledge of the free energy of the trimer.

A comparison of the Becker–Döring and Lothe–Pound theories with exact cluster free energies from Monte Carlo simulations

Thermodynamic functions for small clusters of atoms interacting with 6:12 potentials are calculated using the classical Becker–Döring theory and the revised Lothe–Pound theory. Comparisons are made with the results of Monte Carlo studies on such clusters.

Physical cluster free energy from liquid-state perturbation theory

Toxvaerd's extension of the Barker-Henderson perturbation theory to a nonuniform fluid is used to calculate the total free energy of clusters of Lennard-Jones 12–6 atoms. The theory predicts a minimum free energy density profile for two chosen cluster sizes, 43-atoms and 87-atoms, and the free energy values and profiles agree very well with recent Monte Carlo experiments for such clusters.

Theory and Monte Carlo simulation of physical clusters in the imperfect vapor

A formal physical cluster theory for an imperfect gas, valid for an arbitrary definition of a ``physical cluster,'' is described. The role of the definition of the physical cluster is stressed. For a particular definition of the physical cluster, which may be appropriate in nucleation theory, the Helmholtz free energy of 13-, 43-, 60-, 70-, 80-, 87-, and 100-atom argon clusters is calculated in the classical limit for temperatures ranging from absolute zero to 100°K using Monte Carlo techniques. It is found that a cluster's free energy is almost independent of its definition provided that the definition is reasonable and the temperature is sufficiently low. The results are compared with the predictions using the harmonic approximation.