Article pubs.acs.org/JPCB
Virial Equation of State of Water Based on Wertheim’s Association Theory Hye Min Kim,† Andrew J. Schultz,† and David A. Kofke* Department of Chemical and Biological Engineering, 303 Furnas Hall, University at Buffalo, The State University of New York, Buffalo, New York 14260-4200, United States ABSTRACT: Wertheim’s multidensity formalism for pairwise additive molecular interaction is extended to handle nonadditive contributions and is applied to formulate an equation of state (WEOS) for the Gaussian-charge polarizable model (GCPM) of water, with cluster integrals appearing in the theory calculated via the Mayer sampling Monte Carlo method. At both sub- and supercritical temperatures, the equation of state of GCPM water obtained from WEOS converges well to Monte Carlo simulation data, and performs significantly better than the conventional virial treatment (VEOS). The critical temperature for GCPM water using a fourth-order WEOS is given to within 1.3% of the established value, compared to a 17% error shown by fifth-order VEOS; as seen in previous applications, the critical density obtained from both VEOS and WEOS significantly underestimates the true critical density for GCPM water. Examination of the magnitudes of the computed cluster diagrams at the critical density finds that negligible contributions are made by clusters in which a water molecule has both of its hydrogens involved in association interactions.
1. INTRODUCTION The virial equation of state (VEOS) describes the pressure− volume−temperature (pVT) relation of dilute gases in terms of
Bn represents directly the interaction among n molecules and can be expressed in terms of integrals over the molecules’ positions and orientations.2 It is customary to express these integrals using cluster diagrams:3
In these diagrams, the black lines represent the Mayer f function, fij = exp( −βu(rij)) − 1
where u(rij) is the intermolecular potential between molecule i and molecule j with separation distance rij. Each black point represents an integral of a molecule’s position over all space coordinates r and orientations ω, normalized by Ω. The expressions above assume rigid molecules and classical behavior, although extensions to handle other cases are available.4−6 We have verified with several potential models that VEOS describes the thermodynamic behavior of noncondensed nonpolar fluids successfully,4,7 converging for densities up to the spinodal and (for supercritical temperatures) beyond the critical point. However, VEOS is less effective in predicting the thermodynamic behavior of polar fluids such as water.
Figure 1. Convergence of VEOS for GCPM water at 650 K. Compressibility factor Z is plotted against density reduced by the critical density; ρc of GCPM23 is 0.3344 g/cm3.
a power series in density with temperature-dependent coefficients known as virial coefficients:1 p Z≡ = 1 + B2 ρ + B3ρ2 + B4 ρ3 + B5ρ 4 + ··· pkT (1) where Z is the compressibility factor, p is the pressure, ρ is the number density, k is the Boltzmann constant, T is the absolute temperature, and Bn is the nth virial coefficient. © 2012 American Chemical Society
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Received: July 7, 2012 Revised: November 10, 2012 Published: November 13, 2012 14078
dx.doi.org/10.1021/jp3067475 | J. Phys. Chem. B 2012, 116, 14078−14088
The Journal of Physical Chemistry B
Article
Benjamin et al. reported values of B2 and B3 in 2007 and B4 and B5 in 2009 for the Gaussian charge polarizable model of water (GCPM), using the overlap sampling implementation of the Mayer sampling Monte Carlo (MSMC) method.8,9 Convergence of VEOS is demonstrated in Figure 1, which plots Z − 1 (i.e., the deviation from ideality) against the density. The figure shows that the VEOS4 series (i.e., VEOS truncated after B4ρ3) is converged only up to densities of about half the critical density, at which point it diverges from VEOS5. In contrast, at a comparable, slightly supercritical temperature, VEOS4 for the Lennard-Jones model is known to be converged for densities about up to the critical point.7 For GCPM water, extending the virial series beyond VEOS5 is possible, but computing higher order virial coefficients requires much more CPU time and it is probably not worthwhile. We consider then whether an alternative molecularly detailed equation of state can be applied to describe the thermodynamic properties of associating fluids accurately. Many equations of state for associating fluids have been developed over the years,10−21 and although they are based on molecular concepts of association, they do not incorporate molecular-level details in a way that is done by a treatment such as the virial equation of state. Instead, they identify a few parameters that relate to general notions of molecular association, such as the size of the region available for binding, the binding strength, and the number of such binding sites, and they rely on fitting to experimental data to fix values of these parameters. Such approaches have been very successful in a variety of applications, and they are used frequently in engineering practice. Our aim in this work is different. We develop a formulation that has the character of VEOS, in that it can incorporate all available detail about the intermolecular potential, while yielding an equation of state that (unlike VEOS) is effective in describing associating systems. Such a treatment can then provide a route to truly predictive capabilities for associating systems and their mixtures, at least up to moderate densities. It may also prove valuable in interpreting the behavior of such systems. In previous work, we examined a simple Lennard-Jones model having one or more association sites formed from a conical square-well potential and showed that the Wertheim treatment of associating fluids is much superior to the standard virial approach.22 The Wertheim equation of state (WEOS) is derived on the basis of Wertheim’s association theory13,15 and can be used as an effective alternative to the VEOS in the respect that WEOS maintains a molecular level detail like the VEOS. We derived the WEOS for 1-site, 2-site, and 4-site association models up to the fourth order in density. WEOS showed an excellent convergence to the Monte Carlo simulation results in the NpT ensemble while the VEOS failed significantly to predict the pVT behavior of those association models. In the present work, we examine WEOS for a much more realistic model, the Gaussian charge polarizable model (GCPM) of water due to Paricaud et al.,23 and illustrated in Figure 1. It is well-known that water molecules associate via hydrogen bonds and that this has a profound influence on their behavior. Fixed-charge models such as TIP4P are optimized for the liquid phase, and therefore, they provide a poor description of the vapor.24 Polarizable models such as GCPM water are needed to account for the multibody effects that make water molecules in the vapor behave differently from those in condensed water. GCPM water includes Gaussian-distributed
partial charges and a dipole polarizability, and it has been shown to characterize both the liquid and the vapor phases well. Wertheim’s treatment, however, is formulated for pairwise additive molecular models, and it requires some extension before it can be applied to a multibody potential such as GCPM water. In the next section, we explain how to derive WEOS for GCPM on the basis of Wertheim’s theory. In section 3, we review the Mayer sampling method used as the computational approach for evaluating the coefficients appearing in the WEOS. We present and discuss the simulation results in section 4, and we summarize our work in section 5.
2. WERTHEIM’S EQUATION OF STATE FOR GCPM WATER 2.1. Formalism and Theory. The grand-canonical partition function Ξ is given by ∞
Ξ=
∑ N =0
1 N!
N
∫ ··· ∫ exp(−βuN )·∏ z(i) d1 ... dN i=1
(3)
where uN is total intermolecular potential among N molecules and z(i) is the constant fugacity. uN is given by25 uN =
∑ uij + ∑ i
