Melting of crystallites in a solid porous matrix and the application limits of the Gibbs-Thomson equation.

A thermodynamic model is proposed to describe the melting of lamellar crystallite in a solid medium. This model includes a modification of the Gibbs-Thomson equation to make it applicable to the above-mentioned crystallites. The need for such modification is supported experimentally by studying the impact of the surroundings on the melting point of the crystallites. In particular, the application of the model to nanocrystals in open-porous systems makes it possible to determine the analytical relations for the melting point, the heat of melting, and the inverse effective size of the pores. The fitting of the experimental data with these functional relations then allows for the calculation of the nanocrystalline density, pressure in the nanocrystal, and difference in the surface tension coefficients at the nanocrystal-matrix interface and melt-matrix interface, as well as the difference in the surface entropies per unit area at the nanocrystal-matrix and melt-matrix interfaces.

Communication: Low-temperature approximation of the virial series for the Lennard-Jones and modified Lennard-Jones models.

The regularity of the existing data on the virial coefficients for the Lennard-Jones and modified Lennard-Jones models has allowed a rough extrapolation to the coefficients of higher orders. The corresponding approximation of the infinite virial series has been proposed for the limited temperature interval: 0.4-0.8 of the critical temperature. The loci of zero points of isothermal bulk modulus obtained on the basis of this approximation are close to the vapor-liquid branch of the experimental binodal rather than spinodal. In addition, those points ((dP/dV)T = 0) almost coincide with the divergence points of the approximated virial series that may eliminate some disputable questions about the boundaries of adequacy for the virial equation of state and makes the theoretical isotherms qualitatively similar to the real in the condensation region.

Modified Lennard-Jones model: virial coefficients to the 7th order.

The modified Lennard-Jones potential, which simplifies the numerical simulations and maintains the realistic behavior of its parent, is proposed to a role of the standard interaction model for both the experimental and theoretical studies. The virial coefficients of this model up to the seventh order have been calculated for the range of temperatures kT/ɛ = 0.3-70. In the computations, a technique has been used, that combines the quadrature integration and Mayer Sampling Monte Carlo method (MSMC). Unlike the original MSMC, this technique does not require the reference coefficients of another potential and can be used in a wide range of temperatures for various interaction models.

Condensation of the Lennard-Jones fluid on the basis of the Gibbs single-phase approach.

An accurate expression for the cluster expansion of the configuration integral has been considered based on the Gibbs single-phase approach without additional assumptions and approximations. The computational results for the Lennard-Jones fluid demonstrate the beginning of the condensation process at the singularity points of the isothermal compressibility. It has also been shown that the accuracy boundary of the virial equation of state corresponds to these points, not the virial series singularities. At the high density regime, the inappropriate behavior of the cluster expansion has been discovered that may be related to the infinite limits of the cluster integrals, i.e., their independency on the density.

Equation of state beyond the radius of convergence of the virial expansion.

The well-known problem of the virial expansion low-density limitation is considered within Mayer's cluster expansion method. The expression for the configuration integral and the corresponding equation of state are presented based on this approach but not limited by the convergence radius of the series for density and activity. When taking into account any number of irreducible integrals at the thermodynamic limit, this equation of state demonstrates the exact coincidence with the virial one inside the domain of its convergence but specifies the condensation process directly outside that domain. Thus, the assumption of some researchers that the condensation should appear in the domain where the proof of the virial expansion is limited may now be regarded as confirmed, exclusively using the classical Gibbs statistics.

Construction of subcritical isotherms for model and real gases on the basis of Mayer's cluster expansion.

Existing rigorous statistical approaches still cannot quantitatively describe condensation phenomena in real fluids and even model systems with some simplified interaction potential. Here, we present a method to evaluate the unlimited subcritical set of Mayer's reducible cluster integrals (the power coefficients of virial expansions) by using the information on several virial coefficients and empirical value of saturation activity. As to the requirements on the initial number of known virial coefficients, the calculations for the Lennard-Jones model indicate that only the second virial coefficient is sufficient to reproduce gas isotherms (including the flat phase-transition region) with high accuracy at low temperatures. This remarkable feature allows the simplification of the method for real fluids with an unknown interaction potential: In particular, the calculated isotherms of several real substances (including water) are in good agreement with experiments. Additionally, the obtained results indicate the existence of some important universality which needs serious future exploration.

Some universality in subcritical behavior of real substances and model fluids.

On the basis of the latest advances in Mayer's cluster-based approach, the reduced forms of the well-known virial expansions are derived in terms of scaled reducible and irreducible cluster integrals. This transformation minimizes the dependence on temperature and the effect of parameters specific for each thermodynamic system, thus making the resulting reduced expansions indeed universal on the quantitative level. In particular, the scaling of isotherms and saturation curves for various systems (the Lennard-Jones model, different lattice gases, and real substances with simple nonpolar molecules as well as complex polar ones) confirms the approximate universality of the proposed reduced variables for temperature, pressure, and density at subcritical gaseous states up to the saturation point. In addition, the temperature dependence of the correspondingly scaled second virial coefficients also appears similar for various systems.

Divergence of activity expansions: Is it actually a problem?

For realistic interaction models, which include both molecular attraction and repulsion (e.g., Lennard-Jones, modified Lennard-Jones, Morse, and square-well potentials), the asymptotic behavior of the virial expansions for pressure and density in powers of activity has been studied taking power terms of high orders into account on the basis of the known finite-order irreducible integrals as well as the recent approximations of infinite irreducible series. Even in the divergence region (at subcritical temperatures), this behavior stays thermodynamically adequate (in contrast to the behavior of the virial equation of state with the same set of irreducible integrals) and corresponds to the beginning of the first-order phase transition: the divergence yields the jump (discontinuity) in density at constant pressure and chemical potential. In general, it provides a statistical explanation of the condensation phenomenon, but for liquid or solid states, the physically proper description (which can turn the infinite discontinuity into a finite jump of density) still needs further study of high-order cluster integrals and, especially, their real dependence on the system volume (density).

Virial and high-density expansions for the Lee-Yang lattice gas.

On the basis of the recently established "hole-particle" symmetry of the lattice-gas Hamiltonian, the high-density equation of state has been derived in a form of pressure and density expansions in powers of activity. This equation is proposed as an alternative and complementary to the previously obtained pressure expansion in powers of density. For the well-known Lee-Yang lattice-gas model (a two-dimensional square lattice with a square-well interaction potential), the power coefficients (i.e., cluster and irreducible cluster integrals) up to the seventh order have been evaluated as accurate functions of temperature. The convergence of the expansions in powers of both density and activity to the exact Lee-Yang solution is investigated.

High-density equation of state for a lattice gas.

For the lattice gas models of arbitrary geometry and dimensions with absolute repulsion between particles at zero distance (a hard core identical to a single lattice site) and arbitrary repulsion or attraction at other distances, the "hole-particle" symmetry of the system potential energy has been stated and an equation of state has been derived on the basis of the classical Gibbs statistics. The equation is completely analogous to the well-known virial equation of state, except that it is more accurate at high-density states, while the virial equation has the low-density limitation. Both equations contain the common set of the so-called irreducible integrals, related to the corresponding virial coefficients, and can be used together to describe the behavior of a lattice gas in a wide range of densities.

Adequacy of the virial equation of state and cluster expansion.

The limits for the accuracy of the virial expansion and the problem of its divergence have been investigated using the exact cluster expansion of the configuration integral. In the subcritical temperature regimes the virial equation of state is applicable up to the singularity point of the isothermal compressibility, i.e., to the possible beginning of the condensation process. At supercritical temperatures this equation should be applicable within the region where the cluster expansion is adequate. The problem of the virial series divergence has been found to be irrelevant to the actual behavior of the cluster expansion. Considering the Lennard-Jones fluid as well as the system of hard spheres, the inadequate behavior of the cluster expansion has been discovered in the high density regime. The major reason for this inadequacy should be the basic simplification of the cluster expansion: the integration of irreducible diagrams over the infinite limits.