(475d) Extrapolation to an Infinite-Order Thermodynamic Perturbation Theory and Implications in the Critical Region

In a recent work, we characterized the repulsive and attractive contributions to the Helmholtz free energy of realistic molecular fluids via adaptation of a third-order Weeks-Chandler-Andersen (WCA) perturbation theory [Ghobadi and Elliott, J. Chem. Phys. 139, 234104 (2013)]. To do so, the fused soft chains of different length interacting with WCA potential were simulated at several densities and temperatures, covering the entire phase diagram. Then, perturbation contributions were computed using the configurations of the purely repulsive reference term. The resulting EOS, named as SAFT-γ WCA, provided a robust agreement with full-potential simulation data for the thermophysical properties of pure fluids and mixtures. Consistent with the findings of Avendano et al [J. Phys. Chem. B 115, 11154 (2011) and J. Phys. Chem. B 117, 2717 (2013)], the inclusion of the third-order perturbation term substantially improved the predictions of the EOS for the critical properties. Nevertheless, to achieve statistically meaningful values for the third-order term, relatively long simulation times (~50 ns) were needed. On the other hand, we observed that the temperature and density dependency of the perturbation contributions follows a predictable trend for molecules with different chain length. Furthermore, it appeared that the molecular simulation data for the third-order term can be correlated by scaling the second-order term with a Gaussian function. In this work, we assume that the higher-order perturbation terms follow the same trend. Then, using the properties of power series, we define a renormalization function that takes into account the contribution of the higher-order terms. From a physical point of view, this renormalization function incorporates the impact of the long-range density fluctuations on the thermophysical properties. We demonstrate that by using this renormalization, one can obtain a remarkable agreement with experimental critical pressure, temperature and density without scarifying the accuracy outside the critical region. To capture anomalies of derivative properties such as heat capacity at the critical point, we augmented a correction term, originally proposed by Span and Wagner [J. Phys. Chem. Ref. Data 25, 1509 (1996)], to this normalization. The resulting renormalization function, named as G-function, has only 3 adjustable parameters and provides excellent agreement with experimental data for properties in the critical region. Implementation of the G-function is not constrained to SAFT-γ WCA EOS. We provide an application to the PC-SAFT EOS to support the generality of the proposed renormalization scheme. Comparing to the procedures that are based on Renormalization Group (RG) Theory, application of the G-function is simpler and less computationally expensive. It requires no iteration on the Helmholtz free energy and does not involve numerical integration or differentiation. The extension to mixtures and long-chain molecules is also carried out in a straightforward way without scarifying the accuracy or introducing additional adjustable parameters.