Combined temperature and density series for fluid-phase properties. I. Square-well spheres

Using the Zeno line to assess and refine molecular models

The Zeno line is the locus of points on the temperature-density plane where the compressibility factor of the fluid is equal to one. It has been observed to be straight for a broad variety of real fluids, although the underlying reasons for this are still unclear. In this work, a detailed study of the Zeno line and its relation to the vapor-liquid coexistence curve is performed for two simple model pair-potential fluids: attractive square-well fluids with varying well-widths λ and Mie n-6 fluids with different repulsive exponents n. Interestingly, the Zeno lines of these fluids are curved, regardless of the value of λ or n. We find that for square-well fluids, λ ≈ 1.8 presents a Zeno line, which is the most linear over the largest temperature range. For Mie n-6 fluids, we find that the straightest Zeno line occurs for n between 8 and 10. Additionally, the square-well and Mie fluids with the straightest Zeno line showed the closest quantitative agreement with the vapor-liquid coexistence curve for experimental fluids that follow the principle of corresponding states (e.g., argon, xenon, krypton, methane, nitrogen, and oxygen). These results suggest that the Zeno line can provide a useful additional feature, in complement to other properties, such as the phase envelope, to evaluate molecular models.

Accurate thermodynamics of simple fluids and chain fluids based on first-order perturbation theory and second virial coefficients: uv-theory

We develop a simplification of our recently proposed uf-theory for describing the thermodynamics of simple fluids and fluids comprising short chain molecules. In its original form, the uf-theory interpolates the Helmholtz energy between a first-order f-expansion and first-order u-expansion as (effective) lower and upper bounds. We here replace the f-bound by a new, tighter (effective) lower bound. The resulting equation of state interpolates between a first-order u-expansion at high densities and another first-order u-expansion that is modified to recover the exact second virial coefficient at low densities. The theory merely requires the Helmholtz energy of the reference fluid, the first-order u-perturbation term, and the total perturbation contribution to the second virial coefficient as input. The revised theory-referred to as uv-theory-is thus simpler than the uf-theory but leads to similar accuracy, as we show for fluids with intermolecular pair interactions governed by a Mie potential. The uv-theory is thereby easier to extend to fluid mixtures and provides more flexibility in extending the model to non-spherical or chain-like molecules. The usefulness of the uv-theory for developing equation-of-state models of non-spherical molecules is here exemplified by developing an equation of state for Lennard-Jones dimers.

Evaluation of Osmotic Virial Coefficients via Restricted Gibbs Ensemble Simulations, with Support from Gas-Phase Mixture Coefficients

We present a method for computing osmotic virial coefficients in explicit solvent via simulation in a restricted Gibbs ensemble. Two equivalent phases are simulated at once, each in a separate box at constant volume and temperature and each in equilibrium with a solvent reservoir. For osmotic coefficient BN, a total of N solutes are individually exchanged back and forth between the boxes, and the average distribution of solute numbers between the boxes provides the key information needed to compute BN. Separately, expressions are developed for BN as a series in solvent reservoir density ρ₁, with the coefficients of the series expressed in terms of the usual gas-phase mixture coefficients B_ij. Normally, the B_ij are defined for an infinite volume, but we suggest that the observed dependence of B_ij on system size L can be used to estimate L dependence of the BN, allowing them to be computed accurately at L → ∞ while simulating much smaller system sizes than otherwise possible. The methods for N = 2 and 3 are demonstrated for two-component mixtures of size-asymmetric additive hard spheres. The proposed methods are demonstrated to have greater precision than established techniques, for a given amount of computational effort. The ρ₁ series for BN when applied by itself is (for this noncondensing model) found to be the most efficient in computing accurate osmotic coefficients for the solvent densities considered here.

Algebraic second virial coefficient of the Mie m - 6 intermolecular potential based on perturbation theory

We propose several simple algebraic approximations for the second virial coefficient of fluids whose molecules interact by a generic Mie m - 6 intermolecular pair potential. In line with a perturbation theory, the parametric equations are formulated as the sum of a contribution due to a reference part of the intermolecular potential and a perturbation. Thereby, the equations provide a convenient (low-density) starting point for developing equation-of-state models of fluids or for developing similar approximations for the virial coefficient of (polymeric-)chain fluids. The choice of Barker and Henderson [J. Chem. Phys. 47, 4714 (1967)] and Weeks, Chandler, and Andersen [Phys. Rev. Lett. 25, 149 (1970); J. Chem. Phys. 54, 5237 (1971); and Phys. Rev. A 4, 1597 (1971)] for the reference part of the potential is considered. Our analytic approximations correctly recover the virial coefficient of the inverse-power potential of exponent m in the high-temperature limit and provide accurate estimates of the temperatures for which the virial coefficient equals zero or takes on its maximum value. Our description of the reference contribution to the second virial coefficient follows from an exact mapping onto the second virial coefficient of hard spheres; we propose a simple algebraic equation for the corresponding effective diameter of the hard spheres, which correctly recovers the low- and high-temperature scaling and limits of the reference fluid's second virial coefficient.

On-the-Fly Second Virial Coefficients

A simple but approximate algorithm is described for computing second virial coefficients based on equilibrated molecular configurations that may be generated during any Monte Carlo or molecular dynamics simulation. The algorithm uses simple quadrature based on sampling every binary pair in the configuration and moving their center-center distances from zero to infinity. Comparisons are made in the literature results using more sophisticated sampling and integration for n-alkanes of ethane through n-dodecane. Accuracy is within the error bars determined by block averaging, and temperature effects can be inferred using a single configurational temperature, including perturbative virial coefficients. Predictably, the accuracy is best at the configurational temperature and when the configurational density is lowest. More notably, good accuracy is achieved from configurations at intermediate densities, and the trend at high density conveys valuable insight about conformational changes. The algorithm is simple enough to assign as a homework problem in an introductory molecular modeling course and reinforces the elementary knowledge of pairwise potentials among multisite molecules, numerical integration, and conformational averaging. The result is also quite valuable on its own merits, especially considering thermodynamic integration to compute phase equilibria. Additionally, the incidental data derived from the computation can provide simple but meaningful insights into the nature of multisite interactions, as demonstrated by relating the Mayer averaged potential to an effective Mie potential. Altogether, the argument is made that the computation of the second virial coefficient should be considered to be a routine metric of any molecular simulation, such as the radial distribution function, pressure, or energy.

Accurate first-order perturbation theory for fluids: uf-theory

We propose a new first-order perturbation theory that provides a near-quantitative description of the thermodynamics of simple fluids. The theory is based on the ansatz that the Helmholtz free energy is bounded below by a first-order Mayer-f expansion. Together with the rigorous upper bound provided by a first-order u-expansion, this brackets the actual free energy between an upper and (effective) lower bound that can both be calculated based on first-order perturbation theory. This is of great practical use. Here, the two bounds are combined into an interpolation scheme for the free energy. The scheme exploits the fact that a first-order Mayer-f perturbation theory is exact in the low-density limit, whereas the accuracy of a first-order u-expansion grows when density increases. This allows an interpolation between the lower "f"-bound at low densities and the upper "u" bound at higher liquid-like densities. The resulting theory is particularly well behaved. Using a density-dependent interpolating function of only two adjustable parameters, we obtain a very accurate representation of the full fluid-phase behavior of a Lennard-Jones fluid. The interpolating function is transferable to other intermolecular potential types, which is here shown for the Mie m-6 family of fluids. The extension to mixtures is simple and accurate without requiring any dependence of the interpolating function on the composition of the mixture.

Calculation of high-order virial coefficients for the square-well potential

Accurate virial coefficients B_{N}(λ,ɛ) (where ɛ is the well depth) for the three-dimensional square-well and square-step potentials are calculated for orders N=5-9 and well widths λ=1.1-2.0 using a very fast recursive method. The efficiency of the algorithm is enhanced significantly by exploiting permutation symmetry and by storing integrands for reuse during the calculation. For N=9 the storage requirements become sufficiently large that a parallel algorithm is developed. The methodology is general and is applicable to other discrete potentials. The computed coefficients are precise even near the critical temperature, and thus open up possibilities for analysis of criticality of the system, which is currently not accessible by any other means.