Alternative Approaches to the Equilibrium Properties of Hard-Sphere Liquids. In: Mulero Á, editor. Theory and Simulation of Hard-Sphere Fluids and Related Systems. Berlin: Springer Berlin Heidelberg; 2008. pp. 183-245. [DOI: 10.1007/978-3-540-78767-9_6]

Discontinuous Structural Transitions in Fluids with Competing Interactions

This paper explores how competing interactions in the intermolecular potential of fluids affect their structural transitions. This study employs a versatile potential model with a hard core followed by two constant steps, representing wells or shoulders, analyzed in both one-dimensional (1D) and three-dimensional (3D) systems. Comparing these dimensionalities highlights the effect of confinement on structural transitions. Exact results are derived for 1D systems, while the rational function approximation is used for unconfined 3D fluids. Both scenarios confirm that when the steps are repulsive, the wavelength of the oscillatory decay of the total correlation function evolves with temperature either continuously or discontinuously. In the latter case, a discontinuous oscillation crossover line emerges in the temperature-density plane. For an attractive first step and a repulsive second step, a Fisher-Widom line appears. Although the 1D and 3D results share common features, dimensionality introduces differences: these behaviors occur in distinct temperature ranges, require deeper wells, or become attenuated in 3D. Certain features observed in 1D may vanish in 3D. We conclude that fluids with competing interactions exhibit a rich and intricate pattern of structural transitions, demonstrating the significant influence of dimensionality and interaction features.

Structural and thermodynamic properties of hard-sphere fluids

This Perspective article provides an overview of some of our analytical approaches to the computation of the structural and thermodynamic properties of single-component and multicomponent hard-sphere fluids. For the structural properties, they yield a thermodynamically consistent formulation, thus improving and extending the known analytical results of the Percus-Yevick theory. Approximate expressions linking the equation of state of the single-component fluid to the one of the multicomponent mixtures are also discussed.

Chemical potential of a test hard sphere of variable size in hard-sphere fluid mixtures

A detailed comparison between the Boublík-Mansoori-Carnahan-Starling-Leland (BMCSL) equation of state of hard-sphere mixtures is made with Molecular Dynamics (MD) simulations of the same compositions. The Labík and Smith simulation technique [S. Labík and W. R. Smith, Mol. Simul. 12, 23-31 (1994)] was used to implement the Widom particle insertion method to calculate the excess chemical potential, βμ₀^ex, of a test particle of variable diameter, σ₀, immersed in a hard-sphere fluid mixture with different compositions and values of the packing fraction, η. Use is made of the fact that the only polynomial representation of βμ₀^ex which is consistent with the limits σ₀ → 0 and σ₀ → ∞ has to be of the cubic form, i.e., c₀(η)+c¯₁(η)σ₀/M₁+c¯₂(η)(σ₀/M₁)²+c¯₃(η)(σ₀/M₁)³, where M₁ is the first moment of the distribution. The first two coefficients, c₀(η) and c¯₁(η), are known analytically, while c¯₂(η) and c¯₃(η) were obtained by fitting the MD data to this expression. This in turn provides a method to determine the excess free energy per particle, βa^ex, in terms of c¯₂, c¯₃, and the compressibility factor, Z. Very good agreement between the BMCSL formulas and the MD data is found for βμ₀^ex, Z, and βa^ex for binary mixtures and continuous particle size distributions with the top-hat analytic form. However, the BMCSL theory typically slightly underestimates the simulation values, especially for Z, differences which the Boublík-Carnahan-Starling-Kolafa formulas and an interpolation between two Percus-Yevick routes capture well in different ranges of the system parameter space.

Equation of state and critical point behavior of hard-core double-Yukawa fluids

A theoretical study on the equation of state and the critical point behavior of hard-core double-Yukawa fluids is presented. Thermodynamic perturbation theory, restricted to first order in the inverse temperature and having the hard-sphere fluid as the reference system, is used to derive a relatively simple analytical equation of state of hard-core multi-Yukawa fluids. Using such an equation of state, the compressibility factor and phase behavior of six representative hard-core double-Yukawa fluids are examined and compared with available simulation results. The effect of varying the parameters of the hard-core double-Yukawa intermolecular potential on the location of the critical point is also analyzed using different perspectives. The relevance of this analysis for fluids whose molecules interact with realistic potentials is also pointed out.

Simple relationship between the virial-route hypernetted-chain and the compressibility-route Percus-Yevick values of the fourth virial coefficient

As is well known, approximate integral equations for liquids, such as the hypernetted chain (HNC) and Percus-Yevick (PY) theories, are in general thermodynamically inconsistent in the sense that the macroscopic properties obtained from the spatial correlation functions depend on the route followed. In particular, the values of the fourth virial coefficient B(4) predicted by the HNC and PY approximations via the virial route differ from those obtained via the compressibility route. Despite this, it is shown in this paper that the value of B(4) obtained from the virial route in the HNC theory is exactly three halves the value obtained from the compressibility route in the PY theory, irrespective of the interaction potential (whether isotropic or not), the number of components, and the dimensionality of the system. This simple relationship is confirmed in one-component systems by analytical results for the one-dimensional penetrable-square-well model and the three-dimensional penetrable-sphere model, as well as by numerical results for the one-dimensional Lennard-Jones model, the one-dimensional Gaussian core model, and the three-dimensional square-well model.