On molecular-based equations of state: rigor versus speculations

Fluid Phase Equilibria 182 (2001) 3–15

On molecular-based equations of state: rigor versus speculations Ivo Nezbeda a

E. Hála Laboratory of Thermodynamics, Institute of Chemical Process Fundamentals, Academic Sciences, 6-Suchdol, 165 02 Prague, Czech Republic b Department of Physics, J.E. Purkynˇe University, 400 96 Úst´ı nad Labem, Czech Republic Received 5 August 2000; accepted 22 December 2000

Abstract A general scheme for developing any semi-empirical molecular-based equation of state is formulated along with several rules which reflect the essentials of physics of fluids and which should be observed. Approximations and simplifications used in the implementation of the scheme are analyzed in the light of these rules and examples showing superiority of molecular-based considerations over purely intuitive or empirical ones are presented. © 2001 Elsevier Science B.V. All rights reserved. Keywords: Statistical mechanics; Perturbation expansion; Molecular-based equations

1. Introduction In addition to experimental measurements, the common way to determine the thermodynamic properties of pure fluids and fluid mixtures is by means of equations of state (EOS). There are, in principle, three types of equations: (1) empirical; (2) theoretical; (3) semi-empirical (semi-theoretical). Empirical equations are typically obtained by fitting the known experimental data of the considered real system to a usually arbitrary many-parameter function and should thus be more appropriately called correlation functions (see, e.g. [1]). One advantage of empirical equations is their accuracy but they also suffer from several disadvantages. First of all, they require a very large body of experimental data. Secondly, their application is limited only to the systems for which the parameters were obtained and only to the range of thermodynamic conditions used in the fitting procedure. Finally, they cannot be easily extended to mixtures. On the other hand, theoretical equations are very general and applicable for the entire family of systems obeying the given intermolecular potential model. The main drawback of theoretical equations is that they are valid only for the specific interaction model, which is always only a very approximate representation of the real systems, and their accuracy is usually only fair (see, e.g. [2]). The most suitable method to derive an EOS for practical purposes seems thus a combination of both these approaches: to use a theoretically-based functional form of the EOS and treat some of its parameters E-mail address: [email protected] (I. Nezbeda). 0378-3812/01/$20.00 © 2001 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 8 - 3 8 1 2 ( 0 1 ) 0 0 3 7 5 - 2

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as quantities adjustable to experimental data. This procedure reduces considerably the amount of needed experimental data, removes the direct dependence of the EOS on the chosen intermolecular potential model, and guarantees accuracy while maintaining simultaneously a relative simplicity, transferability to mixtures, and generality. Semi-empirical equations derived in this way bear very often the label “molecular-based” and may possess, under certain circumstances, also the potential of predictability. Typical examples of such equations are perturbed equations, i.e. equations written in the form of a sum of terms which are supposed to reflect contributions from different types of intermolecular interactions (see, e.g. [3–5]). The immediate questions which arise are the following: (i) are all semi-empirical equations truly molecular-based? (ii) How can we decide what is and what is not a molecular-based EOS? And, (iii) when such an EOS can be used for predictions at all? It is the purpose of this paper to address these questions. A theoretically justified general scheme for developing any semi-empirical EOS is outlined, and several rules based on physical considerations which should be observed when developing a truly molecular-based EOS are formulated. Examples for each step in the perturbation calculation scheme are presented, different approximations used in the implementation of the scheme and their suitability are analyzed, and superiority of molecular-based equations is demonstrated. 2. Theory 2.1. Preliminary considerations From the molecular point of view, we must begin with the total intermolecular energy U = U (q1 , . . . , qN ) which is a function of generalized coordinates of all molecules. The first approximation we usually make is that of pairwise additivity,
u(qi , qj ), (1) U= i
where u is an intermolecular energy between molecules i and j . Since the assumption of pairwise additivity is only approximate, the pair potential models used in practical calculations are not approximations of the correct pair interaction but effective potentials accounting (to a certain extent) also for non-additivity and other effects, including our lack of knowledge. A typical way to construct a realistic pair potential is to view the molecule as a hard body with distributed interaction sites of two kinds (neutral, typically but not necessarily Lennard–Jones-like, and Coulombic) and to write an effective realistic (transferable) intermolecular pair potential in the form of a site–site potential [6–8] 

 q i qj e 2 u(1, 2) ≡ u(R12 , Ω1 , Ω2 ) = (2) uneutral (rij ) + = uneutral (1, 2) + uCoul (1, 2), rij i∈{1}j ∈{2} where the set (R12 , Ωi ) defines, respectively, the mutual position and orientation of a pair of molecules, rij denotes the separation between site i on molecule 1 and site j on molecule 2, qi are partial charges, and e the elementary electron charge. In order to incorporate explicitly cooperative phenomena at the level of pair potentials, polarizibility may be added to Eq. (2) which however does not change qualitatively the considered approach and this possibility will not therefore be considered here.

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Not all practical calculations begin necessarily with the most accurate realistic interaction model available but with a physically plausible model made up of several terms reflecting contributions from different types of interactions, u(1, 2) = ua (1, 2) + ub (1, 2) + · · · .

(3)

Neither the realistic model (2) nor the composed model (3) are amenable to a simple straightforward theoretical treatment. Model (2) is evidently too complex, and as regards model (3), it is also practically impossible for theory to treat all terms contributing to u(1, 2) at the same level of accuracy. Consequently, in both cases, we have to resort to a perturbation expansion. 2.2. Perturbation expansion Given an intermolecular pair potential u, the perturbation expansion method proceeds as follows [9–11]: 1. u is decomposed into a reference part and a perturbation part u = uref + upert .

(4)

2. The Helmholtz free energy, A, is expanded in powers of (βupert ) A = Aref + β A1 ({upert }) + β 2 A2 ({upert }) + · · · ,

(5)

where β is the inverse temperature, β = 1/kB T , Aref the Helmholtz free energy of the reference fluid defined by uref , and the correction terms Ai are functionals of the structure of the reference fluid. 3. After the correction terms are evaluated, the EOS can be derived in the form:     βP ∂(βA/N) βP ≡ρ = + z1 + z2 + · · · , (6) ρ ∂ρ ρ ref V where P is the pressure, V the volume, and ρ the number density, ρ = N/V . 3. Development of a molecular-based EOS 3.1. Decomposition of u(1,2) The decomposition given by Eq. (4) is not unique and is dictated by both physical considerations (convergence of the expansion) and mathematical considerations (feasibility of the evaluation of the individual terms). It is well known that the evaluation of the perturbation terms Ai beyond the first-order poses severe problems [9–11]. It means that in practical calculations, the perturbation expansion may provide an EOS only in the form   βP βP = + z1 . (7) ρ ρ ref Consequently, to consider this first-order EOS as a reasonable approximation to reality, the higher-order terms must be negligible which means that the perturbation expansion must be fast converging. The question is how to meet this requirement.

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It was well established already long time ago that the perturbation expansion is fast converging if the structure of the reference and considered fluids are nearly identical (very similar). For non-polar fluids, this condition leads to a reference defined by short-range repulsive interactions [10,12]. For polar and associating fluids, the situation has not been so clear and numerous theoretical attempts to deal with these fluids have been only partially successful often producing even contradictory results. Only recently the effect of the short- and long-range forces on the structure of these fluids has systematically been examined in detail [13–15] with the result very similar to that known for non-polar fluids. It has been found that the structure of polar and associating fluids, defined in terms of the site–site correlation functions, is determined predominantly by short-range interactions which may be, unlike the case of non-polar fluids, both repulsive and attractive. In other words, the reference system must incorporate all the short-range forces whereas the long-range Coulombic interactions, no matter how strong they may be, provide only a small correction to the properties of this reference and may thus be treated as a perturbation. We give two examples of the decomposition considering both forms of the potential models, Eqs. (2) and (3). Example 1. As a typical example of potential of type (3) we take the model considered by Muller and Gubbins [4] in their study of water uMG (1, 2) = uLJ (1, 2) + uHB (1, 2) + uDD (1, 2),

(8)

where the individual terms represent, respectively, the neutral Lennard–Jones interaction, hydrogen bonding, and dipole–dipole interaction. Following only purely practical arguments, Muller and Gubbins considered the short-range hydrogen bonding as a perturbation and defined the reference system in this way uMG,ref = uLJ (1, 2) + uDD (1, 2),

(9)

Fig. 1. Comparison of the second virial coefficient for the potential model (8) (solid line) with those for the reference potential models (9) (dashed line) and (10) (filled circles).

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In contrast to this choice, when we follow the physical arguments given above, we are guided to define the reference system by the short-range part of Eq. (9), i.e. utheor,ref = uLJ (1, 2) + uHB (1, 2).

(10)

A difference in the convergence of the expansion about these two qualitatively different reference systems should be expected. To demonstrate the effect of the choice of the reference, we have evaluated the second virial coefficient for all three potential models, Eqs. (8)–(10), and results are shown in Fig. 1. As it is seen, over the entire considered temperature range, the physical reference (10) approximates the virial coefficient of the uMG fluid extremely well, whereas the performance of the reference (9) rapidly deteriorates with decreasing temperature and deviates very significantly from the correct result. Example 2. The realistic site–site model (2) is composed of a number of equivalent terms none of which can be simply neglected. The construction of a short-range reference must therefore employ a cut-off procedure; e.g. a brute-force cut-off setting, u(1, 2) ≡ 0 for R12 > Rcut , or a smooth cut-off by means of a switch function, see, e.g. [16]. In order to make expansion of Eq. (5) about a short-range system, theoretically tractable without resorting to computer simulations, we followed the idea suggested in [16]. Thus, in [15,17] we defined the reference model in such a way that the perturbation may be written in terms of the leading multipole moments interaction. Thus, for the systems made up of molecules with a permanent dipole moment (dipolar and associating fluids) we have ushort-range,ref = u(1, 2) − S(R12 ; R1 , R2 )uDD (1, 2),

(11)

where R12 is the center-to-center distance between the molecules, and S, S ≡ 0 for R12 < R1 and S ≡ 1 for R12 > R2 , is the switch function which gradually turns on the dipole–dipole interaction over the interval (R1 , R2 ). In other words, by using the switch function in deducting the dipole–dipole interaction, the force field at short intermolecular separations remains intact (for details see [17]). Similarly, for quadrupolar fluids we have ushort-range,ref = u(1, 2) − S(R12 ; R1 , R2 )uQQ (1, 2).

(12)

It has been shown [15,17] that the deduction of the leading multipole–multipole interaction is sufficient to make the potential short-ranged and equivalent to cut-off methods. In Figs. 2 and 3 we compare computer simulation results for the site–site correlation functions of typical polar fluids, acetone and carbon dioxide modeled by realistic site–site potentials [18,19], with those of the respective short-range systems defined above (for details see [15]). As it is seen, the correlation functions of the full and reference systems are nearly indistinguishable. In order to demonstrate that the properly defined short-range reference captures also the thermodynamic properties of the original system very well, we show in Fig. 4 the results for the internal energy of water for temperatures ranging from the room temperature to the supercritical one. (The results are taken from [17] where the realistic water was defined by the TIP4P potential and the associated short-range reference by deducting the dipole–dipole interaction, see Eq. (11)). It is easy to verify that the difference in the internal energy between the TIP4P water and its reference does not exceed 5%.

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Fig. 2. Comparison of the site–site correlation functions of a realistic model of acetone at ambient conditions (filled circles) with those for the corresponding short-range reference (solid line).

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Fig. 3. The same as Fig. 2 for carbon dioxide.

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Fig. 4. Comparison of the internal energy of the TIP4P water (symbols) with that of the associated short-range reference (11) (dashed lines) at different temperatures: T = 298 K (filled circles); T = 353 K (open circles); T = 550 K (filled triangles); T = 700 K (open triangles).

3.2. Description of the reference To accomplish the third step of the perturbation scheme, both the thermodynamic and structural properties of the reference must first be available, preferably in a closed analytic form. For realistic potential models, the short-range reference system is still too complex to be amenable to a direct analytical description. This problem is therefore solved by another perturbation expansion which usually results in a mapping of the properties of the reference onto those of an appropriate simple model. The procedure may be exemplified by the well-known theory for non-polar fluids [9,10]. For these fluids, the reference is defined by soft-repulsive bodies whose properties are then approximated by those of hard-spheres of a certain diameter or, in a more general case, of certain hard bodies [20]. This approximation leads then to an EOS of the form     βP βP βP = + z ≈ + z. (13) ρ ρ short-range,ref ρ hard body Provided that we deal with large flexible molecules, then the hard body term would correspond to that for a model of a flexible chain (see, e.g. [21]). It is worth reminding the reader at this point that the form of this equation does not mean that the perturbation expansion is carried out about a hard body reference. The compressibility factor of the hard body fluid in this equation only approximates the unknown compressibility factor of the short-range reference. The size and shape of the reference hard bodies and their dependence on the thermodynamic conditions reflect then the degree of accuracy of the approximation of the original short-range system.

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To give an example, the reference system for the simple Lennard–Jones fluid within the Barker–Henderson theory is defined by [10]    σ 12  σ 6 uBH,ref (R) = 4 − , for R < σ, (14) R R and the properties of this reference system are approximated by those of the fluid of hard-spheres of diameter  σ d= [1 − exp(−βuBH,ref )]R 2 dR. (15) 0

For polar and associating fluids, the situation is more complex and although it has not been solved yet completely, some encouraging results have already been achieved. The problem is to find a simple theoretically tractable model which would approximate reasonably accurately both the structural and thermodynamic properties of the short-range reference. A path pursued in the last decade is based on primitive models (for a review see [22]) and the thermodynamic perturbation theory (TPT) [23] developed particularly for these models. The primitive models approximate by simple means the force field generated by the short-range reference. Thus, the short-range repulsions are approximated by the hard-sphere interaction and the residual Coulombic attraction by a square-well interaction which may give rise to hydrogen bonding. Within this model, the molecule is thus viewed as a fused-hard-sphere body with additional interaction sites of two kinds, let us say + and − (for details see [22]). The unlike sites attract one another whereas the like sites repel one another. Furthermore, for water and some other associating fluids, the parameters of the model must be set up so as to respect the condition of steric incompatibility, i.e. the fact that (i) only one hydrogen bond may be established between the pair of molecules, and (ii) one (+ or −) interaction site may be engaged in establishing one hydrogen bond only. These are also the necessary conditions for the TPT to be applicable. Examples of possible primitive models for water and methanol are shown in Fig. 5. To summarize, there are several steps which ultimately lead to an EOS for polar and associating fluids   βP βP = + z1 (16) ρ ρ short-range,ref   βP βP ≈ + z1 (17) ρ ρ primary,model   βP βP ≈ + zassociation + z1 . (18) ρ ρ hard body

Fig. 5. Schematic representation of the molecules of water and methanol within the family of the primitive models.

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We remind again that the last decomposition of the compressibility factor of the reference system into the hard body and associating parts results from the application of the TPT to the primitive model and does not mean in any case that the perturbation expansion for the fluid of interest is carried out about the hard body reference. In other words, the correction z1 must be evaluated over the properties of the primitive model. 3.3. Evaluation of the correction terms Within the first-order perturbation expansion, the correction to the reference Helmholtz free energy, A, is given by the ensemble average of the perturbation part of the total internal energy [10–12],  1 2 1 A = U ref = ρ [gref (1, 2) − 1] u(1, 2) d(1) d(2) − ρN × constant 2 2 1 = ρN[I (ρ, T ) − constant], (19) 2 where g(1, 2) is the pair correlation function. Very rarely the perturbation potential is given by a single term (which may be the case of simple fluids like argon). Here u(1, 2) is usually made up of several terms and we then face the problem of evaluating several perturbation integrals, typically I = Idispersion + Ilong-range .

(20)

Relatively simple and yet physically plausible evaluation of the perturbation integral I seems the most difficult problem for the entire perturbation theory to solve. 3.3.1. Mean-field approximation The simplest and most frequently used approximation is that of the mean field. This crude approximation ignores completely the structure of the reference fluid and sets gref (1, 2) ≡ 1 outside the impenetrable core of the molecules. The perturbation integral is thus zero regardless of the form of the perturbation potential and hence z1,mean field = −βρ 2 × constant,

(21)

which yields an augmented van der Waals equation. The augmented van der Waals equation, consisting of a reference term and the mean field term, does not account for any specific interactions between the molecules and cannot thus evidently be able to describe properly the behavior of a larger group of compounds. It is more appropriate (and very usual) to apply the mean field approximation only to the dispersion term which yields then the EOS in the form   βP βP = − βρ 2 × constant + z1 (22) ρ ρ short-range,ref 3.3.2. Treatment of dispersion forces The dispersion forces for simple fluids are well defined by quantum mechanics and may be written as a series in terms of the inverse even powers of the intermolecular separation, (1/R)−n , n ≥ 6. Since the correlation function of the reference is the key quantity in the perturbation integral I , approximations are applied to the form of the dispersion forces. It may seem acceptable, from the physical point of

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Fig. 6. Comparison of the density dependence of the perturbation integral I for the (1/R)6 dispersion forces (solid lines) with that for the square-well attraction (dashed lines) of different range λ.

view, to replace the continuous dependence of the dispersion forces either by a constant or a simply varying attractive field, that is, e.g. by a square-well or triangular-well. Surprisingly, the result is not very encouraging as it is seen from Fig. 6. In this figure we compare the density dependence of the perturbation integral for the simple fluid obtained using the hard-sphere radial distribution function for the reference and exact R −6 interaction with that for the square-well attraction of different range (for details see [24]). We see that in the exact case, I is a simple monotonically increasing function of density whereas the density dependence of I for the square-well attraction is rather complex and unpredictable as the range varies. Consequently, instead of simplifying the result, we get more complicated density dependence and this artifact must then be later compensated by additional (empirical) terms added to the EOS. The same result as given above holds true not only for the triangular-well attraction, but also in the case of other reference systems. In [24], the perturbation integral was evaluated directly by Monte Carlo simulations for the short-range reference for water and square-well attraction and similar results as above were obtained. These findings seem to disqualify empirical attempts to improve EOS by approximating the dispersion force contribution by functions obtained, e.g. for the simple square-well fluid as is the case of some versions of the SAFT equation [25–28]. Such an approximation is not justified at all and, as we have shown above, it may even worsen the results.

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3.3.3. Treatment of specific forces and SAFT equation As for the evaluation of the perturbation integral for interactions other than the dispersion ones, no general recipe can be offered. But one important remark seems in order. From the definition of integral, I it is evident that the reference and perturbation contributions are interrelated and it is therefore not possible to build purely formally a term developed for one reference into an equation based on another reference. Such a formal evaluation immediately debases the molecular basis of the development of the EOS and turns the resulting equation into another correlation formula. Unfortunately, this is a very frequent practice and a typical example is again the SAFT equation [29]. Since this equation, based on a decomposition of the Helmoltz free energy into individual contributions originating in the molecular shape, hydrogen bonding, and dispersion forces (and, possibly, other interactions), has become frequently used in last years, few remarks seem appropriate. When the parameters of the SAFT equation are adjusted by fitting a set of experimental data, it usually produces good results. But as regards to predictions, its performance is known to be to a certain extent unpredictable. This defect has both technical and principal reasons. As regards the technical aspect, the equation uses for the association part the simplest type of primitive models available [30] which is known to capture only qualitatively the structure of real associating fluids. Thus, in order to approximate thermodynamics, its parameters are either set to unrealistic values or treated as adjustable which violates the necessary conditions for applicability of the TPT at all. As regards the principle problems, in the decomposition of the Helmholtz free energy none of the terms is considered as a reference term and all contributions are therefore evaluated independently using semi-empirical formulas available on the market [cf. Eq. (18) and the remark below it]. Consequently, the SAFT equation as is cannot be viewed as a perturbed, molecular-based equation but there are clear ways to convert it into such an equation and make it more reliable.

4. Conclusions The power and usefulness of molecular-based equations has been verbally proclaimed dozens times and nobody has questioned results and recommendations of, and conclusions resulting from, molecular studies. However, in practice, all these achievements seem to have been ignored so far. As a typical example, it may serve recent attempts to derive an accurate EOS for the Lennard–Jones fluid. The empirical equations of Nicolas et al. [31] and of Johnson et al. [32] required about fifty parameters to fit experimental data and yet they are inferior to the molecular-based EOS of Kolafa and Nezbeda [33] with much fewer adjustable parameters. On the other hand, one can find in the literature many equations dressed-up in molecular-like appearance but which are purely empirical in nature. Potential danger in use of such molecular-like equations is that they are often used to unjustifiably and hence misleadingly interpret and assess effects of various types of interactions upon the properties of fluids. Being aware of all these facts, and in the light of seemingly never ending activity in the field of equations of state, we have attempted in this paper to summarize the basic unquestionable theoretical findings which seem relevant for applications, particularly for developing a molecular-based EOS. These findings have been summarized into a general perturbation scheme and certain rules which should be observed in implementation of the scheme in order to respect the essentials of physics of fluids. A few examples have been given to demonstrate superiority of molecular-based approximations over intuitive ones.

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Acknowledgements This research was supported by the Grant Agency of the Czech Republic (Grant No. 203/99/0134) and the Grant Agency of the Academy of Sciences (Grant No. A4072712). References [1] A. Saul, W. Wagner, J. Phys. Chem. Ref. Data 16 (1987) 893–984. [2] A. Chialvo, Yu. V. Kalyuzhnyi, P.T. Cummings, in: K. Hutchenson, N. Foster (Eds.), Supercritical Fluid Science and Technology, Vol. 608, 1995. [3] I.G. Economou, M.D. Donohue, Ind. Eng. Chem. Res. 31 (1992) 2388–2394. [4] E.A. Muller, K.E. Gubbins, Ind. Eng. Chem. Res. 34 (1995) 3662–3673. [5] J.R. Elliott, S.J. Suresh, M.D. Donuhue, Ind. Eng. Chem. Res. 29 (1990) 1476–1485. [6] W.I. Jorgensen, J. Am. Chem. Soc. 103 (1981) 335–340. [7] W.L. Jorgensen, C.J. Swenson, J. Am. Chem. Soc. 107 (1985) 569–578. [8] W.I. Jorgensen, J. Phys. Chem. 90 (1986) 1276–1284. [9] J.P. Hansen, I. McDonald, Theory of Simple Liquids, Academic Press, New York, 1986. [10] T. Boubl´ık, I. Nezbeda, K. Hlavatý, Statistical Thermodynamics of Simple Liquids and Their Mixtures, Elsevier, Amsterdam, 1980. [11] C.G. Gray, K.E. Gubbins, Theory of Molecular Fluids, Vol. 1, Clarendon Press, Oxford, 1984. [12] L.L. Lee, Molecular Thermodynamics of Nonideal Fluids, Butterworths, Boston, 1988. [13] I. Nezbeda, J. Kolafa, Mol. Phys. 97 (1999) 1105–1116. [14] J. Kolafa, I. Nezbeda, Mol. Phys. 98 (2000) 1505–1520. [15] I. Nezbeda, J. Kolafa, M. L´ısal, Mol. Phys., in press. [16] I. Nezbeda, J. Kolafa, Czech. J. Phys. 40 (1990) 138–150. [17] I. Nezbeda, M. L´ısal, Mol. Phys. 99 (2001) 291–300. [18] P. Jedlovszky, G. Pálinkás, Mol. Phys. 84 (1995) 217–233. [19] J.G. Harris, K.H. Yung, J. Phys. Chem. 99 (1995) 12021–12024. [20] T. Boubl´ık, I. Nezbeda, Coll. Czech. Chem. Commun. 51 (1986) 2301–2432. [21] D. Ghonasgi, V. Perez, J. Chem. Phys. 101 (1994) 6880–6887. [22] I. Nezbeda, J. Mol. Liquids 73/74 (1997) 317–336. [23] M.S. Wertheim, J. Stat. Phys. 35 (1984) 19–35. [24] I. Nezbeda, Fluid Phase Equilib. 180 (2001) 165–171. [25] M. Banaszak, Y.C. Chiew, M. Radosz, Phys. Rev. E 48 (1993) 3760–3765. [26] F.W. Tavares, J. Chang, S.I. Sandler, Mol. Phys. 86 (1995) 1451–1471. [27] Y.H. Fu, S.I. Sandler, Ind. Eng. Chem. Res. 34 (1995) 1897–1909. [28] A. Gil-Villegas, A. Galindo, P.J. Whitehead, S.J. Mills, G. Jackson, A.N. Burgess, J. Chem. Phys. 106 (1997) 4168–4186. [29] W.G. Chapman, K.E. Gubbins, G. Jackson, M. Radosz, Ind. Eng. Chem. Res. 29 (1990) 1709–1721. [30] J. Kolafa, I. Nezbeda, Mol. Phys. 61 (1987) 161–175. [31] J.J. Nicolas, K.E. Gubbins, W.B. Streett, D.J. Tildesley, Mol. Phys. 37 (1979) 1429–1438. [32] J.K. Johnson, J.A. Zollweg, K.E. Gubbins, Mol. Phys. 78 (1993) 591–618. [33] J. Kolafa, I. Nezbeda, Fluid Phase Equilib. 100 (1994) 1–34.