For mixtures where σ22/σ11 = 1, the single-component hard-sphere reference has given similar results for the phase envelopes as the other references. Fig. 9 displays phase envelopes of mixtures with size-asymmetric molecules. Only results with the additive and non-additive hard-sphere references are shown, since the single-component reference failed to predict two phases. Fig. 10 shows that the pressure-density isotherms from the single-component reference deviate by a large amount from the results from theN P T Monte Carlo simulations, while the simulations display agreement with the other two references (solid lines). This explains why the single-component reference does not give two phases.
The failure of the single-component reference for high σ22/σ11-ratios was already seen for the cross-virial coefficient discussed in Sec. IV A. However, arguably the most important cause of its mispredictions can be attributed to the representation of a1. The variable x0 is defined for the different references in Eq. (43). For a single-component LJ fluid, the typical interval of x0 is 1.02< x0 <1.11, where the highest ratio is encountered at T∗ = 10. The same interval is encountered when evaluatingx0,ij in additive and non-additive hard-sphere mixtures. For the single-component hard-sphere reference, on the other hand, x0,ij can be both above and below 1. For the mixture displayed in Fig. 9b, we find that 0.7<x0,ij<1.55 for the single-component reference. In principle,a1,ijas defined by the integral in Eq. (36) can be This is the author’s peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset. PLEASE CITE THIS ARTICLE AS DOI:10.1063/1.5142771
0.0 0.2 0.4 0.6 0.8
Additive Ref.: a1, ij
MC: a1, 11
(b) First order perturbation term
0.0 0.2 0.4 0.6 0.8
(c) Second order perturbation term
0.0 0.2 0.4 0.6 0.8
(d) Third order perturbation term
FIG. 11. Comparison of perturbation theory with an additive reference (lines) with canonical MC simulations for the reduced perturbation terms. An LJ mixture with σ22/σ11 = 1.5, 11 = 22 is simulated at T∗ = 1. N2 ∈ {150,250,350}, corresponding to mole fractions of Type 2 molecules of 30, 50, 70 %. In a) the configurational energy contribution from the 11, 22 and 12 pair interactions are plotted for a composition of x1 = 0.5. b)-d) Mixture perturbation terms. One standard deviation of the sample mean is used for the error bars of the canonical MC results of a3 (d). The error bars are not included for a1 (b) anda2 (c) as the magnitude of the error is smaller than the symbol size.
evaluated by numerical integration. This is usually not done in practical implementations of perturbation theories, as it gives unfeasible computational times. Instead, the correlations presented in Sec. II B 1 are used. By comparing to a numerical integration of Eq. (36) (not shown), we find that these correlations are grossly inaccurate in the interval where This is the author’s peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset. PLEASE CITE THIS ARTICLE AS DOI:10.1063/1.5142771
the single-component reference operates in mixtures where σ22/σ11 deviate from one. To use the single-component reference for such mixtures, these correlations must be improved.
However, as the inherent accuracy of this reference is limited, as discussed in Sec. II B 1, there are few arguments to proceed along this route.
Fig. 9 shows that the predictions from the perturbation theories with the additive and non-additive references lie above the simulations results by Vrabec et al.47To verify whether these simulations were accurate, we performed independent GEMC simulations that were found to be in excellent agreement with previous simulation results (not shown).
For particles with σ22/σ11-ratios that differ from one, the radial distribution function of the mixture can differ significantly from that of the pure fluid, in particular at high densities. It is then questionable whether using the pure component radial distribution function in the expression fora1 (Eq. (36)) is a good assumption. We tested this assumption by obtaining the independent terms from the different interactions, a1,ij, by use of MC simulations (see Sec. III A) and used this to compute a1 of the mixture numerically with Eq. (36). The resulting values for a1,ij (symbols) differ from the approximation facilitated by Eq. (39), which gives only one solid-line for all of the i−j pairs, as shown in Fig. 11a.
However, when weighted with the mole-fractions, Fig. 11b demonstrates that the a1-value from the perturbation theory (lines) is in excellent agreement with the a1 obtained with MC simulations (symbols, See Sec. III A for details). This suggests that replacing the pair correlation function of the mixture with that of the pure component at the same effective hard-sphere diameter gives a reasonable representation of a1, even though the a1,ij-values differ. However, we cannot rule out the possibility that small deviations ina1 are responsible for some of the deviations in Fig. 9.
Figs. 11c and 11d reveal large deviations betweena2 anda3 from the perturbation theory, and the corresponding values from the MC simulations, in particular at high densities. At T∗ = 1 and for a fixed density, both a2 and a3 are uniquely defined in the perturbation theory described in Sec. II, since the α-value of the mixture is fixed. This explains why the solid lines for all the compositions overlap in Figs. 11c and 11d. The simulation results however, do not reproduce this behavior, and there is a clear distinction between a2 and a3 values obtained at different compositions. This shows that the expressions for a2 and a3 fail to reproduce the qualitative behavior of the perturbation terms, even for moderate differences in the particles sizes. The overprediction of the phase equilibrium pressures in This is the author’s peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset. PLEASE CITE THIS ARTICLE AS DOI:10.1063/1.5142771
Fig. 9 is largely due to the mispredictions of a2 at higher densities (cf. Fig. 11c): we tested that settinga3 to zero in the perturbation theories did not alter the phase envelopes, whereas also setting a2 to zero changed it significantly (green curve in Fig. 9).
V. CONCLUSION
This work addresses the fundamentals of perturbation theory for mixtures. The starting point was the SAFT-VR Mie equation of state, which is a perturbation theory for Mie fluids that uses a third-order Barker–Henderson expansion of the Helmholtz energy. Three different hard-sphere reference systems were compared: 1) single-component, 2) additive mixture and 3) non-additive mixture. Binary mixtures of Lennard-Jones (LJ) fluids were investigated to evaluate the accuracy of the different formulations, where the ratios of the well-depths () and position where the interaction potential equals zero (σ) were varied.
Only the non-additive sphere reference reproduced the exact result for the hard-sphere contribution to the cross-virial coefficient. The additive hard-hard-sphere reference devi-ated less than 3% from the exact results. The situation could be improved by incorporating an extra term in the perturbation theory that is usually neglected, which reduced the er-ror to 0.02%. The pure-component reference gave poor predictions when the σ-ratio of the binary mixture deviated from unity.
Accounting for non-additivity was found to be important to accurately predict the sol-ubility of mixtures with high -ratios and when non-additivity was explicitly incorporated into the mixing rule forσ. In these cases, the non-additive hard-sphere reference is preferred, but the single-component reference was most accurate when the σ-ratio was one.
For all cases investigated except for the single-component reference, comparison to Monte Carlo simulations in the canonical ensemble showed that the first order perturbation term, a1, was represented to a high accuracy. To approximate the radial distribution function of the mixture by that of the pure-component fluid at the corresponding hard-sphere diameter was found to be a good assumption for a1. We found significant deviations between theory and simulations for the second and third order perturbation terms, a2 and a3, for mixtures with - and σ-ratios that deviated from unity. In these mixtures, the perturbation theory failed to predict the phase equilibrium envelopes regardless of reference, in particular close to critical states.
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Further development of the perturbation theory for such mixtures is an important task for the future, as mixtures described by Mie-fluids with - and σ-ratios similar to those addressed in the present work are important in many natural and industrial applications.
ACKNOWLEDGMENTS
The authors thank Erich A. Müller, Amparo Galindo and George Jackson for helpful discussions. Ailo Aasen and Øivind Wilhelmsen have been supported by the ENERSENSE initiative for which they are grateful. Morten Hammer, Ailo Aasen, Åsmund Ervik and Øivind Wilhelmsen have been supported by the HYVA project, which is part of the Strate-gic Institute Programme of SINTEF Energy Research funded through the Basic Research Funding scheme of the Research Council of Norway.
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