Critical isotherms from virial series using asymptotically consistent approximants

Abstract

The low-density equation of state of a fluid along its critical isotherm is considered. An asymptotically consistent approximant is formed having the correct leading-order scaling behavior near the vapor-liquid critical point, while retaining the correct low-density behavior as expressed by the virial equation of state. The formulation is demonstrated for the Lennard-Jones fluid, and models for helium, water, and n-alkanes. The ability of the approximant to augment virial series predictions of critical properties is explored, both in conjunction with and in the absence of criticalproperty data obtained by other means. Given estimates of the critical point from molecular simulation or experiment, the approximant can refine the critical pressure or density by ensuring that the critical isotherm remains well-behaved from low density to the critical region. Alternatively, when applied in the absence of other data, the approximant remedies a consistent underestimation of the critical density when computed from the virial series alone. V C 2014 American Institute of Chemical Engineers AIChE J, 60: 3336-3349, 2014 Keywords: virial equation of state, critical phenomena, crossover, approximant, Lennard-Jones P c . When solved using a sufficiently high-order virial series, these predictions for T c and P c are often close to simulation data and/or experiments, but those for q c are consistently about 10% too low in such a comparison. 4,8 Underestimation of the critical density by the virial series is connected to the nonclassical behavior of real fluids at the critical point, for which critical scaling laws assert that q c is a branch-point singularity of the function P(q) along the critical isotherm. In the context of critical phenomena, the molar In this section, critical isotherms are constructed using given values of the critical density q c , critical pressure P c , Values are given in Lennard-Jones units. Virial coefficients are from Schultz et al. 5 q c,AJ is the corrected critical density given by the smallest positive root of (7), which takes the following as inputs: d 5 4.789; P c , q c , and T c as predicted by the virial series at each order J (given above); and the corresponding virial coefficients (taken from an interpolation). Numbers in parentheses specify the 68% uncertainty on the last digit, propagated from uncertainty in the virial coefficients (with the exception of the simulation values). a Predicted using (7) with P c,sim , T c,sim , and the first seven virial coefficients taken at T c,sim (from an interpolation) as inputs.

Communication: Analytic continuation of the virial series through the critical point using parametric approximants

“…(4) at T = T c differs from the one in Ref. 17 in that it relies on knowledge of B 0 , and is not restricted to ρ < ρ c .…”

“…(4) allows the approximant to reduce to a critical isotherm approximant when T = T c , similar to the one given in Ref. 17, such that P = P c − A(ρ)(1 − ρ/ρ c ) δ . The approximant given by Eq.…”

“…17 Here, we are interested in an analytic continuation of the virial series that incorporates the first three scaling paths given in Eq. (2), so that we may obtain an equation of state valid at low density, the critical region, and intermediate regions.…”

Calculation of high-order virial coefficients for the square-well potential

“…Figure 3 shows the temperature dependence of B 4 and illustrative of the other coefficients as well. Accordingly, attempts to evaluate the critical properties for the SW model from these coefficients do not succeed as well as for the LJ model [5]. The use of approximants [5,7] may improve this outcome, and the coefficients reported here should be very useful in formulating such treatments.…”

“…This is partly because high-order virial coefficients are needed to investigate the convergence properties of the series, and to improve the convergence by, for example, resummation via rational functions or other forms [1][2][3][4][5][6][7][8], but such coefficients are difficult to calculate. Much work has been devoted to the hard-sphere model, for which orders N ≤ 10 have been calculated [9][10][11], before a breakthrough in the algorithm was made [12] that gave access to higher orders [12][13][14].…”