Global Free Energy of Square-Well Fluids for all Densities and all Widths via Singular Value Decomposition

The thermodynamics of square-well (SW) fluids of variable width has been a subject of interest for decades, for both its intrinsic value and applications. The free-energy of SW systems is known from molecular simulations by... [ view full abstract ]

The thermodynamics of square-well (SW) fluids of variable width has been a subject of interest for decades, for both its intrinsic value and applications. The free-energy of SW systems is known from molecular simulations by various authors [1,2,3,4], following the high-temperature perturbation theory of Barker and Henderson [5]. Nevertheless, appropriate analytic expressions, valid for all widths, for the terms a_n in the perturbation expansion, which are functions of width and density, have not been found previously. Here, we propose a procedure, based on the technique of singular value decomposition [6,7], which identifies separate components depending on density and width, and which admits a simple rendering in terms of orthonormal functions. 

Simulation data of a_n, n = 1,…4, supplemented with known limits at extreme values of the attractive SW range [8–10], were sampled to obtain matrices factorizable into singular values and corresponding orthogonal vectors. Keeping just the first few terms of the decompositions, we obtained typical errors less than 1% over the whole region of density and range. Moreover, we were able to generalize the discrete component and coefficient vectors by smooth functions that allow us to calculate the free energy over the whole interval of fluid densities (from zero to random close-packing) and SW ranges –from 1 to the van der Waals or Kac limit at infinite range [11,12].

A global free-energy model as the one here presented can be applied in several fields such as: Statistical Associating Fluid Theory (SAFT), density functional theory (DFT) and modeling the phase separation in systems made of nano- and mesogenic particles.

This work was supported by CONACYT (Mexico) project FDC 2015-02-1450.

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