Formation free energy of an i-mer at spinodal.

In statistical mechanics, the formation free energy of an i-mer can be understood as the Gibbs free energy change in a system consisting of pure monomers after and prior to the formation of the i-mer. For molecules interacting via Lennard-Jones potential, we have computed the formation free energy of a Stillinger i-mer [F. H. Stillinger, J. Chem. Phys. 38, 1486 (1963)] and a ten Wolde-Frenkel (tWF) [P. R. ten Wolde and D. Frenkel, J. Chem. Phys. 109, 9901 (1998)] i-mer at spinodal at reduced temperatures from 0.7 to 1.2. It turns out that the size of a critical Stillinger i-mer remains finite and its formation free energy is on the order of kBT, and the size of a critical tWF i-mer remains finite and its formation free energy is even higher. This can be explained by Binder's theory [K. Binder, Phys. Rev. A 29, 341 (1984)] that for a system, when approaching spinodal, if the Ginzburg criterion is not satisfied, a gradual transition will take place from nucleation to spinodal decomposition, where the free-energy barrier height is on the order of kBT.

Formation free energies of clusters at high supersaturations.

The Helmholtz free energy of a constrained supersaturated vapor with a cluster size distribution consisting of clusters of various sizes is modeled as a mixture of hard spheres of various sizes attracting each other. This model naturally takes into account monomer-monomer and monomer-cluster interactions, so it implicitly pertains to nonideal gases, unlike prior work. Based on this model, the expressions for the equilibrium concentration and the formation free energies of clusters in a metastable supersaturated vapor have been derived. These results indicate that the widely used formula, ni = n1exp(-ßΔGi), that computes the formation free energy of a cluster does not work at high supersaturations. As an example, the formation free energies of clusters with Stillinger's physical cluster definition in metastable, highly supersaturated vapors interacting via Lennard-Jones potential are studied using these expressions. Noticeable differences have been found for both the formation free energies of clusters and sizes of the critical clusters computed from our proposed expressions vs those from the formula ni = n1exp(-ßΔGi).

The free energy of the metastable supersaturated vapor via restricted ensemble simulations. III. An extension to the Corti and Debenedetti subcell constraint algorithm.

In order to improve the sampling of restricted microstates in our previous work [C. Nie, J. Geng, and W. H. Marlow, J. Chem. Phys. 127, 154505 (2007); 128, 234310 (2008)] and quantitatively predict thermal properties of supersaturated vapors, an extension is made to the Corti and Debenedetti subcell constraint algorithm [D. S. Corti and P. Debenedetti, Chem. Eng. Sci. 49, 2717 (1994)], which restricts the maximum allowed local density at any point in a simulation box. The maximum allowed local density at a point in a simulation box is defined by the maximum number of particles Nm allowed to appear inside a sphere of radius R, with this point as the center of the sphere. Both Nm and R serve as extra thermodynamic variables for maintaining a certain degree of spatial homogeneity in a supersaturated system. In a restricted canonical ensemble, at a given temperature and an overall density, series of local minima on the Helmholtz free energy surface F(Nm, R) are found subject to different (Nm, R) pairs. The true equilibrium metastable state is identified through the analysis of the formation free energies of Stillinger clusters of various sizes obtained from these restricted states. The simulation results of a supersaturated Lennard-Jones vapor at reduced temperature 0.7 including the vapor pressure isotherm, formation free energies of critical nuclei, and chemical potential differences are presented and analyzed. In addition, with slight modifications, the current algorithm can be applied to computing thermal properties of superheated liquids.

Radial density distribution of the metastable supersaturated vapor via restricted ensemble simulations.

Extensive restricted canonical ensemble Monte Carlo simulations [D. S. Corti and P. Debenedetti, Chem. Eng. Sci. 49, 2717 (1994)] for the supersaturated Lennard--Jones (LJ) vapor were performed. These simulations were conducted at different densities and reduced temperatures from 0.7 to 1.0 and the radial density distribution functions were obtained, most of which are unavailable via integral equation theory due to phase separations. Among different constraints imposed on the system studied, the one with the local minimum of the excess free energy was taken to be the one that approximates the equilibrium state of the metastable LJ vapor. For the slightly saturated state points, where integral equation theory does have a solution, compared with our simulations, differences of the radial density distribution functions were found and they are attributed to ignoring density fluctuations in the integral equation theory.

The free energy of the metastable supersaturated vapor via restricted ensemble simulations. II. Effects of constraints and comparison with molecular dynamics simulations.

Extensive restricted canonical ensemble Monte Carlo simulations [D. S. Corti and P. Debenedetti, Chem. Eng. Sci. 49, 2717 (1994)] were performed. Pressure, excess chemical potential, and excess free energy with respect to ideal gas data were obtained at different densities of the supersaturated Lennard-Jones (LJ) vapor at reduced temperatures from 0.7 to 1.0. Among different constraints imposed on the system studied, the one with the local minimum of the excess free energy was taken to be the approximated equilibrium state of the metastable LJ vapor. Also, a comparison of our results with molecular dynamic simulations [A. Linhart et al., J. Chem. Phys. 122, 144506 (2005)] was made.

Study of thermal properties of the metastable supersaturated vapor with the integral equation method.

Pressure, excess chemical potential, and excess free energy data for different densities of the supersaturated argon vapor at reduced temperatures from 0.7 to 1.2 are obtained by solving the integral equation with perturbation correction to the radial distribution function [F. Lado, Phys. Rev. 135, A1013 (1964)]. For those state points where there is no solution, the integral equation is solved with the interaction between argon atoms modeled by Lennard-Jones potential plus a repulsive potential with one controlling parameter, alpha exp(-rsigma) and in the end, all the thermal properties are mapped back to the alpha=0 case. Our pressure data and the spinodal obtained from the current method are compared with a molecular dynamics simulation study [A. Linhart et al., J. Chem. Phys. 122, 144506 (2005)] of the same system.

The free energy of the metastable supersaturated vapor via restricted ensemble simulations.

Pressure, excess chemical potential, and excess free energy, with respect to ideal gas data at different densities of the supersaturated Lennard-Jones particle vapor at the reduced temperature 0.7 are obtained by the restricted canonical ensemble Monte Carlo simulation method [D. S. Corti and P. Debenedetti, Chem. Eng. Sci. 49, 2717 (1994)]. The excess free energy values depend upon the constraints imposed on the system with local minima exhibited for densities below the spinodal density and monotonic variation for densities larger than the spinodal density. The results are compared with a molecular dynamics simulation study [A. Linharton et al., J. Chem. Phys. 122, 144506 (2005)] on the same system. The current study verifies the conclusion drawn by the simulation work that clustering of Lennard-Jones atoms exists even in the vicinity of spinodal. Our method gives an alternative to molecular dynamic simulations for the determination of equilibrium properties of a metastable fluid, especially close to the spinodal, and does not require a very large system to carry out the simulation.

Equilibrium adsorption on single and aggregated nanospheres.

The mean field density functional theory has been used to explore Ar adsorption onto single and double identical carbon dioxide nanospheres. We have studied adsorption from subsaturation up to saturation at several temperatures. For the single sphere case, the adsorption excesses and density profiles approach those of plane cases as the spherical substrate size increases; at each temperature, the transition from thin-film to thick-film adsorption is strongly dependent on the size of substrate and the adsorption behavior approaches to that of a plane when the substrate size goes to a large value. For the double sphere case, the cluster formed within the region analogous to the pendular region formed by two contacting identical nanospheres of radius of 1.7 nm has also been studied. The two sphere structure favorites the cluster forming in the interstitial region but does not affect the adsorption transition.