Closure for the Ornstein-Zernike equation with pressure and free energy consistency

The Ornstein-Zernike (OZ) integral equation theory is a powerful approach to simple liquids due to its low computational cost and the fact that, when combined with an appropriate closure equation, the theory is thermodynamically complete. However, approximate closures proposed to date exhibit pressure or free energy inconsistencies that produce inaccurate or ambiguous results, limiting the usefulness of the Ornstein-Zernike approach. To address this problem, we combine methods to enforce both pressure and free energy consistency to create a new closure approximation and test it for a single-component Lennard-Jones fluid. The closure is a simple power series in the direct and total correlation functions for which we have derived analytical formulas for the excess Helmholtz free energy and chemical potential. These expressions contain a partial molar volumelike term, similar to excess chemical potential correction terms recently developed. Using our bridge approximation, we have calculated the pressure, Helmholtz free energy, and chemical potential for the Lennard-Jones fluid using the Kirkwood charging, thermodynamic integration techniques, and analytic expressions. These results are compared with those from the hypernetted chain equation and the Verlet-modified closure against Monte Carlo and equations-of-state data for reduced densities of ${\rho }^{*}<1$ and temperatures of $T^{*}=1.5$, 2.74, and 5. Our closure shows consistency among all thermodynamic paths, except for one expression of the Gibbs-Duhem relation, whereas the hypernetted chain equation and the Verlet-modified closure exhibit consistency between only a few relations. Accuracy of the closure is comparable to the Verlet-modified closure and a significant improvement to results obtained from the hypernetted chain equation.

Figure 1

Absolute pressure $p{\sigma }^3/\epsilon$ as a function of density $\rho {\sigma }^3$. Green, black, and blue lines are from the HNC, VM, and Eq. (11) bridge corrections, respectively. Red crosses are from the equations of state in [31]. Cyan crosses are available MC data from [29] and [30] for $T^{*}=2.74$ and 5.

Figure 2

Coefficients $a$ (red, filled) and $b_1$ (blue, unfilled) for Eq. (11) vs $\rho {\sigma }^3$ at $T^{*}=1.5$ (diamonds), 2.74 (squares), and 5 (circles).

Figure 3

Helmholtz free energy per particle $A^e/\epsilon$ as a function of density $\rho {\sigma }^3$. Green, black, and blue lines are from the HNC, VM, and Eq. (11) bridge corrections, respectively. Red crosses are from the equations of state in [31]. Cyan crosses are MC simulation data taken from [29] and [30] for $T^{*}=2.74$ and 5.

Figure 4

Excess chemical potential ${\mu }^e/\epsilon$ as a function of density $\rho {\sigma }^3$. Green, black, and blue lines are from the HNC, VM, and Eq. (11) bridge corrections, respectively. Red crosses are from the equations of state in [31].