(561d) Inferring High Order TPT Contributions from Perturbative Virial Coefficients

In a series of recent papers we have developed a predictive thermodynamic perturbation theory (TPT) of bulk and interfacial properties for which each contribution to the theory could be validated with molecular simulation. We showed that the repulsive and attractive contributions could be rigorously identified with appropriate reference and perturbative potentials and that transferable characterization of chain molecules, polar molecules, and hydrogen bonding molecules could be obtained. Coincidentally, two findings from that work were particularly important, meriting further specific attention in the present work: (1) Selecting the Weeks–Chandler–Andersen (WCA)² reference potential leads to computed attractive perturbations that are peaked near the critical density. (2) The third order contribution has a similar appearance to the second order term, except that it is slightly more peaked near the critical density. Recently, simulations of the third and fourth order TPT contributions have come available for square well spheres. Once again, we see a tendency for the fourth order term to be similar to the third order term, except for being slightly more peaked near the critical density. To illustrate, simulation results for the ratios of perturbation contributions of fused Lennard-Jones chains and square well spheres closely follow a Gaussian distribution with the peak near the critical density. These observations suggest the possibility of recursive behavior in the higher order perturbation contributions that may be amenable to some form of simple extrapolation. We have previously shown that this extrapolation leads substantially improves the characterization of the binodal curve in the critical region.

In general, the behavior in the critical region has been a subject of longstanding concern. There is a general consensus that the behavior in the critical region is anomalous because of non-analytic behavior that may be characterized by recursive behavior in the form of the renormalization group (RG) theory.(ref: Chandler, Kiselev, White) Naturally, a simple extrapolation of the perturbation contributions results in an analytic function for the equation of state (EOS). Nevertheless, it would be valuable to understand how much of the anomalous behavior in the critical region can be attributed to a simple analytic continuation of the TPT contributions, and what remains to be explained by non-analytic contributions. Briefly, we find that the “flatness” of the binodal curve in the critical region can be explained to within 99% of the critical temperature using an analytic extrapolation, including accurate characterization of the critical temperature, pressure, and density.

In the present work, we show that the slope and intercept of the Gaussian function at zero density are dictated by the second and third virial coefficients. Taking the peak density of each Gaussian as a constant for a given fluid results in the ability to characterize the higher order TPT contributions without further simulation or adjustable parameters. In the case of the square well (SW) model, the second and third virial coefficients are known exactly to infinite order in temperature. The perturbative virial coefficients can be obtained exactly by appropriate expansion of terms involving the exponential of reciprocal temperature. We explore how these exact coefficients compare with what may be obtained by a simple extrapolation. For Lennard-Jones chains, the perturbative virial coefficients must be computed by careful consideration of cluster integral expansions of appropriate order. Recent work by Elliott, Schultz, and Kofke has facilitated this computation and results to fourth order are available for LJ chains. We review that work and show how predictions using those coefficients provide good agreement with simulation results for the TPT coefficients. In this sense, a high order TPT is obtained from virial coefficients. This EOS is shown to be applicable to liquids as well as vapors with near quantitative agreement of the binodal curve to within 99% of the critical temperature.