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On the fluid phase behaviour of fluid binary mixtures using the Yukawa fluid molecular model
Fluid Phase Equilibria 293 (2010) 59–65
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Fluid Phase Equilibria journal homepage: www.elsevier.com/locate/fluid
On the fluid phase behaviour of fluid binary mixtures using the Yukawa fluid molecular model Osvaldo H. Scalise a,∗ , Douglas Henderson b a b
Instituto de Física de Líquidos y Sistemas Biológicos (IFLYSIB), UNLP-CONICET-CICPBA, C.C. 565, 1900 La Plata, Argentina Department of Chemistry and Biochemistry, Brigham Young University, Provo, UT 84602-5700, USA
a r t i c l e
i n f o
Article history: Received 5 November 2009 Received in revised form 15 February 2010 Accepted 17 February 2010 Available online 24 February 2010 Keywords: Perturbation theory Yukawa molecular model Fluid phase diagrams of binary mixtures
a b s t r a c t By expanding Ginoza’s mean spherical approximation (MSA) results in an inverse-temperature expansion, Henderson et al. obtained explicit results for the thermodynamic functions of a pure Yukawa fluid. We have recently published explicit results for the coefficients in an inverse-temperature expansion of the thermodynamic functions for the MSA for mixtures of Yukawa fluids. Attention is drawn to the fact that the MSA in the Ginoza formulation, does not always yield a convergent solution. The expansion used in this paper will always yield a result. In this work we present our investigations of the fluid phase diagram of Yukawa binary mixtures by considering an expansion of the MSA Helmholtz free energy up to the fifth order of the inverse-temperature expansion. The calculated fluid phase diagrams for Yukawa binary mixtures are similar to those of real mixtures. © 2010 Elsevier B.V. All rights reserved.
1. Introduction The knowledge of the fluid phase equilibrium of fluid binary mixtures is both an interesting and important problem to be solved. In many fields of applications, such as physical chemistry and chemical engineering, the information about the phase equilibrium of binary mixtures at high pressures and temperatures is very useful. In some applications of chemical engineering and the petroleum industry for instance, it is important in the design of fluid separation devices such as distillation columns and extraction trains. Such a knowledge, also, has been found useful in astrophysical and geophysical studies. This is an interesting problem to be solved in statistical mechanics since it involves developing new methods for calculating the different kinds of the fluid phase diagrams that are exhibited by real binary mixtures by starting from predefined molecular models. Experimentally, great effort has been made for investigating the fluid phase equilibrium behaviour of fluid binary mixtures at high pressures and temperatures [1]. Theoretically, the thermodynamic perturbation theory approach, which is an inverse-temperature expansion, has been found very useful in describing fluid properties of a singlecomponent fluid. A good description of the thermodynamic functions of such fluids has been obtained using second-order perturbation theory [2–4]. However, the study of mixtures is
∗ Corresponding author. Tel.: +54 221 425 4904; fax: +54 221 425 7317. E-mail addresses: ohs@iflysib.unlp.edu.ar, o [email protected] (O.H. Scalise). 0378-3812/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.fluid.2010.02.019
less complete and consequently, the theory of the properties of mixtures is nowadays still an active field of research. Thus, the theoretical study of fluid phase equilibria for binary mixtures has turned out to be a very important and useful field. It will provide information, for example, on the relationship between the intermolecular forces and the different types of phase transitions that take place in fluid binary mixtures such as gas–gas, liquid–liquid, and liquid–liquid–gas equilibrium. The mean spherical approximation (MSA) [5] yields good analytical results for the Yukawa fluid, which is a very reasonable molecular model for a single-component simple fluid. The original MSA solution for that fluid, due to Waisman [6], is analytic but not explicit. Ginoza [7] simplified that solution obtaining an analytic, but still not quite explicit, solution for that fluid. Henderson et al. [8] obtained explicit results for the thermodynamic functions of a single-component Yukawa fluid by expanding Ginoza’s result in an inverse-temperature expansion. Ginoza, and subsequently Blum et al. [9,10], have given expressions for the properties of a multicomponent mixture of Yukawa fluids. Ginoza’s expressions are for what he calls a factorizable mixture (Section 2). Blum et al. have generalized these results to non-factorizable mixtures. In the Ginoza–Blum studies, implicit, but not explicit, results were obtained. Curiously, the question of whether the Ginoza–Blum equations actually yield a convergent solution has not been investigated. To obtain numerical results from the implicit expression requires an iteration, which may be time consuming if extensive calculations are needed, as is the case in a study of phase equilibria. Additionally, it would be rather frustrating to arrive at a region of non-convergence in the midst of such
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O.H. Scalise, D. Henderson / Fluid Phase Equilibria 293 (2010) 59–65
a calculation. It is an advantage to have explicit results for the study of phase equilibria. Ginoza and Blum et al. did not report any numerical results. We report some numerical results here and consider the question of the convergence of the Ginoza–Blum expressions. In a previous paper [11], Ginoza’s results were extended to the case of an m-component fluid mixture of Yukawa fluids. Ginoza’s solution was considered and explicit results using an inversetemperature expansion up to the fifth order were obtained for both the coefficients of the expansion and the Helmholtz free energy (HFE) of the mixture. A fifth order expansion seems sufficient to give accurate results. In this paper we show that using the HFE and pressure expressions up to the fifth order of the inverse-temperature series is enough for predicting, qualitatively at least, the fluid phase diagrams of a fluid binary mixture of Yukawa fluids. The results show fluid phase diagrams that are similar to those of real binary mixtures. The next section presents a brief description of the theoretical approach; an interested reader, however, is referred to our previous publication [11] where a full description of the explicit formulae is given. In Section 3 results and discussion of binary mixtures that differ in the values of the Yukawa potential are presented. Finally, conclusions are presented in Section 4.
Ginoza obtained the following equation: ( + z) =
=
K =ˇ
uY (ri , rj ) =
⎩ − r exp ij
−zij (rij − ij ) ij
,
rij ≥ ij ,
(1)
(5)
with xi = Ni /N, the mole fraction of the species i. Ni is the number of particles of species i and N is the total number of molecules. Following Ginoza, we have:
ii ε¯
.
(6)
The parameter ε¯ is defined in Eq. (9). It may be noted that K = ˇLB /Z¯ 2 , with ˇ = 1/kT with k and T, the Boltzmann’s constant and temperature respectively, and LB =
xi xj LB ij
(7)
i
and
rij < ij
xi εii ,
i=1
2.1. Molecular model
(4)
Results up to the fifth order for n are given in Ref. [11]. It is assumed that for a binary mixture of Yukawa fluids with the same decay parameter z and having equal molecular diameters , i.e. differing only in the value of the intensity of the attractive energy interaction, that K is defined as follows:
Zi =
ij ij
K n n .
n
⎧ ⎨ ∞,
(3)
where = 3 /6 and = N/V , with V the volume of the fluid; see Ref. [11] for details. The generalization of Eq. (3) for the non-factorizable case has been given by Blum et al. Eq. (3), and its generalization, does not yield an explicit result. Iteration is required. An explicit solution of Eq. (3) can be obtained [8] using a series in powers of K:
2. Theory
In modelling the interaction of the molecules in a fluid binary mixture, it is assumed that the constituent molecules are rigid spheres of equal size interacting via the Yukawa pair potential. Thus, the pair potential between two molecules, say i and j, reads as follows:
6K D2 (),
Z¯ =
Zi xi .
(8)
i
where ij is the strength of the attracting interaction between molecules i and j, and zij is the range of the potential. For different pair of molecules we have assumed that the value of the decay parameter does not change and so zij = z. Further, we have assumed for the parameter value z = 1.8/ since it mimics a Lennard–Jones 12–6 potential [12]. The separation of the centers of molecules, i and j, is rij and i is the diameter of a molecule of species i and it is assumed that ij = (i + j )/2. In this paper we assume that all species have the same diameter so that ij = for all i and j. In later papers we will remove this restriction.
The superscript LB denotes that Eq. (2), the Lorentz–Berthelot (LB) mixing rule has been used for computing the cross interaction energy term. Further, and for this factorizable binary mixture, the following average energy is defined: ε¯ =
xi ii .
(9)
i
As may be noted, the following equation must be satisfied:
Zi2 xi = 1,
(10)
i
2.2. Yukawa mixtures
with
In this section a brief description of the theoretical approach is given as follows. We have assumed that a non-electrolyte fluid mixture is formed by molecules interacting via Eq. (1). Ginoza used the MSA and obtained an implicit solution for a simple non-electrolyte fluid mixture of molecules that interact via the Yukawa intermolecular potential. The MSA analytic solution so obtained was for the case of a factorizable mixture assuming that ij = , zij = z, with
ij =
i j .
(2)
xi = 1.
(11)
i
2.3. Thermodynamic properties The excess Helmholtz free energy (HFE) of the Yukawa binary mixture can be obtained using the following expansion: A =
K n An ,
(12)
n
Eq. (2) is one of the Berthelot mixing rules. The case of a mixture whose molecules all have the same diameter is of particular interest because molecules may be exchanged at no cost in the core energies and is, therefore, a stringent test of a perturbation expansion.
with 4 − 3 A0 = . NkT (1 − )2
(13)
O.H. Scalise, D. Henderson / Fluid Phase Equilibria 293 (2010) 59–65
Eq. (13) is obtained from the Carnahan–Starling [13] expression for the hard sphere pressure. The coefficients An are given by 1 En , n
An =
(14)
for n > 0. In Eq. (14) the En are the coefficients of the excess energy which is given by the following expansion: E =
K n En .
(15)
n
Full expressions for the En coefficients are given in Ref. [11]. The pressure is given by p=
K n pn .
(16)
n
Full expressions for the pn coefficients, for n > 0 are given in Ref. [11], being the p0 term given by, p0 V 1 + + 2 − 3 = . NkT (1 − )2
(17)
The Gibbs free energy is obtained using the expressions for the HFE and the pressure given above: G = A + pV.
(18)
3. Results and discussions Because we have assumed for the z parameter of the Yukawa pair potential has the same value, z = 1.8/, for the two components of the mixture (this parameter value mimics the Lennard–Jones (LJ) potential), the pure components of a Yukawa fluid binary mixture differ only in the values for the negative energy intensity value parameters ii . The energy ij is always given by Eq. (2). This restriction will be removed in later papers. First we compare, for two cases, the results of the series expansion for the Helmholtz free energy, A, with the results obtained from an iterative solution of Ginoza’s formulae. In the left panel of
61
Fig. 1 we consider the pure fluid case. In the right panel of this figure we give results for an extremely asymmetric mixture, where 11 = 0 and 22 is non-zero, for an equimolar mixture, x = 1/2. The quantity A is the excess Helmholtz free energy over that of a hard sphere fluid. There is no significant difference between the iterative and series expansion methods in the region for which the Ginoza formulae yield a convergent result. However, ultimately diverges and there is no solution for the MSA. We believe that this is the first time that the question of the convergence Ginoza equations has been studied. In any case, for the pure fluid this failure to converge occurs well within the thermodynamically unstable region between the coexisting vapour and liquid phases and is of no interest in a study of an equilibrium pure fluid. In a sense, the series expansion is an analytic continuation of the iterative solution. Whether there will always be solutions of the Ginoza–Blum formulae in the region of thermodynamic stability is, of course, unknown for all types of mixtures. However, the series expansion of the MSA results provides a useful method of accessing regions for which the Ginoza–Blum formulae fail to converge. Apart from the question of convergence, an iterative solution is not so difficult for a simple calculation. Moreover, for the study of phase equilibria, it is time consuming to make repeated iterations and the possibility of non-convergence in the midst of an extensive calculation would be frustrating. The explicit method presented here can be a real advantage from both points of view. We now present results for both the critical behaviour and the phase equilibria obtained using the molecular model and the theory given in the previous section. We have calculated (a) the critical points of pure components, (b) the vapour pressure lines of the pure components, (c) critical locus of the mixture, (d) the gas–liquid coexistence curves (isotherms) of the mixture and (e) the gas–liquid–liquid equilibrium curves of the mixture. We have investigated the following cases of binary mixtures for 22 = 1 with (a) 11 = 0.5, (b) 11 = 0.25 and (c) 11 = 0.005. The investigated binary mixture cases exhibit critical lines which are characteristic of type III behaviour in the classification of the phase diagrams given by Scott and van Konynenburg [14]. Mix-
Fig. 1. The Helmholtz free energy A, in excess of that of a hard sphere fluid as a function of ˇ times the relevant value of for three densities, ∗ = d3 = 0.3, 0.6, and 0.9. The left panel give results for a pure fluid and the right panel gives results for an highly asymmetric mixture for which 11 = 12 = 0 and 22 is non-zero and x = 1/2. The solid curves gives the results of the exact MSA in the Ginoza formulation and the broken curves gives the results of our HFE. The solid curves terminate when the Ginoza expression no longer yields a solution for .
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O.H. Scalise, D. Henderson / Fluid Phase Equilibria 293 (2010) 59–65
Fig. 2. p∗ –T ∗ projection of the p∗ –T ∗ –x thermodynamic surface of an equal diameter binary mixture of Yukawa fluids of equal diameter. C1 and C2 are the critical points of the pure components of the mixture. The solid lines that terminate at C1 and C2 show the vapour pressure equilibrium lines respectively for components 1 and 2. Critical curves shown are for mixtures of Yukawa fluids with ε22 = 1, and respectively for (a) ε11 = 0.005 (dotted line), (b) ε11 = 0.25 (dash dotted line), (c) ε11 = 0.5 (dashed line) and (d) ε11 = 0.5 shown by the short dotted line. The short dash dotted curve is the projection of the gas–liquid–liquid equilibrium line labelled as (e) and ending up the upper critical end point (UCEP).
tures which exhibit this type of behaviour include those systems that show the so called gas–gas immiscibility [1]. The classical conditions for a critical point of a binary mixture are [1]:
∂2 G ∂x2 ∂3 G ∂x3
and
∂4 G ∂x4
= 0,
(19)
= 0,
(20)
> 0.
(21)
p,T
p,T
p,T
Fig. 3. A, B, C, D, E, and F are projections of the corresponding critical loci cuts at respectively given values of temperatures shown by the arrows. Captions are as in Fig. 2.
The critical lines emerging at the critical point of the less volatile component of the mixture are calculated for 22 = 1 and using respectively for (a) 11 = 0.005, (b) 11 = 0.25 and for (c and d) 11 = 0.5. Fig. 2 also shows the three-phase line that ends at the upper critical end point (UCEP). At that point, three of the fluid phases become into two phases as the temperature of the mixture increases. It does coincide in a critical point where the critical locus meets at the three-phase line as is shown in Fig. 2. In Fig. 3, which is similar to Fig. 2, arrows are shown, for a reason that will become clear below, at the following values of the temperature, T ∗ = 0.65, T ∗ = 0.675, T ∗ = 0.95, and T ∗ = 1.25. Usually the phase diagrams of mixtures are given as the p∗ –T ∗ projection of the three dimensional p∗ –T ∗ –x surface. However, for our calculations we have found that it is useful to work with different cuts of the p∗ –T ∗ –x surface. By cutting the p∗ –T ∗ plane with a perpendicular plane at, for instance, T ∗ = 0.65, the critical points A and B shown in Fig. 3 result from the intersection of the plane at T ∗ = 0.65 and the two critical curves labelled (c) and (d) of Fig. 2. Clearly it may be noted that the two critical points A and B are
Two fluid phases (say a and b) of the fluid binary mixture of say, species i (i = 1, 2) are in equilibrium if , the chemical potential of species i at a given pressure (p), and temperature (T ), satisfy the following equations: (a)
i
(b)
= i ,
T (a) = T (b) ,
i = 1, 2,
(22) (23)
and p(a) = p(b) .
(24)
We have calculated the critical points and the fluid phase equilibria of the Yukawa binary mixtures by respectively solving numerically Eqs. (19)–(24) for G, the Gibbs free energy and the pressure p, of the mixture given in the above section. The reduced quantities given in terms of the Yukawa potential parameters are the density ∗ = N 3 /V , temperature T ∗ = kT/22 , and pressure p∗ = p 3 /22 . Fig. 2 shows the two vapour pressure lines of pure components 1 and 2 that end at C1 and C2 , the corresponding critical points. These are calculated using the expression for p given in Section 2 and numerically solving the equations [∂p/∂ ]T,x = 0, [∂2 p/∂ 2 ]T,x = 0 for x = 0, 1, respectively.
Fig. 4. p∗ –x cut at T ∗ = 0.65 of the p∗ –T ∗ –x thermodynamic surface. The isotherm shows three different phase equilibrium: (a) the dotted lines ending at the critical point B, (b) the short dotted lines ending at the critical point A and (c) the dashed lines shown at the region of lower pressure values. The projection of the three-phase line is shown by the dotted line; see Fig. 3.
O.H. Scalise, D. Henderson / Fluid Phase Equilibria 293 (2010) 59–65
63
Fig. 5. p∗ –x cut at T ∗ = 0.675 of the p∗ –T ∗ –x thermodynamic surface. Captions are as in Fig. 4. D and C are the corresponding two critical points of the binary mixture.
those marked in Fig. 3. Then, for calculating the two different phase equilibria (shown by both the parts that terminate at C1 and C2 ), it may be useful to take as the starting values for the pressures, values which are a bit lower than those of the corresponding critical points of Fig. 3. Then, with the starting values of p∗ , T ∗ and guessing the starting values for the mole fractions of the gas phase xg and the liquid phase xl , the searching procedure for the gas–liquid equilibrium may start for each gas–liquid equilibrium by solving the Eqs. (22)–(24) given above. The gas–liquid equilibrium in the lower pressure region, shown by the dashed line, may be obtained using the starting value of the pressure a lower value than that of the pressure of three-phase equilibrium line (see Figs. 3 and 4). In Fig. 4 we present the p∗ –x cut at T ∗ = 0.65, i.e. the isotherm of the mixtures calculated at that temperature. Fig. 5 shows the isotherm for the mixture calculated at T ∗ = 0.675 that also presents two critical points that correspond to the gas–liquid equilibrium shown by the dotted and the short-dashed lines. As it also may be noted, D and C are the critical points also shown in Fig. 2. The procedure we used for calculating the total isotherm is the same as the one described above.
Fig. 6. p∗ –x cut at T ∗ = 0.95 of the p∗ –T ∗ –x thermodynamic surface. The isotherm presents only one critical point, E, see Fig. 3, which is an ending point of the gas and liquid branches shown by the dashed and dotted lines respectively.
Fig. 7. p∗ –T ∗ –x thermodynamic surface for an equal diameter binary Yukawa mixture with ε22 = 1 and ε11 = 0.5; see Fig. 3. One loci of the critical points of the mixture is shown by the long solid curve. A second loci of critical points is shown by the short dotted curve. The calculated isotherms shown in black are at T ∗ = 0.65, 0.675, 0.8, 0.9 and 0.95.
The isotherm calculated at T ∗ = 0.9 presents only one critical point for the mixture labelled as E, as is shown in both Figs. 3 and 6. For calculating the gas–liquid phase equilibrium it is useful to start at lower values of the pressure than that of the critical point E, and proceed with the searching to obtain the gas and liquid branches of the isotherm as explained before. Finally, for the case of a binary mixture of Yukawa fluids with ε11 = 0.5 and ε22 = 1, we show in Fig. 7 the calculated p∗ –T ∗ –x thermodynamic surface. As is shown, it is similar to the p∗ –T ∗ –x diagram of the ethane + methanol real mixture (see Fig. 6.17 in Ref. [1] for the p∗ –T ∗ –x ethane + methanol system). As shown in Figs. 2–7, this is a case of Type III in the Scott and van Konynenburg classification of phase diagrams of fluid binary mixtures [14].
Fig. 8. p∗ –x cut of the p∗ –T ∗ –x thermodynamic surface at T ∗ = 1.25 for an equal diameter binary Yukawa mixture with ε22 = 1 and ε11 = 0.005. The lines shown by the dashed and dotted curves are the two branches of the phase equilibrium ending at F, the critical point of the mixture; see Fig. 3.
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O.H. Scalise, D. Henderson / Fluid Phase Equilibria 293 (2010) 59–65
become more alike, i.e. in our case when values of ε11 increases towards the value of ε22 our results shown in Fig. 2 (curve (b)) present a behaviour found in binary systems exhibiting gas–gas equilibrium of second kind although a minimum value in the temperature is not predicted by the theory. In Figs. 8 and 9 the calculated isotherms at T ∗ = 1.25 and T ∗ = 1.245 are also shown for the case of a fluid Yukawa binary mixture with ε11 = 0.005. In Fig. 10 we show the p∗ –T ∗ –x diagram in which the coexistence curve show gas–gas equilibrium behaviour. 4. Conclusions
Fig. 9. p∗ –x cut of the p∗ –T ∗ –x thermodynamic surface for an equal diameter binary Yukawa mixture at T ∗ = 1.245 with ε22 = 1 and ε11 = 0.005. Dashed and dotted curves are the two branches of the phase equilibrium ending at G, the critical point of the mixture; see Fig. 3.
We have also investigated two more cases of fluid Yukawa binary mixtures with ε11 = 0.005 and ε22 = 1, and for ε11 = 0.25 and ε22 = 1. With our algorithm it is difficult to consider the extremely asymmetric case where 11 is identically zero. Hence, we chose a small value for 11 (11 = 0.005) to mimic this case. These two types of binary mixtures present the behaviour found in real systems that show the gas–gas equilibrium [1]. Further, the investigated binary Yukawa mixtures of this work present the behaviour found in real systems. That is, one that shows the increasing of the miscibility between the components as they become more alike; i.e. when the corresponding critical curves starting at the critical point of the less volatile components move towards a region of lower temperatures values (see Fig. 2). The results shown in Fig. 2 (curve (a)) present the behaviour known as gas–gas equilibrium of first kind [1]. As the components
In this work we have investigated the fluid phase behaviour of some binary Yukawa mixtures using a theoretical approach developed for pure fluids [8] and recently extended to the case of mixtures [11] assuming that molecules interact via the Yukawa potential. To our knowledge, the fact that the MSA may not always yield a convergent solution, at least in the Ginoza formulation, has not been studied, or even pointed out, earlier. The non-convergence of the exact MSA equation for a pure Yukawa fluid, with z = 1.8, occurs outside the region of thermodynamic stability and is of little practical interest. There is no guarantee that this will be the case for a given mixture. The series expansion used here will always yield a convergent result. For the binary mixture’s cases investigated in this work we conclude that our calculations predict the fluid behaviour found in real binary systems that develop Type III behaviour in the classification of Scott and van Konynenburg [14] showing gas–gas, gas–liquid and gas–liquid–liquid equilibrium as have been found in real binary systems. It is interesting to note that such three-phase equilibrium behaviour found in real systems, such as an ethane–methanol system (see Fig. 7 and Ref. [1]), can be predicted using the fifth order MSA Helmholtz free energy expansion. All the systems that have been considered have molecules with equal diameter. The case of mixtures of fluids whose molecules that have unequal diameters will be considered in future publications. List of symbols A∗ reduced Helmholtz free energy, A∗ = ˇA/N k Boltzmann constant N total number of molecules in the system p pressure p∗ reduced pressure, p 3 /22 T absolute temperature T∗ reduced temperature, kT/22 V volume List of abbreviations HFE Helmholtz free energy LJ Lennard–Jones 12:6 potential model xg mole fraction of the gas phase xl mole fraction of the liquid phase Y Yukawa potential Greek letters packing fraction ˇ 1/kT
density
∗ reduced density, N 3 /V hard spheres diameter chemical potential of phase (i) (i)
Fig. 10. p∗ –T ∗ –x thermodynamic surface for an equal diameter binary Yukawa mixture with ε22 = 1 and ε11 = 0.005. Isotherms shown in black are at T ∗ = 1.25 and at T ∗ = 1.275. The critical loci is shown by the dashed line. The vapour pressure of the less volatile component of the mixture ending at C2 , the critical point, is shown by the solid curve.
Acknowledgements DH acknowledges the support of the National Institutes of Health through Grant No. 3R01GM076013-0451. OHS is member
O.H. Scalise, D. Henderson / Fluid Phase Equilibria 293 (2010) 59–65
of Carrera del Investigador Científico, Comisíon de Investigaciones Científicas de la Provincia de Buenos Aires (CICPBA). References [1] J.S. Rowlinson, F.L. Swinton, Liquid and Liquid Mixtures, 3rd ed., Butterworth, London, 1982. [2] J.A. Barker, D. Henderson, J. Chem. Phys. 47 (8) (1967) 2856–2861. [3] J.A. Barker, D. Henderson, J. Chem. Phys. 47 (11) (1967) 4714–4721. [4] J.A. Barker, D. Henderson, Rev. Mod. Phys. 48 (4) (1976) 587–671. [5] J.L. Lebowitz, J.K. Percus, Phys. Rev. 144 (1) (1966) 251–258. [6] E. Waisman, Mol. Phys. 25 (1) (1973) 45–48.
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[7] M.J. Ginoza, Phys. Soc. Jpn 54 (8) (1985) 2783–2785. [8] D. Henderson, L. Blum, J.P. Noworyta, J. Chem. Phys. 102 (12) (1995) 4973–4975. [9] L. Blum, F. Vericat, J.N. Herrera-Pacheco, J. Stat. Phys. 66 (1–2) (1992) 249–262. [10] L. Blum, J.A. Hernando, J. Phys. Conden. Matter 14 (2002) 11933–11944. [11] D. Henderson, O.H. Scalise, Collect. Czech. Chem. Commun. 73 (3) (2008) 424–438. [12] D. Henderson, E. Waisman, J.L. Lebowitz, L. Blum, Mol. Phys. 35 (1) (1978) 241–255. [13] N.F. Carnahan, K.E. Starling, J. Chem. Phys. 51 (2) (1969) 635–636. [14] R.L. Scott, P.H. van Konynenburg, Discuss. Faraday Soc. 49 (1970) 87–97; P.H. van Konynenburg, R.L. Scott, Phil. Trans. R. Soc. London Ser. A 298 (1442) (1980) 495–540.
Contents lists available at ScienceDirect
Fluid Phase Equilibria journal homepage: www.elsevier.com/locate/fluid
On the fluid phase behaviour of fluid binary mixtures using the Yukawa fluid molecular model Osvaldo H. Scalise a,∗ , Douglas Henderson b a b
Instituto de Física de Líquidos y Sistemas Biológicos (IFLYSIB), UNLP-CONICET-CICPBA, C.C. 565, 1900 La Plata, Argentina Department of Chemistry and Biochemistry, Brigham Young University, Provo, UT 84602-5700, USA
a r t i c l e
i n f o
Article history: Received 5 November 2009 Received in revised form 15 February 2010 Accepted 17 February 2010 Available online 24 February 2010 Keywords: Perturbation theory Yukawa molecular model Fluid phase diagrams of binary mixtures
a b s t r a c t By expanding Ginoza’s mean spherical approximation (MSA) results in an inverse-temperature expansion, Henderson et al. obtained explicit results for the thermodynamic functions of a pure Yukawa fluid. We have recently published explicit results for the coefficients in an inverse-temperature expansion of the thermodynamic functions for the MSA for mixtures of Yukawa fluids. Attention is drawn to the fact that the MSA in the Ginoza formulation, does not always yield a convergent solution. The expansion used in this paper will always yield a result. In this work we present our investigations of the fluid phase diagram of Yukawa binary mixtures by considering an expansion of the MSA Helmholtz free energy up to the fifth order of the inverse-temperature expansion. The calculated fluid phase diagrams for Yukawa binary mixtures are similar to those of real mixtures. © 2010 Elsevier B.V. All rights reserved.
1. Introduction The knowledge of the fluid phase equilibrium of fluid binary mixtures is both an interesting and important problem to be solved. In many fields of applications, such as physical chemistry and chemical engineering, the information about the phase equilibrium of binary mixtures at high pressures and temperatures is very useful. In some applications of chemical engineering and the petroleum industry for instance, it is important in the design of fluid separation devices such as distillation columns and extraction trains. Such a knowledge, also, has been found useful in astrophysical and geophysical studies. This is an interesting problem to be solved in statistical mechanics since it involves developing new methods for calculating the different kinds of the fluid phase diagrams that are exhibited by real binary mixtures by starting from predefined molecular models. Experimentally, great effort has been made for investigating the fluid phase equilibrium behaviour of fluid binary mixtures at high pressures and temperatures [1]. Theoretically, the thermodynamic perturbation theory approach, which is an inverse-temperature expansion, has been found very useful in describing fluid properties of a singlecomponent fluid. A good description of the thermodynamic functions of such fluids has been obtained using second-order perturbation theory [2–4]. However, the study of mixtures is
∗ Corresponding author. Tel.: +54 221 425 4904; fax: +54 221 425 7317. E-mail addresses: ohs@iflysib.unlp.edu.ar, o [email protected] (O.H. Scalise). 0378-3812/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.fluid.2010.02.019
less complete and consequently, the theory of the properties of mixtures is nowadays still an active field of research. Thus, the theoretical study of fluid phase equilibria for binary mixtures has turned out to be a very important and useful field. It will provide information, for example, on the relationship between the intermolecular forces and the different types of phase transitions that take place in fluid binary mixtures such as gas–gas, liquid–liquid, and liquid–liquid–gas equilibrium. The mean spherical approximation (MSA) [5] yields good analytical results for the Yukawa fluid, which is a very reasonable molecular model for a single-component simple fluid. The original MSA solution for that fluid, due to Waisman [6], is analytic but not explicit. Ginoza [7] simplified that solution obtaining an analytic, but still not quite explicit, solution for that fluid. Henderson et al. [8] obtained explicit results for the thermodynamic functions of a single-component Yukawa fluid by expanding Ginoza’s result in an inverse-temperature expansion. Ginoza, and subsequently Blum et al. [9,10], have given expressions for the properties of a multicomponent mixture of Yukawa fluids. Ginoza’s expressions are for what he calls a factorizable mixture (Section 2). Blum et al. have generalized these results to non-factorizable mixtures. In the Ginoza–Blum studies, implicit, but not explicit, results were obtained. Curiously, the question of whether the Ginoza–Blum equations actually yield a convergent solution has not been investigated. To obtain numerical results from the implicit expression requires an iteration, which may be time consuming if extensive calculations are needed, as is the case in a study of phase equilibria. Additionally, it would be rather frustrating to arrive at a region of non-convergence in the midst of such
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a calculation. It is an advantage to have explicit results for the study of phase equilibria. Ginoza and Blum et al. did not report any numerical results. We report some numerical results here and consider the question of the convergence of the Ginoza–Blum expressions. In a previous paper [11], Ginoza’s results were extended to the case of an m-component fluid mixture of Yukawa fluids. Ginoza’s solution was considered and explicit results using an inversetemperature expansion up to the fifth order were obtained for both the coefficients of the expansion and the Helmholtz free energy (HFE) of the mixture. A fifth order expansion seems sufficient to give accurate results. In this paper we show that using the HFE and pressure expressions up to the fifth order of the inverse-temperature series is enough for predicting, qualitatively at least, the fluid phase diagrams of a fluid binary mixture of Yukawa fluids. The results show fluid phase diagrams that are similar to those of real binary mixtures. The next section presents a brief description of the theoretical approach; an interested reader, however, is referred to our previous publication [11] where a full description of the explicit formulae is given. In Section 3 results and discussion of binary mixtures that differ in the values of the Yukawa potential are presented. Finally, conclusions are presented in Section 4.
Ginoza obtained the following equation: ( + z) =
=
K =ˇ
uY (ri , rj ) =
⎩ − r exp ij
−zij (rij − ij ) ij
,
rij ≥ ij ,
(1)
(5)
with xi = Ni /N, the mole fraction of the species i. Ni is the number of particles of species i and N is the total number of molecules. Following Ginoza, we have:
ii ε¯
.
(6)
The parameter ε¯ is defined in Eq. (9). It may be noted that K = ˇLB /Z¯ 2 , with ˇ = 1/kT with k and T, the Boltzmann’s constant and temperature respectively, and LB =
xi xj LB ij
(7)
i
and
rij < ij
xi εii ,
i=1
2.1. Molecular model
(4)
Results up to the fifth order for n are given in Ref. [11]. It is assumed that for a binary mixture of Yukawa fluids with the same decay parameter z and having equal molecular diameters , i.e. differing only in the value of the intensity of the attractive energy interaction, that K is defined as follows:
Zi =
ij ij
K n n .
n
⎧ ⎨ ∞,
(3)
where = 3 /6 and = N/V , with V the volume of the fluid; see Ref. [11] for details. The generalization of Eq. (3) for the non-factorizable case has been given by Blum et al. Eq. (3), and its generalization, does not yield an explicit result. Iteration is required. An explicit solution of Eq. (3) can be obtained [8] using a series in powers of K:
2. Theory
In modelling the interaction of the molecules in a fluid binary mixture, it is assumed that the constituent molecules are rigid spheres of equal size interacting via the Yukawa pair potential. Thus, the pair potential between two molecules, say i and j, reads as follows:
6K D2 (),
Z¯ =
Zi xi .
(8)
i
where ij is the strength of the attracting interaction between molecules i and j, and zij is the range of the potential. For different pair of molecules we have assumed that the value of the decay parameter does not change and so zij = z. Further, we have assumed for the parameter value z = 1.8/ since it mimics a Lennard–Jones 12–6 potential [12]. The separation of the centers of molecules, i and j, is rij and i is the diameter of a molecule of species i and it is assumed that ij = (i + j )/2. In this paper we assume that all species have the same diameter so that ij = for all i and j. In later papers we will remove this restriction.
The superscript LB denotes that Eq. (2), the Lorentz–Berthelot (LB) mixing rule has been used for computing the cross interaction energy term. Further, and for this factorizable binary mixture, the following average energy is defined: ε¯ =
xi ii .
(9)
i
As may be noted, the following equation must be satisfied:
Zi2 xi = 1,
(10)
i
2.2. Yukawa mixtures
with
In this section a brief description of the theoretical approach is given as follows. We have assumed that a non-electrolyte fluid mixture is formed by molecules interacting via Eq. (1). Ginoza used the MSA and obtained an implicit solution for a simple non-electrolyte fluid mixture of molecules that interact via the Yukawa intermolecular potential. The MSA analytic solution so obtained was for the case of a factorizable mixture assuming that ij = , zij = z, with
ij =
i j .
(2)
xi = 1.
(11)
i
2.3. Thermodynamic properties The excess Helmholtz free energy (HFE) of the Yukawa binary mixture can be obtained using the following expansion: A =
K n An ,
(12)
n
Eq. (2) is one of the Berthelot mixing rules. The case of a mixture whose molecules all have the same diameter is of particular interest because molecules may be exchanged at no cost in the core energies and is, therefore, a stringent test of a perturbation expansion.
with 4 − 3 A0 = . NkT (1 − )2
(13)
O.H. Scalise, D. Henderson / Fluid Phase Equilibria 293 (2010) 59–65
Eq. (13) is obtained from the Carnahan–Starling [13] expression for the hard sphere pressure. The coefficients An are given by 1 En , n
An =
(14)
for n > 0. In Eq. (14) the En are the coefficients of the excess energy which is given by the following expansion: E =
K n En .
(15)
n
Full expressions for the En coefficients are given in Ref. [11]. The pressure is given by p=
K n pn .
(16)
n
Full expressions for the pn coefficients, for n > 0 are given in Ref. [11], being the p0 term given by, p0 V 1 + + 2 − 3 = . NkT (1 − )2
(17)
The Gibbs free energy is obtained using the expressions for the HFE and the pressure given above: G = A + pV.
(18)
3. Results and discussions Because we have assumed for the z parameter of the Yukawa pair potential has the same value, z = 1.8/, for the two components of the mixture (this parameter value mimics the Lennard–Jones (LJ) potential), the pure components of a Yukawa fluid binary mixture differ only in the values for the negative energy intensity value parameters ii . The energy ij is always given by Eq. (2). This restriction will be removed in later papers. First we compare, for two cases, the results of the series expansion for the Helmholtz free energy, A, with the results obtained from an iterative solution of Ginoza’s formulae. In the left panel of
61
Fig. 1 we consider the pure fluid case. In the right panel of this figure we give results for an extremely asymmetric mixture, where 11 = 0 and 22 is non-zero, for an equimolar mixture, x = 1/2. The quantity A is the excess Helmholtz free energy over that of a hard sphere fluid. There is no significant difference between the iterative and series expansion methods in the region for which the Ginoza formulae yield a convergent result. However, ultimately diverges and there is no solution for the MSA. We believe that this is the first time that the question of the convergence Ginoza equations has been studied. In any case, for the pure fluid this failure to converge occurs well within the thermodynamically unstable region between the coexisting vapour and liquid phases and is of no interest in a study of an equilibrium pure fluid. In a sense, the series expansion is an analytic continuation of the iterative solution. Whether there will always be solutions of the Ginoza–Blum formulae in the region of thermodynamic stability is, of course, unknown for all types of mixtures. However, the series expansion of the MSA results provides a useful method of accessing regions for which the Ginoza–Blum formulae fail to converge. Apart from the question of convergence, an iterative solution is not so difficult for a simple calculation. Moreover, for the study of phase equilibria, it is time consuming to make repeated iterations and the possibility of non-convergence in the midst of an extensive calculation would be frustrating. The explicit method presented here can be a real advantage from both points of view. We now present results for both the critical behaviour and the phase equilibria obtained using the molecular model and the theory given in the previous section. We have calculated (a) the critical points of pure components, (b) the vapour pressure lines of the pure components, (c) critical locus of the mixture, (d) the gas–liquid coexistence curves (isotherms) of the mixture and (e) the gas–liquid–liquid equilibrium curves of the mixture. We have investigated the following cases of binary mixtures for 22 = 1 with (a) 11 = 0.5, (b) 11 = 0.25 and (c) 11 = 0.005. The investigated binary mixture cases exhibit critical lines which are characteristic of type III behaviour in the classification of the phase diagrams given by Scott and van Konynenburg [14]. Mix-
Fig. 1. The Helmholtz free energy A, in excess of that of a hard sphere fluid as a function of ˇ times the relevant value of for three densities, ∗ = d3 = 0.3, 0.6, and 0.9. The left panel give results for a pure fluid and the right panel gives results for an highly asymmetric mixture for which 11 = 12 = 0 and 22 is non-zero and x = 1/2. The solid curves gives the results of the exact MSA in the Ginoza formulation and the broken curves gives the results of our HFE. The solid curves terminate when the Ginoza expression no longer yields a solution for .
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Fig. 2. p∗ –T ∗ projection of the p∗ –T ∗ –x thermodynamic surface of an equal diameter binary mixture of Yukawa fluids of equal diameter. C1 and C2 are the critical points of the pure components of the mixture. The solid lines that terminate at C1 and C2 show the vapour pressure equilibrium lines respectively for components 1 and 2. Critical curves shown are for mixtures of Yukawa fluids with ε22 = 1, and respectively for (a) ε11 = 0.005 (dotted line), (b) ε11 = 0.25 (dash dotted line), (c) ε11 = 0.5 (dashed line) and (d) ε11 = 0.5 shown by the short dotted line. The short dash dotted curve is the projection of the gas–liquid–liquid equilibrium line labelled as (e) and ending up the upper critical end point (UCEP).
tures which exhibit this type of behaviour include those systems that show the so called gas–gas immiscibility [1]. The classical conditions for a critical point of a binary mixture are [1]:
∂2 G ∂x2 ∂3 G ∂x3
and
∂4 G ∂x4
= 0,
(19)
= 0,
(20)
> 0.
(21)
p,T
p,T
p,T
Fig. 3. A, B, C, D, E, and F are projections of the corresponding critical loci cuts at respectively given values of temperatures shown by the arrows. Captions are as in Fig. 2.
The critical lines emerging at the critical point of the less volatile component of the mixture are calculated for 22 = 1 and using respectively for (a) 11 = 0.005, (b) 11 = 0.25 and for (c and d) 11 = 0.5. Fig. 2 also shows the three-phase line that ends at the upper critical end point (UCEP). At that point, three of the fluid phases become into two phases as the temperature of the mixture increases. It does coincide in a critical point where the critical locus meets at the three-phase line as is shown in Fig. 2. In Fig. 3, which is similar to Fig. 2, arrows are shown, for a reason that will become clear below, at the following values of the temperature, T ∗ = 0.65, T ∗ = 0.675, T ∗ = 0.95, and T ∗ = 1.25. Usually the phase diagrams of mixtures are given as the p∗ –T ∗ projection of the three dimensional p∗ –T ∗ –x surface. However, for our calculations we have found that it is useful to work with different cuts of the p∗ –T ∗ –x surface. By cutting the p∗ –T ∗ plane with a perpendicular plane at, for instance, T ∗ = 0.65, the critical points A and B shown in Fig. 3 result from the intersection of the plane at T ∗ = 0.65 and the two critical curves labelled (c) and (d) of Fig. 2. Clearly it may be noted that the two critical points A and B are
Two fluid phases (say a and b) of the fluid binary mixture of say, species i (i = 1, 2) are in equilibrium if , the chemical potential of species i at a given pressure (p), and temperature (T ), satisfy the following equations: (a)
i
(b)
= i ,
T (a) = T (b) ,
i = 1, 2,
(22) (23)
and p(a) = p(b) .
(24)
We have calculated the critical points and the fluid phase equilibria of the Yukawa binary mixtures by respectively solving numerically Eqs. (19)–(24) for G, the Gibbs free energy and the pressure p, of the mixture given in the above section. The reduced quantities given in terms of the Yukawa potential parameters are the density ∗ = N 3 /V , temperature T ∗ = kT/22 , and pressure p∗ = p 3 /22 . Fig. 2 shows the two vapour pressure lines of pure components 1 and 2 that end at C1 and C2 , the corresponding critical points. These are calculated using the expression for p given in Section 2 and numerically solving the equations [∂p/∂ ]T,x = 0, [∂2 p/∂ 2 ]T,x = 0 for x = 0, 1, respectively.
Fig. 4. p∗ –x cut at T ∗ = 0.65 of the p∗ –T ∗ –x thermodynamic surface. The isotherm shows three different phase equilibrium: (a) the dotted lines ending at the critical point B, (b) the short dotted lines ending at the critical point A and (c) the dashed lines shown at the region of lower pressure values. The projection of the three-phase line is shown by the dotted line; see Fig. 3.
O.H. Scalise, D. Henderson / Fluid Phase Equilibria 293 (2010) 59–65
63
Fig. 5. p∗ –x cut at T ∗ = 0.675 of the p∗ –T ∗ –x thermodynamic surface. Captions are as in Fig. 4. D and C are the corresponding two critical points of the binary mixture.
those marked in Fig. 3. Then, for calculating the two different phase equilibria (shown by both the parts that terminate at C1 and C2 ), it may be useful to take as the starting values for the pressures, values which are a bit lower than those of the corresponding critical points of Fig. 3. Then, with the starting values of p∗ , T ∗ and guessing the starting values for the mole fractions of the gas phase xg and the liquid phase xl , the searching procedure for the gas–liquid equilibrium may start for each gas–liquid equilibrium by solving the Eqs. (22)–(24) given above. The gas–liquid equilibrium in the lower pressure region, shown by the dashed line, may be obtained using the starting value of the pressure a lower value than that of the pressure of three-phase equilibrium line (see Figs. 3 and 4). In Fig. 4 we present the p∗ –x cut at T ∗ = 0.65, i.e. the isotherm of the mixtures calculated at that temperature. Fig. 5 shows the isotherm for the mixture calculated at T ∗ = 0.675 that also presents two critical points that correspond to the gas–liquid equilibrium shown by the dotted and the short-dashed lines. As it also may be noted, D and C are the critical points also shown in Fig. 2. The procedure we used for calculating the total isotherm is the same as the one described above.
Fig. 6. p∗ –x cut at T ∗ = 0.95 of the p∗ –T ∗ –x thermodynamic surface. The isotherm presents only one critical point, E, see Fig. 3, which is an ending point of the gas and liquid branches shown by the dashed and dotted lines respectively.
Fig. 7. p∗ –T ∗ –x thermodynamic surface for an equal diameter binary Yukawa mixture with ε22 = 1 and ε11 = 0.5; see Fig. 3. One loci of the critical points of the mixture is shown by the long solid curve. A second loci of critical points is shown by the short dotted curve. The calculated isotherms shown in black are at T ∗ = 0.65, 0.675, 0.8, 0.9 and 0.95.
The isotherm calculated at T ∗ = 0.9 presents only one critical point for the mixture labelled as E, as is shown in both Figs. 3 and 6. For calculating the gas–liquid phase equilibrium it is useful to start at lower values of the pressure than that of the critical point E, and proceed with the searching to obtain the gas and liquid branches of the isotherm as explained before. Finally, for the case of a binary mixture of Yukawa fluids with ε11 = 0.5 and ε22 = 1, we show in Fig. 7 the calculated p∗ –T ∗ –x thermodynamic surface. As is shown, it is similar to the p∗ –T ∗ –x diagram of the ethane + methanol real mixture (see Fig. 6.17 in Ref. [1] for the p∗ –T ∗ –x ethane + methanol system). As shown in Figs. 2–7, this is a case of Type III in the Scott and van Konynenburg classification of phase diagrams of fluid binary mixtures [14].
Fig. 8. p∗ –x cut of the p∗ –T ∗ –x thermodynamic surface at T ∗ = 1.25 for an equal diameter binary Yukawa mixture with ε22 = 1 and ε11 = 0.005. The lines shown by the dashed and dotted curves are the two branches of the phase equilibrium ending at F, the critical point of the mixture; see Fig. 3.
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O.H. Scalise, D. Henderson / Fluid Phase Equilibria 293 (2010) 59–65
become more alike, i.e. in our case when values of ε11 increases towards the value of ε22 our results shown in Fig. 2 (curve (b)) present a behaviour found in binary systems exhibiting gas–gas equilibrium of second kind although a minimum value in the temperature is not predicted by the theory. In Figs. 8 and 9 the calculated isotherms at T ∗ = 1.25 and T ∗ = 1.245 are also shown for the case of a fluid Yukawa binary mixture with ε11 = 0.005. In Fig. 10 we show the p∗ –T ∗ –x diagram in which the coexistence curve show gas–gas equilibrium behaviour. 4. Conclusions
Fig. 9. p∗ –x cut of the p∗ –T ∗ –x thermodynamic surface for an equal diameter binary Yukawa mixture at T ∗ = 1.245 with ε22 = 1 and ε11 = 0.005. Dashed and dotted curves are the two branches of the phase equilibrium ending at G, the critical point of the mixture; see Fig. 3.
We have also investigated two more cases of fluid Yukawa binary mixtures with ε11 = 0.005 and ε22 = 1, and for ε11 = 0.25 and ε22 = 1. With our algorithm it is difficult to consider the extremely asymmetric case where 11 is identically zero. Hence, we chose a small value for 11 (11 = 0.005) to mimic this case. These two types of binary mixtures present the behaviour found in real systems that show the gas–gas equilibrium [1]. Further, the investigated binary Yukawa mixtures of this work present the behaviour found in real systems. That is, one that shows the increasing of the miscibility between the components as they become more alike; i.e. when the corresponding critical curves starting at the critical point of the less volatile components move towards a region of lower temperatures values (see Fig. 2). The results shown in Fig. 2 (curve (a)) present the behaviour known as gas–gas equilibrium of first kind [1]. As the components
In this work we have investigated the fluid phase behaviour of some binary Yukawa mixtures using a theoretical approach developed for pure fluids [8] and recently extended to the case of mixtures [11] assuming that molecules interact via the Yukawa potential. To our knowledge, the fact that the MSA may not always yield a convergent solution, at least in the Ginoza formulation, has not been studied, or even pointed out, earlier. The non-convergence of the exact MSA equation for a pure Yukawa fluid, with z = 1.8, occurs outside the region of thermodynamic stability and is of little practical interest. There is no guarantee that this will be the case for a given mixture. The series expansion used here will always yield a convergent result. For the binary mixture’s cases investigated in this work we conclude that our calculations predict the fluid behaviour found in real binary systems that develop Type III behaviour in the classification of Scott and van Konynenburg [14] showing gas–gas, gas–liquid and gas–liquid–liquid equilibrium as have been found in real binary systems. It is interesting to note that such three-phase equilibrium behaviour found in real systems, such as an ethane–methanol system (see Fig. 7 and Ref. [1]), can be predicted using the fifth order MSA Helmholtz free energy expansion. All the systems that have been considered have molecules with equal diameter. The case of mixtures of fluids whose molecules that have unequal diameters will be considered in future publications. List of symbols A∗ reduced Helmholtz free energy, A∗ = ˇA/N k Boltzmann constant N total number of molecules in the system p pressure p∗ reduced pressure, p 3 /22 T absolute temperature T∗ reduced temperature, kT/22 V volume List of abbreviations HFE Helmholtz free energy LJ Lennard–Jones 12:6 potential model xg mole fraction of the gas phase xl mole fraction of the liquid phase Y Yukawa potential Greek letters packing fraction ˇ 1/kT
density
∗ reduced density, N 3 /V hard spheres diameter chemical potential of phase (i) (i)
Fig. 10. p∗ –T ∗ –x thermodynamic surface for an equal diameter binary Yukawa mixture with ε22 = 1 and ε11 = 0.005. Isotherms shown in black are at T ∗ = 1.25 and at T ∗ = 1.275. The critical loci is shown by the dashed line. The vapour pressure of the less volatile component of the mixture ending at C2 , the critical point, is shown by the solid curve.
Acknowledgements DH acknowledges the support of the National Institutes of Health through Grant No. 3R01GM076013-0451. OHS is member
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of Carrera del Investigador Científico, Comisíon de Investigaciones Científicas de la Provincia de Buenos Aires (CICPBA). References [1] J.S. Rowlinson, F.L. Swinton, Liquid and Liquid Mixtures, 3rd ed., Butterworth, London, 1982. [2] J.A. Barker, D. Henderson, J. Chem. Phys. 47 (8) (1967) 2856–2861. [3] J.A. Barker, D. Henderson, J. Chem. Phys. 47 (11) (1967) 4714–4721. [4] J.A. Barker, D. Henderson, Rev. Mod. Phys. 48 (4) (1976) 587–671. [5] J.L. Lebowitz, J.K. Percus, Phys. Rev. 144 (1) (1966) 251–258. [6] E. Waisman, Mol. Phys. 25 (1) (1973) 45–48.
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[7] M.J. Ginoza, Phys. Soc. Jpn 54 (8) (1985) 2783–2785. [8] D. Henderson, L. Blum, J.P. Noworyta, J. Chem. Phys. 102 (12) (1995) 4973–4975. [9] L. Blum, F. Vericat, J.N. Herrera-Pacheco, J. Stat. Phys. 66 (1–2) (1992) 249–262. [10] L. Blum, J.A. Hernando, J. Phys. Conden. Matter 14 (2002) 11933–11944. [11] D. Henderson, O.H. Scalise, Collect. Czech. Chem. Commun. 73 (3) (2008) 424–438. [12] D. Henderson, E. Waisman, J.L. Lebowitz, L. Blum, Mol. Phys. 35 (1) (1978) 241–255. [13] N.F. Carnahan, K.E. Starling, J. Chem. Phys. 51 (2) (1969) 635–636. [14] R.L. Scott, P.H. van Konynenburg, Discuss. Faraday Soc. 49 (1970) 87–97; P.H. van Konynenburg, R.L. Scott, Phil. Trans. R. Soc. London Ser. A 298 (1442) (1980) 495–540.