Journal of Molecular Liquids 164 (2011) 162–170

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Journal of Molecular Liquids

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Simple analytic equations of state for the generalized hard-core Mie(α, β) andMie(α, β) fluids from perturbation theory

Hervé Guérin ⁎4 rue Commandant Faurax 69006 Lyon, France

⁎ Retired from Ecole Supérieure de Chimie Physique Evard du 11 novembre 1918, 69100 Villeurbanne, France

E-mail address: herve.guerin0257@orange.fr.

0167-7322/$ – see front matter © 2011 Elsevier B.V. Alldoi:10.1016/j.molliq.2011.09.003

a b s t r a c t

a r t i c l e i n f o

Article history:Received 4 April 2011Received in revised form 27 July 2011Accepted 2 September 2011Available online 15 September 2011

Keywords:Equation of stateHard-core Lennard–Jones potentialMie potentialsPerturbation theory

New, simple and analytic perturbation theory equations of state for generalized hard-core Mie HCMie(α, β)and Mie(α, β) fluids are proposed. They are based on the second-order Barker–Henderson perturbation the-ory in the macroscopic compressibility approximation and the new analytical expression of the radial distri-bution function of hard spheres, gHS(r), developed by Sun in terms of a polynomial expansion of basefunctions adapted to the square-well and Sutherland potentials [Can. J. Phys. 83 (2005) 55], the combinationof which yields the HCMie(α, β) and Mie(α, β) functions. The compressibility factors, the residual internalenergies and the radial distribution function at contact with the hard core are then obtained from this equa-tion of state for the HCLJ(12, 6) potential, which is a particular case of the HCMie(α, β) potentials with α=12and β=6. The results are in good agreement with the existing Monte Carlo (MC) simulation data, and com-pare favorably with those obtained from five other equations of state, three of which contain numerical co-efficients fitted to the Monte Carlo results. For the Mie(α, 6) (α=8, 10, 12), fluids, the present equation ofstate is a good representation of recent molecular dynamics (MD) simulations of the pressure and internalenergy. It is more accurate than the statistical associating fluid theory of variable range (SAFT-VR Mie(n,6)) theory for n=8, and 10, while for n=12 the SAFT-VR theory is best. For the Mie(14, 7) fluid, which isoutside the range of application of the SAFT-VR theory, the results for the pressure are in good agreementwith the analytical equation of state obtained from the MC simulation data.

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1. Introduction

Recently new simple analytic expressions for the hard-spherefluid (HS) radial distribution function (RDF), gHS(r), were derivedby Sun [1–4], which provide convenient analytic expressions for thethermodynamic properties of hard-core fluids within the frameworkof the Barker–Henderson (BH) perturbation of fluids [5–12]. ThisHS-RDF is expressed in terms of polynomial expansions of non-linear base functions adapted to some well-known hard-core inter-molecular potentials. For example, they were applied to potentialmodels in inverse powers of the interparticle distance r such as−r−γ, which include the square-well (SW, γ=0) [1], the Sutherland(SU(γ), γ≥6) [1] and the triangular-well (γ=−1) [2] potentials aswell as the important class of hard-core multi-Yukawa potentials[3,4]. The method was also recently extended to the SW chain fluids[13].

These new expressions of gHS(r) should be also particularly conve-nient to study hard-core fluids with potentials having different com-binations of tail segments, for which piecewise integrationprocedures could be implemented. For example, in this way one

could study double SW, SWwith a triangular tail (or linear extension)referred to as the square-well linear extension (SWLE) potential de-fined in Ref. [14] or a SW with a more flexible tail composed of theproduct of linear and Sutherland (SU) functions called the extendedsquare-well (ESW) potential introduced in Ref. [15].

Important potentials to be studied by this method are the general-ized hard-core Lennard–Jones or Mie function HCMie(α,β) andMie(α, β) built up from the potentials studied in Ref. [1]. TheHCMie(α, β) is composed of a SWwith a Mie(α, β) tail, which is itselfa linear combination of two Sutherland potentials: SU(α), u(α)(r)=−ε(σ/r)α, and SU(β), u(β)(r)=−ε(σ/r)β. It is given by (Fig. 1)

u rð Þ ¼ þ∞; r b σ;¼ −ε; σ b r b rmin;

¼ Cεσr

� �α− σ

r

� �β� �¼ −Cu αð Þ rð Þ þ Cu βð Þ rð Þ; r N rmin;

ð1Þ

where σ, rmin and −ε correspond respectively to the zero and mini-mum of the full Mie(α, β) potential (Fig. 1) so that

rmin ¼ σ α=βð Þ1= α−βð Þ; C ¼ β

α−βαβ

� �α= α−βð Þ: ð2Þ

Fig. 1. Comparison of the hard-core Mie(α, β) potential (continuous curve) and theMie(α, β) potential (dashed curve for rb rmin and continuous curve for rN rmin).

163H. Guérin / Journal of Molecular Liquids 164 (2011) 162–170

The general Mie(α, β) potentials are a linear combination of theSU(α) and SU(β) potentials given by

uM rð Þ ¼ Cεσr

� �α− σ

r

� �β� �: ð3Þ

They have varying short-range repulsive interactions, which is ofprime importance to determine the properties of simple fluids. Forthe usual values α=12, β=6 we have rmin=21/6σ and C=4 andthe usual HCLJ(12, 6) and LJ(12, 6) potentials. It is the purpose ofthis article to study the thermodynamic properties of these twotypes of fluids using the BH perturbation theory with the new expres-sions of gHS(r) used by Sun for the SW and SU(γ) potentials in Ref. [1].

The HCLJ(12, 6) was first introduced by Stell and Weis [16] tocompare various approximate theories of simple fluids withoutbeing hindered by the approximations introduced by the presenceof a soft core, and it was used recently to study the osmotic pressurein aqueous protein/salt systems under various conditions [17]. It alsoserved as a testing ground to generate molecular-based equations ofstate (EOS) from the generalized perturbation theory, which uses asreference more realistic potentials such as the SW or attractive Yuka-wa potentials [18]. In this way Sowers and Sandler [18] developedtwo accurate analytic EOS based on the HS and SW reference poten-tials (denoted here respectively SSHS and SSSW), whose free energyperturbation terms were modeled by reasonable analytic functions(polynomial or Gaussian) of the reduced density ρ*=ρσ3 (ρ=N/V)with several adjustable parameters fitted to the extensive MonteCarlo (MC) simulations performed by these authors [18]. Two otheranalytic EOS for the HCLJ(12,6) potential (denoted here SWLE andESW) were also derived by Shen and Lu [14] and Farrokhpour andParsafar [15] based respectively on the SWLE and ESW potentialsmentioned above, whose parameters (depth, range and position ofthe tail) are chosen so that there are equal area constraints with theHCLJ(12,6) for the negative parts of the potentials. For these twoEOS, the free energy and then other thermodynamic functions wereobtained from the BH perturbation theory for which it was not neces-sary to have an analytic expression of gHS(r) since the perturbationintegrals were obtained from the compressibility equation forrb1.3σ and gHS(r)=1 for rN1.3σ.

Finally new EOS for SU(γ) and HCLJ(12,6) fluids were alsoobtained recently by Diez, Largo and Solana (DLS) [19], who parame-terized the MC simulation data of the first- and second-order pertur-bative terms in the inverse temperature expansion of the free energyusing the NVT ensemble of the reference HS fluid [19].

Two of these EOS (SSHS, SSSW) [18], having adjustable parame-ters obtained from computer simulations, lose in physical meaningand cannot be extended to the generalized HCMie(α, β) fluids. Twoother EOS (SWLE, ESW) [14,15] use different potentials, whoseshapes are constrained to mimic the original HCLJ, and this entailssome systematic errors, moreover gHS(r)=1 for rN1.3σ; howeverthey could be generalized to the HCMie(α, β). For the recent DLSEOS, only the first-order term of the free energy could be extendedto deal with the HCMie(α, β), since the potential appears explicitlyin its parameterized expression, while the second-order term is di-rectly fitted by a polynomial in ρ* to the corresponding MC simulationdata for the HCLJ(12,6) potential.

There are many equations of state for the LJ(12,6) fluid, whichwere reviewed elsewhere [20–24]. However, one of the most impor-tant applications of the Mie(α, β) potential models nowadays is in itsuse as elementary pieces in modern molecular-based equations ofstate (EOS), such as the SAFT-VR theory [25] applicable to a largeclass of fluids. In this theory, chain fluid EOS have been developedbased on perturbation theory applied to the monomer LJ(12,6) [26]and Mie(α, β) segments for βbα, β≤6, α≤12 [26–30], in whichthe perturbation terms were evaluated from the mean-value theoremusing an effective packing fraction and the local compressibility ap-proximation [11] for the second-order term. We shall then use themonomer part of these EOS as comparison with the equations we de-velop here and show that they are valid for a wider range of values ofα and β (βbα, βN3, αN3). It is also worthy to note that, although theSAFT-VR theory can be applied to the SW and SU(γ) potentials sepa-rately, it cannot be used as such for the HCMie(α,β) fluids consideredhere, which combines both type of potentials.

We obtain first, in Section 2, the analytic expressions for the thermo-dynamic properties of the HCMie(α, β) fluids free from the drawbacksof the above-mentioned EOS. Moreover some structural propertiessuch as the values of the radial distribution function (RDF) at contactwith the hard core, g(σ), can be obtained by this method as well asfrom the DLS EOS (Section 3). This quantity is important to calculatethe chain contribution in the SAFT-VR theory of hard-core chain fluids[25] once themonomer HCMie(α, β) segments thermodynamic quanti-ties are known. Secondly, in Section 4 these expressions are extended tothe Mie(α, β) fluids by using a temperature-dependent hard core,whose diameter is given by the Barker–Henderson formula [6,11].

The accuracy of these new expressions is then evaluated in Sec-tion 5, first for the HCLJ(12, 6) fluid against the MC simulation dataof Sowers and Sandler [18] and compared to the five EOS briefly de-scribed above (SSHS, SSSW, SWLE, ETW and DLS). The values of g(σ) obtained from our method and the DLS EOS are also comparedto the MC simulation data of Stell andWeis [16]. Secondly, The resultsfor the Mie(α, 6) (α=8, 10, 12) potentials are compared to the mo-lecular dynamics (MD) simulations of Galliéro et al. [30] and theSAFT-VR theory. Finally for the Mie(14, 7) potential, for which theSAFT-VR theory is no longer valid, the results are compared to thoseobtained from an analytical equation of state obtained from the MCsimulation data [31].

2. Thermodynamic properties of the HCMie(α, β) fluids from theBH perturbation theory

In the BH perturbation of fluids [5–12], the residual (res) Helm-holtz free energy Fres, from which other thermodynamic propertiescan be derived, is expressed as a power series in the inverse reducedtemperature, T*=kBT/ε, where kB is the Boltzmann constant and −εis the potential minimum. When truncated at the second order, itmay be written as

Fres

NkBT¼ F−F id

NkBT¼ FHS

NkBTþ F1NkBT

1T� þ

F2NkBT

1T�2 ; ð4Þ

164 H. Guérin / Journal of Molecular Liquids 164 (2011) 162–170

where N is the number of molecules and the subscript id stands forideal gas. The HS and the expansion terms are respectively given by

FHS

NkBT¼ 4η−3η2

1−ηð Þ2 ; ð5Þ

F1NkBT

¼ 12η∫∞1gHS xð Þu� xð Þx2dx; ð6Þ

F2NkBT

¼ −6ηKHS∫∞1gHS xð Þ u� xð Þ½ �2x2dx; ð7Þ

where x=r/σ is the reduced distance, u*(x)=u(x)/ε is the reducedHCMie(α, β) potential, η=πρ*/6 is the packing fraction, ρ*=ρσ 3 isthe reduced density, ρ=N/V is the particle density and KHS is theHS isothermal compressibility given by

KHS ¼ kBT∂ρ∂P

� �HS

¼ 1−ηð Þ41þ 4ηþ 4η2−4η3 þ η4 : ð8Þ

Eqs. (5) and (8) are respectively the Carnahan–Starling (CS) [8,32]expressions of the corresponding quantities. The second-order termcorresponds to the macroscopic compressibility approximation(MCA). The local compressibility approximation could also be used;however it is only marginally better and leads to more complicatedequations so that the former approximation is chosen for simplicity.

In order to evaluate the integrals in Eqs. (6) and (7), we use theanalytical expression of gHS(x) proposed by Sun in Ref. [1]

gHS xð Þ ¼ 0; x b 1;

¼ 1þ ∑3

m¼1

ηm

1−ηð Þm gm xð Þ; 1b x b 3;

¼ 1; x N 3;

ð9Þ

where the base functions gm(x) are given by

gm xð Þ ¼ ∑3

n¼0Amn x−x−4

� �n; 1b x b 2;

¼ ∑3

n¼0Bmn x−256x−7

� �n: 2 b x b 3:

ð10Þ

The constant numerical coefficients, Amn and Bmn, are given in Ref.[1] and Table 1; they were obtained by fitting the MC simulation dataand the numerical results of the Percus–Yevick approximation for thefirst two coordination shells. As Sun showed that the use of the Bmn

coefficients made only negligible differences for the SW and SU(γ)potentials [1], we restrict ourselves to the first coordination shell(1b×b2), which means that we shall only use here the Amn coeffi-cients. Hence we take gm(x)=0 and gHS(x)=1 for xN2.

We now introduce expression (9) for gHS(x) into Eqs. (6) and (7)and obtain

F1NkBT

¼ 12η ∫∞1u� xð Þx2dxþ ∑

3

m¼1

ηm

1−ηð Þm ∫2

1gm xð Þu� xð Þx2dx

� ; ð11Þ

Table 1Coefficients Amn and Bmn appearing in the analytic expressions (9) and (10) of gHS(r) [1].

m/n 0 1 2 3

Amn 1 2.5 −0.2975 −1.4243 0.41412 2 −4.0464 −2.1422 2.07673 0.5 −6.3781 10.13 −3.6507

Bmn 1 −0.34 −0.3317 0.5555 −0.14022 1 2.8897 −3.1078 0.68223 −0.29 −2.619 2.1554 −0.3989

F2NkBT

¼ −6ηKHS ∫∞1u� xð Þ½ �2x2dxþ ∑

3

m¼1

ηm

1−ηð Þm ∫2

1gm xð Þ u� xð Þ½ �2x2dx

� :

ð12Þ

Introducing now expressions (1) for the HCMie(α, β) potentialand (10) for gm(x) into the above equations and using the binomialexpansion

x−x−4� �n ¼ ∑

n

ℓ¼0

−1ð Þℓn!ℓ! n−lð Þ! x

n−5ℓ; ð13Þ

we obtain the following analytical expressions for F1/(NkBT) (βN3,αNβ)

F1NkBT

¼ f1SW xminð Þ−Cf1

αð Þ xminð Þ þ Cf1ðβÞ xminð Þ; ð14Þ

f1SW xminð Þ ¼ −12η

xmin3−13

þ ∑3

m¼1

ηm

1−ηð Þm Lm3ð Þ xmin;1ð Þ

" #; ð15Þ

f1γð Þ xminð Þ ¼ −12η − xmin

3−γ

3−γþ ∑

3

m¼1

ηm

1−ηð Þm Lm3−γð Þ 2; xminð Þ

" #;γ ¼ α;β;

ð16Þ

where xmin=rmin/σ (see Fig. 1) and the Lm(k)(a, b) are auxiliary func-tions stemming respectively from the piecewise integrations from 1to xmin and xmin to 2. They are given by

Lmkð Þ a; bð Þ ¼ ∑

3

n¼0∑n

ℓ¼0

Amn−1ð Þℓn!

ℓ! n−ℓð Þ!×an−5ℓþk−bn−5ℓþk

n−5ℓþ k: ð17Þ

Before we give the expression of F2/(NkBT), some remarks aboutthe above equations are in order: f1SW(xmin) gives the contributionto F1/(NkBT) of the SW part of the HCMie potential; in fact it is the ex-pression given by Sun [1] for a SW of width λ=xmin, and the argu-ments xmin and 1 of the function Lm(3)(xmin, 1) representsrespectively the upper and lower limits of integration. The contribu-tion of the attractive SU(γ) potential, −ε(σ/r)γ, from xmin to infinityis represented by f1(γ)(xmin). The integrations are first done from xmin

to 2 (first coordination shell)with the analytical expressions of gHS(x),which gives the functions Lm(3−γ)(2, xmin), (γ=α, β). For xN2, gm(x)=0 and gHS(x)=1, and the other successive coordination shells are in-cluded in the calculation through the first integral of Eq. (11), whichgives the terms in −xmin

(3−γ)/(3−γ).In an analogous way and using the relations

u� xð Þ½ �2 ¼ 1; σ b r b rmin;

¼ C2 x−2α−2x−α−β þ x−2β� �

; r N rmin;ð18Þ

we obtain the MCA second-order term as

F2NkBT

¼ KHS

2f1SW xminð Þ þ C2f1

2αð Þ xminð Þ þ C2f12βð Þ xminð Þ−2C2f1

αþβð Þ xminð Þh i

:

ð19Þ

The compressibility factor Z (i.e. the EOS) and the residual internalenergy (Ures=U−Uid) are then obtained from the classic thermody-namic relationships

Z ¼ PVNkBT

¼ η∂∂η

FNkBT

� �;

Ures

NkBT¼ −T� ∂

∂T�Fres

NkBT

� �; ð20Þ

165H. Guérin / Journal of Molecular Liquids 164 (2011) 162–170

which gives

Z ¼ ZHS þ z1T� þ

z2T�2 ; zi ¼ η

∂ F i=NkBTð Þ∂η ; i ¼ 1;2: ð21Þ

ZHS is the CS–HS compressibility factor [32] given by

ZHS ¼ 1þ ηþ η2−η3

1−ηð Þ3 : ð22Þ

The first order term z1 is then given by

z1 ¼ z1SW xminð Þ−Cz1

αð Þ xminð Þ þ Cz1βð Þ xminð Þ; ð23Þ

where

z1SW xminð Þ ¼ −12η

xmin3 −13

þ ∑3

m¼1

ηm

1−ηð Þm 1þ m1−η

� �Lm

3ð Þ xmin;1ð Þ" #

;

ð24Þ

z1γð Þ xminð Þ ¼ −12η − xmin

3−γ

3−γþ ∑

3

m¼1

ηm

1−ηð Þm 1þ m1−η

� �Lm

3−γð Þ 2; xminð Þ" #

:

ð25Þ

z1SW(xmin) represents the contribution of the SW part of the potential ofwidth xmin, while z1(γ)(xmin) corresponds to the attractive SU(γ) poten-tial, −ε(σ/r)γ, starting from xmim [1]. Note also that we always use theattractive Sutherland potentials with γ=α, β, introducing the appro-priate signs and coefficients in the various thermodynamic functions.

The MCA second-order term for the compressibility factor is thengiven by

z2 ¼ KHS

2z1

SW xminð Þ þ C2z12αð Þ xminð Þ þ C2z1

2βð Þ xminð Þ−2C2z1αþβð Þ xminð Þ

h i

þ ηKHS

∂KHS

∂η

!F2

NkBT;

ð26Þ

where

ηKHS

∂KHS

∂η ¼ −4η 2þ 5η−η2� �

1−ηð Þ5 KHS: ð27Þ

We note that the MCA second-order terms, F2/(NkBT) and z2,are conveniently expressed here only with the first-order termsf1SW(xmin), z1SW(xmin), f1

(γ)(xmin) and z1(γ)(xmin).

Finally, the residual internal energy, Ures=U−Uid, can be obtainedfrom

Ures

NkBT¼ F1

NkBT1T� þ 2

F2NkBT

1T�2 ð28Þ

or

Ures

Nε¼ Ures

NkBTT� ¼ U1

Nεþ U2

Nε1T� ¼ F1

NkBTþ 2

F2NkBT

1T� : ð29Þ

3. Radial distribution function at contact g(σ) for the HCMie(α, β)

The RDF at contact for the HCMie(α, β) can also be derived from thismethod. In the logarithmic expansion approximation (LEA) [11] it is givenby

g σð Þ ¼ gHS σð Þ exp g1 σð ÞgHS σð Þ

� �; gHS σð Þ ¼ 1−η=2

1−ηð Þ3 ; ð30Þ

where gHS(σ) is the CS–HS contact value and g1(σ) is the first-order termin the expansion in inverse powers of T* of g(r)=gHS(r)+g1(r) in whichthe factor 1/T* is incorporated into g1 for convenience (note that this g1 isdifferent from the base function g1 in Eq. (9)). One can obtain g1(σ) fromthe SAFT-VR theory as [25]

g1 σð Þ ¼ z14ηT� þ

1T�∫

∞1x3

du� xð Þdx

gHS xð Þdx: ð31Þ

The SW part of the potential, u*(x)=−1, does not contribute tothe above integral I, which is then given by

I ¼ −Cα∫∞xmin

gHS xð Þx2−αdxþ Cβ∫∞xmin

gHS xð Þx2−βdx: ð32Þ

In order to discover the physical interpretation of this equation,we note that the first-order contribution to the free energy of the at-tractive SU(γ) potential, −εx−γ, is given by [1]

f1γð Þ 1ð Þ ¼ −12η∫∞

1gHS xð Þx2−γdx; ð33Þ

where the argument of f1(γ) corresponds to the lower bound, x=1, of

the integral. Then each of the integrals of Eq. (32) is proportional tothe contributions of the range [xmin, ∞] to the first-order free energyof the SU potentials in the Mie tail. Hence they are given by

∫∞xmin

gHS xð Þx2−γdx ¼ − f1γð Þ xminð Þ12η

; γ ¼ α;β: ð34Þ

Introducing now Eqs. (34) and (32) into Eq. (31), we canwrite g1(σ)as

g1 σð Þ ¼ z14ηT�−

C12ηT� −αf1

αð Þ xminð Þ þ βf1βð Þ xminð Þ

h i: ð35Þ

This expression is coherent with the one given by Gil-Villegas et al.[25] for the simple SU(γ) potential, −εx−γ, which is

g1γð Þ σð Þ ¼ z1

γð Þ

4ηT�−γ

12ηT� f1γð Þ; ð36Þ

where z1(γ) and f1

(γ) are respectively the first-order terms of the com-pressibility factor and free energy of the attractive SU(γ) potential.We also note that in Eq. (35) the SW part of the HCMie potential istaken into account by the z1 term, which is obtained from the entireHCMie potential. Now combining Eqs. (16), (23)–(25), (30), and(35), we can calculate g1(σ) and g(σ) analytically.

Eq. (35) can also be used to calculate g1(σ) within the frameworkof the DLS EOS by using the parameterized expressions of the SU(γ)potentials given by [19]

f1γð Þ xminð Þ ¼ 1

2∫∞xmin

∂N0 x;ρ�ð Þ∂x −x−γ �

dx: ð37Þ

The function N0(x, ρ*), representing the coordination number inthe HS reference fluid, was determined from the fitting of the simula-tion data for bNiN0=N within the interval 1b×b3. Ni is the number ofintermolecular distances in the range (ri, ri+1) and ⟨⟩0 indicates anaverage performed with the reference HS fluid. It is of the form [19]

N0 x;ρ�ð Þ ¼ N0 xð Þρ� þ ΔN0 x;ρ�ð Þ; ð38Þ

N0 xð Þ ¼ 43π x3−1� �

; ð39Þ

ΔN0 x;ρ�ð Þ ¼ ∑4

i¼2∑8

j¼1aij x−1ð Þjρ�i; ð40Þ

Table 2Comparison with the MC simulations [18] of the residual internal energies −Ures/(Nε)and compressibility factors Z of the HCLJ(12, 6) fluid calculated in this work (TW) andfrom two other EOS's: Sowers and Sandler with HS reference SSHS [18] and second-order parameterizations of Diez, Largo and Solana with long range corrections DLS(∞) [19] at T*=1.35.

T* ρ* −Ures/Nε Z

MC TW SSHS DLS(∞) MC TW SSHS DLS(∞)

1.35 0.05 0.841 0.858 0.856 0.8540.10 0.848 0.762 0.704 0.771 0.716 0.729 0.728 0.7190.20 1.628 1.498 1.412 1.506 0.496 0.494 0.513 0.4840.30 2.344 2.239 2.136 2.233 0.353 0.298 0.350 0.3110.40 3.044 3.005 2.886 2.971 0.258 0.181 0.258 0.2330.50 3.758 3.799 3.669 3.734 0.299 0.215 0.292 0.3040.60 4.541 4.620 4.483 4.529 0.528 0.517 0.562 0.6030.70 5.378 5.457 5.324 5.362 1.206 1.256 1.238 1.2500.80 6.253 6.302 6.181 6.235 2.516 2.650 2.571 2.4360.90 7.122 7.151 7.042 7.152 4.996 4.929 4.931 4.476

AAD 0.071 0.120 0.054 0.051 0.024 0.082AAD% 3.26% 5.69% 2.85% 9.07% 2.27% 5.91%

166 H. Guérin / Journal of Molecular Liquids 164 (2011) 162–170

where the numerical coefficients aij are given in Ref. [19]. The first-order term z1 in Eq. (31) is then given by

z1 ¼ ρ� ∂∂ρ�

F1NkBT

� �; ð41Þ

where F1/(NkBT) is the first-order free energy of the HCMie(α, β)written here as [19]

F1NkBT

¼ 12∫∞1

∂N0 x;ρ�ð Þ∂x u� xð Þdx: ð42Þ

In this equation N0(x, ρ*) is given by Eqs. (38)–(40) and u*(x) rep-resents the HCMie(α, β) potential defined by Eq. (1) in reduced units.The second-order term, F2/(NkBT), used for the calculation of z2, Z andUres/(Nε) is given by a fifth order polynomial in ρ* which fits the cor-responding simulation data [19].

4. Thermodynamic properties of the Mie(α, β) fluids from the BHperturbation theory

Although the SAFT-VR theory of Gil-Villegas et al. [25] was derivedfor both the SW and SU potentials separately, it cannot be applied assuch to the combination of these potentials as they appear in theHCMie(α, β) of Eq. (1). However it was applied to the pure LJ(12,6)potential [26] and more generally to the Mie(α, β) potentials givenby Eq. (3) with βbα, α≤12, β≤6 [27, 30]; this is the SAFT-VR Mie(α, β) theory. As the formalism described above can be easily extend-ed to the general Mie(α, β) without any restriction on the values of αand β (except βbα and βN3,αN3), it is then of interest in this sectionto obtain the thermodynamic functions of the Mie(α, β) fluids, so thatthe results can be compared to the SAFT-VR Mie(α, β) theory wherepossible.

For the Mie(α, β) potentials given by Eq. (3), we introduce first atemperature-dependent hard core, whose diameter is given by theBH theory [6,11,25] as

σBH ¼ σ−∫σ

0exp −uM=kBTð Þdr; ð43Þ

where σ is the zero of the potential uM. A new packing fractionηBH=πρσBH

3 /6 is defined and the first- and second-order expressionsof the free energy are then given by

F1NkBT

¼ C −f1αð Þ xmin ¼ 1ð Þ þ f1

βð Þ xmin ¼ 1ð Þh i

; ð44Þ

F2NkBT

¼ 12C2KHS 1þ 2KηBH

2� �

f12αð Þ xmin ¼ 1ð Þ þ f1

2βð Þ xmin ¼ 1ð Þ−2f1αþβð Þ xmin ¼ 1ð Þ

h i;

ð45Þ

where f1(γ)(xmin=1) is given by Eq. (16) with xmin=1 and η=ηBH,since the SU potentials starts now at x=r/σBH=1 The factor 1+2KηBH

2 with K=(1/0.493)2 was introduced by Betancourt-Cárdenaset al. [33] in their recent derivation of a new equation of state for theLJ(12, 6) fluid to improve the MCA free energy second-order term assuggested initially by Zhang [34]. Their method to obtain this LJ(12,6) EOS is similar to ours except for the use of the more complicat-ed analytical expression for the RDF gHS(r) given by Largo and Solana[35] and Chang and Sandler [36]. As their results for the integrals inEqs. (6) and (7) were too complicated for practical use, they werefitted to power series in ηBH specifically designed for the LJ(12, 6) po-tential, while here the simplicity of the Sun expressions of gHS(r) givenby Eqs. (9) and (10) [1] allows to express these integrals in simple an-alytic forms for any α and βbα greater than 3. It should also be men-tioned here that Sun expressions of gHS(r) were used by Sun et al. [37]for the LJ(12, 6) potential in the context of the Ross variational theory[38] for which the effective diameter σBH is chosen so as to minimize

the Helmholtz free energy. However this method was not generalizedto the Mie(α, β) potentials, although this could be done without re-strictions on the values of α and β except the usual ones.

For the first and second-order terms of the compressibility factor,we obtain from Eq. (20) with η=ηBH

z1 ¼ C −z1αð Þ xmin ¼ 1ð Þ þ z1

βð Þ xmin ¼ 1ð Þh i

; ð46Þ

z2 ¼ 12C2KHS 1þ 2KηBH

2� �

z12αð Þ xmin ¼ 1ð Þ þ z1

2βð Þ xmin ¼ 1ð Þ−2z1αþβð Þ xmin ¼ 1ð Þ

h i

þ F2NkBT

ηBH

KHS

∂KHS

∂ηBHþ 4KηBH

2

1þ 2KηBH2

!;

ð47Þ

where z1(γ)(xmin=1) is given by Eq. (25) with xmin=1 and η=ηBH.

For the residual internal energy, Eq. (28) is modified to take intoaccount the derivative with respect to T* of σBH(T*). It is given by

Ures

NkBT¼ F1

NkBT1T� þ 2

F2NkBT

1T�2 −3

1σBH

∂σBH

∂T�� �

4ηBHT�gHS σBHð Þ þ z1 þ

z2T�

h i;

ð48Þ

where gHS(σBH) is the RDF at contact given by Eq. (30) with η=ηBH,σBH is numerically calculated from Eq. (43), and ∂σBH/∂T* from

∂σBH

∂T ¼ −∫σ

0

uM rð ÞkBT

2 exp −uM rð ÞkBT

� �dr; ð49Þ

with T*=kBT/ε, as explained in detail by Lafitte et al. [27] in theirSAFT-VR Mie theory. We note here that for the LJ(12,6) potential con-venient and accurate analytical expressions for both quantities weredevised by Cotterman et al. [39] and Sun [40]. We shall not usethem here to treat all the Mie(α, β) potentials including the LJ(12, 6)on the same footing.

5. Numerical results

The accuracy of the above formulae giving the values of the resid-ual internal energies Ures/(Nε) and compressibility factors Z are firsttested in Tables 2–7 and Figs. 2 and 3 for the HCLJ(12, 6) fluid againstthe MC simulation data of Sowers and Sandler [18] and the resultsobtained from the five EOS briefly described above. Each table corre-sponds respectively to the temperatures T*=1.35, 1.75, 2.00, 3.00,4.00 and 6.00 with ρ* varying from 0.05 to 0.9. They indicate a) theMC simulations results, b) the values from the above equations de-rived in this work (TW), c) the values from the SSHS EOS [18],

Table 3Same as Table 2 for T*=1.75.

T* ρ* −Ures/Nε Z

MC TW SSHS DLS(∞) MC TW SSHS DLS(∞)

1.75 0.05 0.912 0.921 0.916 0.9180.10 0.765 0.727 0.696 0.737 0.855 0.852 0.847 0.8440.20 1.498 1.447 1.397 1.461 0.752 0.741 0.753 0.7300.30 2.212 2.185 2.115 2.191 0.711 0.680 0.723 0.6830.40 2.941 2.952 2.862 2.938 0.749 0.710 0.776 0.7440.50 3.711 3.752 3.645 3.711 0.993 0.902 0.969 0.9670.60 4.513 4.580 4.463 4.515 1.364 1.365 1.406 1.4320.70 5.366 5.425 5.309 5.352 2.288 2.259 2.250 2.2620.80 6.240 6.277 6.173 6.228 3.771 3.800 3.741 3.6480.90 7.115 7.132 7.042 7.145 6.213 6.243 6.246 5.909

AAD 0.039 0.073 0.016 0.027 0.022 0.062AAD% 1.61% 3.21% 0.90% 2.41% 1.53% 2.64%

Table 4Same as Table 2 for T*=2.00.

T* ρ* −Ures/Nε Z

MC TW SSHS DLS(∞) MC TW SSHS DLS(∞)

2.00 0.05 0.939 0.947 0.940 0.9440.10 0.737 0.713 0.693 0.722 0.904 0.903 0.896 0.8960.20 1.459 1.426 1.391 1.442 0.845 0.844 0.854 0.8330.30 2.172 2.162 2.107 2.173 0.854 0.841 0.879 0.8420.40 2.909 2.930 2.853 2.924 0.975 0.934 0.994 0.9610.50 3.680 3.732 3.635 3.701 1.229 1.194 1.255 1.2490.60 4.504 4.563 4.454 4.508 1.773 1.725 1.764 1.7840.70 5.360 5.411 5.303 5.348 2.760 2.684 2.680 2.6900.80 6.237 6.266 6.170 6.225 4.290 4.288 4.238 4.1600.90 7.112 7.125 7.042 7.142 6.995 6.800 6.804 6.514

AAD 0.032 0.058 0.014 0.042 0.042 0.076AAD% 1.22% 2.33% 0.58% 1.80% 1.64% 2.04%

Table 6Same as Table 2 for T*=4.00.

T* ρ* −Ures/Nε Z

MC TW SSHS DLS(∞) MC TW SSHS DLS(∞)

4.00 0.05 1.027 1.033 1.026 1.0310.10 0.662 0.661 0.683 0.672 1.079 1.076 1.068 1.0710.20 1.349 1.351 1.372 1.376 1.207 1.202 1.204 1.1940.30 2.070 2.081 2.080 2.111 1.415 1.404 1.424 1.4000.40 2.826 2.852 2.822 2.876 1.719 1.723 1.756 1.7320.50 3.636 3.663 3.604 3.668 2.233 2.222 2.259 2.2470.60 4.474 4.504 4.428 4.487 3.022 2.996 3.023 3.0260.70 5.342 5.364 5.284 5.334 4.172 4.188 4.193 4.1960.80 6.222 6.229 6.160 6.214 6.124 6.008 5.991 5.9530.90 7.099 7.098 7.042 7.132 8.813 8.763 8.772 8.630

AAD 0.014 0.035 0.025 0.025 0.028 0.045AAD% 0.41% 1.14% 1.02% 0.65% 0.85% 1.02%

Table 7Same as Table 2 for T*=6.00.

T* ρ* −Ures/Nε Z

MC TW SSHS DLS(∞) MC TW SSHS DLS(∞)

6.00 0.05 1.054 1.060 1.055 1.0590.10 0.642 0.644 0.681 0.655 1.133 1.131 1.125 1.1280.20 1.321 1.327 1.368 1.354 1.322 1.320 1.320 1.3140.30 2.043 2.054 2.074 2.091 1.598 1.592 1.604 1.5880.40 2.810 2.826 2.815 2.859 1.990 1.987 2.010 1.9930.50 3.621 3.639 3.597 3.656 2.576 2.567 2.593 2.5830.60 4.467 4.484 4.421 4.479 3.447 3.423 3.444 3.4440.70 5.339 5.348 5.280 5.330 4.736 4.693 4.700 4.6990.80 6.222 6.217 6.158 6.210 6.648 6.586 6.580 6.5510.90 7.100 7.089 7.042 7.129 9.392 9.421 9.432 9.336

AAD 0.011 0.041 0.027 0.019 0.020 0.023AAD% 0.35% 1.77% 1.18% 0.46% 0.53% 0.55%

10

167H. Guérin / Journal of Molecular Liquids 164 (2011) 162–170

which have not been published before, and finally d) the values fromthe second-order DLS EOS, which were only given graphically [19].The values obtained for the same state points from the SWLE EOSare given in Ref. [14], those obtained from the ETW are given in Ref.[15], while the SSSW values are displayed in Refs. [15,18], hence wedo not reproduce them here for brevity. In this way the AAD (absoluteaverage deviations), both in absolute value and percentage, calculat-ed from our EOS can easily be compared in Table 8 to the AAD corre-sponding to the five other EOS.

Concerning the calculation of F1/(NkBT) in the DLS EOS fromEq. (42) for the HCLJ(12, 6) potential, some further comments arenecessary. The integration in Eq. (42) was performed in Ref. [19]only up to x=3 because the simulation data used to fit N0(x) corre-spond to the interval 1b×b3. However, the final results obtainedfrom for Ures/(Nε) and Z are compared to the MC simulation results

Table 5Same as Table 2 for T*=3.00.

T* ρ* −Ures/Nε Z

MC TW SSHS DLS(∞) MC TW SSHS DLS(∞)

3.00 0.05 0.999 1.005 0.998 1.0030.10 0.685 0.678 0.686 0.689 1.022 1.019 1.011 1.0140.20 1.380 1.376 1.378 1.398 1.090 1.083 1.088 1.0740.30 2.097 2.108 2.088 2.132 1.223 1.216 1.243 1.2130.40 2.852 2.878 2.831 2.892 1.470 1.459 1.502 1.4740.50 3.648 3.686 3.614 3.679 1.883 1.878 1.924 1.9120.60 4.482 4.523 4.436 4.494 2.554 2.570 2.602 2.6110.70 5.352 5.380 5.290 5.339 3.667 3.684 3.687 3.6930.80 6.234 6.241 6.163 6.217 5.405 5.432 5.405 5.3550.90 7.105 7.107 7.042 7.135 8.193 8.106 8.114 7.925

AAD 0.018 0.034 0.022 0.019 0.025 0.047AAD% 0.60% 0.73% 0.78% 0.58% 1.07% 1.24%

of Sowers and Sandler [18], which include long range corrections,i.e. the corrections due to the cut-off of the potentials. For the SU(γ)potential, −εx−γ, this correction is given by [41]

ΔUres

Nε¼ 2πρ� 3

3−γ

3−γ; ð50Þ

which is found very small for the SU(γ) potential [41], but for the SUpotentials occurring in the HCMie(α, β): +Cεx−α and −Cεx−β, thecorrections are no longer completely negligible, since they are

0

2

4

6

8

0 0,2 0,4 0,6 0,8 1ρ∗

Ζ

Fig. 2. Compressibility factor Z for the HCLJ(12, 6) fluid. Points: MC simulation datafrom Ref. [18] for T*=1.35 (♦); T=2 (▲); T*=3 (□) and T*=6 (○). The continuouslines are the results from this work.

0

2

4

6

8

10

12

0 0,2 0,4 0,6 0,8 1ρ∗

U*

Fig. 3. Same as Fig. 2 for the residual internal energy U*=−Ures/Nε of the HCLJ(12, 6)fluid. For clarity, each curve and the corresponding simulation data have been shiftedupwards by a unity with respect to those immediately below.

Table 9Values of the HCLJ(12, 6) RDF at contact g(σ) from Eqs. (30) and (35) calculated in thiswork (TW) and the DLS(∞) EOS with long-range corrections, and comparison to theMC simulations data of Stell and Weis [16].

ρ* T* MC TW DLS(∞)

0.10 1.20 2.490 2.078 2.0450.10 1.35 2.181 1.944 1.9170.20 1.60 1.94 1.775 1.7750.60 1.60 2.57±0.02 2.500 2.5790.85 0.85 3.94±0.04 3.869 3.7250.90 0.75 4.44±0.04 4.246 4.0260.91 1.35 4.90±0.04 4.749 4.6100.91 2.74 5.15±0.05 5.025 4.952AAD 0.18 0.25

Table 10Comparison with the molecular dynamics (MD) simulations [30] of the residual inter-nal energies −Ures/(Nε) and reduced pressure P* of the Mie(8, 6) fluid calculated inthis work (TW) and from the SAFT-VR Mie (8, 6) theory.

T* ρ* −Ures/Nε P*

168 H. Guérin / Journal of Molecular Liquids 164 (2011) 162–170

multiplied by the factor C. These corrections can be introduced inEq. (42) by integrating from 3 up to infinity the term (1/2)4πρ*x2u*(x), which comes from Eqs. (38) and (39). This gives for βN3 andαNβ

124πCρ�∫∞

3x2 x−α−x−β� �

dx ¼ 2πCρ� − 33−α

3−αþ 33−β

3−β

!: ð51Þ

This term corresponds to the sum of the corrections given byEq. (50) for the HCMie SU potentials, hence to include in the DLS for-malism the correction due to the cut-off of the potential at x=3, wejust have to integrate from x=xmin up to infinity the term (1/2)4πρ*x2u*(x), and up to x=3 the term given by Eq. (40) containingthe fitted numerical coefficients aij. The values labeled DLS(∞)reported in Tables 2–7, include this integration up to infinity andthe results obtained are in closer agreement with the Sowers andSandler MC simulation data [18] than those obtained by integratingonly up to x=3.

Close inspection of these tables and in particular Table 8 revealsthat overall the best EOS for Z is the SSHS followed closely by theDLS(∞), the SSSW and our EOS developed in this work (TW), whilefor U the best one is the DLS(∞), followed closely by ours (TW) andSSHS. However, Tables 2–7 show that at higher temperatures(T*≥3) our EOS (TW) becomes the best one for both U and Z. Thisis due to the fact that at lower temperatures in a region slightly intothe two-phase region (for example T*=1.35, ρ*=0.4) Z is smalland a slight deviation entails a large AAD expressed in %, while forthe DLS(∞) and SSHS obtained from fitting the MC results this isavoided. For the DLS(∞) EOS the results are too low at high densitiesfor Z, which increases the corresponding AAD. Finally the EOSobtained by adjusting potential shapes to the HCLJ function (ESW

Table 8Total absolute average deviations for the compressibility factors Z and internal energyUres/(Nε), AAD(Z) and AAD(Ures), for the HCLJ(12, 6) EOS developed in this work (TW)and five other EOS from the MC simulation data in Tables 2–7. DLS(∞): second-orderDiez, Largo and Solana parameterizations with long range corrections (see text) [19];SSHS: Sowers and Sandler EOS with a HS reference [18]; SSSW: Sowers and SandlerEOS with a SW reference [15,18]; SWLE: square-well linear extension [14]; ESW: ex-tended square-well [15].

EOS TW DLS(∞) SSHS SSSW SWLE ESW

AAD(Z) 0.030 0.056 0.027 0.047 0.053 0.1622.49% 2.23% 1.31% 2.35% 6.05% 9.99%

AAD(Ures) 0.031 0.026 0.060 0.159 0.173 0.4681.24% 1.22% 2.48% 5.78% 5.42% 10.68%

and SWLE) are less accurate. Of course our EOS can be extended toother values of α and β.

The RDF at contact g(σ) is calculated for the HCLJ(12, 6) fromEqs. (30) and (35), where z1, f1(α)(xmin) and f1(β)(xmin) are takenfrom our EOS (TW) with Eqs. (16), (23)–(25). For the DLS(∞) methodEqs. (37)–(42) are used, in which the integration up to infinity is alsoapplied to the SU(α) and SU(β) potentials. The values thus obtainedare compared to the Stell and Weis MC simulation data [16] inTable 9, where it can be seen that our EOS (TW) is slightly moreaccurate.

Secondly, the reduced pressure P*=Pσ3/ε=Zρ*T* (σ defines theposition where uM changes sign) and residual internal energy,−Ures/Nε, for the Mie(8, 6) fluids calculated from Eqs. (43)–(49)are compared in Table 10 to the MD simulation results of Galliéro etal. [30] and those of the SAFT-VR Mie(8, 6) theory recalculated herewith the same ρ*=ρσ3 for 12 state points corresponding to four dif-ferent isotherms very close to Tr=T*/Tc*=0.65, 0.75, 0.85, and 1.5,and three isobars very close to Pr.=P*/Pc*=1.98, 13.86, and 29.7where Tc* and Pc* are respectively the critical reduced temperatureand pressure as explained in Ref. [30]. The first nine states (fromtop down in Table 10) are subcritical (liquid), while the last threestates are supercritical. As can be seen from Table 10, our EOS pro-vides an overall good representation of the internal energy and pres-sure, which are of better accuracy than the SAFT-VR Mie(8, 6) values,except that our results for P* are too low for state points with highdensities and low temperatures. This was also observed for theBetancourt-Cárdenas et al. EOS of the LJ(12,6) [33], which is similarto ours as explained above. For the Mie(10, 6) and Mie(12, 6) fluidsat the same state points defined as above, we only indicate here(Table 11) for brevity the AAD obtained by our method, the SAFT-VR theory, and the Betancourt-Cárdenas et al. EOS [33] of the LJ(12,6). It can be noted that the accuracy of the SAFT-VR theory increases

MD TW SAFTVR MD TW SAFTVR

1.184 0.7831 6.854 6.751 7.276 0.356 0.499 0.5941.184 0.8884 7.459 7.344 7.810 2.358 2.297 2.5431.176 0.9711 7.739 7.717 8.089 5.040 4.577 5.2701.359 0.7272 6.323 6.220 6.699 0.332 0.507 0.5591.364 0.8545 7.103 6.980 7.410 2.366 2.410 2.5451.358 0.9443 7.438 7.386 7.750 5.058 4.862 5.2611.549 0.6652 5.746 5.625 6.060 0.345 0.541 0.5671.541 0.8213 6.764 6.629 7.028 2.356 2.462 2.5411.544 0.9118 7.166 7.073 7.422 5.081 5.054 5.2872.721 0.1479 1.267 1.138 1.267 0.333 0.364 0.3562.722 0.6228 4.845 4.592 4.846 2.349 2.612 2.6152.726 0.7731 5.648 5.395 5.638 5.046 5.351 5.364AAD 0.125 0.247 0.168 0.206

Table 11Absolute average deviation (AAD) from the molecular dynamics (MD) simulation datafor −Ures/Nε and P* obtained in this work (TW), the SAFT-VR Mie(α, 6) theory forα=8, 10 and 12 and the Betancourt-Cárdenas et al. EOS [33] (BEOS) valid only forthe LJ(12, 6). The twelve state points defined in Ref. [30] and Table 10 are used.

Mie AAD(Ures) AAD(P*)

TW SAFT-VR BEOS TW SAFT-VR BEOS

(8,6) 0.125 0.247 0.168 0.206(10,6) 0.099 0.153 0.176 0.201(12,6) 0.103 0.042 0.069 0.178 0.109 0.207

169H. Guérin / Journal of Molecular Liquids 164 (2011) 162–170

with α, while it stays about the same for our method, and for α=12the SAFT-VR theory is the most accurate.

Finally, for the Mie(14,7) fluid, which is outside the range of theSAFT-VR Mie theory, the compressibility factor Z is compared inFig. 4 to the results obtained from an accurate analytical EOS fittedto the MC simulation data [31] for T*=0.7, 1, 2, 4, and 6. In particularT*=1 is very close to the critical isotherm, since for the Mie(14, 7)fluid the critical point is at: Tc*=0.9941, ρc*=0.3278, Pc*=0.090and Zc=0.276 [31]. It can be seen that the results from the presentEOS are in good agreement with those obtained from this analyticEOS with a total AAD=0.060 for the 39 state points shown in Fig. 4.Again the pressure is too low for high densities and low temperatureswhere the perturbation theory itself becomes as expected lessaccurate.

In conclusion, by combining the second-order BH perturbationtheory in the macroscopic compressibility approximation (MCA)with the simple and analytic form of the HS-RDF, gHS(r), proposedby Sun [1] in Eqs. (9) and (10), we obtain accurate analytic EOS forfluids, whose potentials combine the SW and SU(γ=α, β) potentialssuch as the HCMie(α, β) and Mie(α, β) potentials. This was alreadyshown in the application of this method to the SW chain fluids [13],where it was noted that it is more accurate than the SAFT-VR theory.The numerical coefficients Amn defining the HS-RDF are unique, whilefor the SAFT-VR the coefficients defining the parameterization of theeffective packing fraction are different for each type of potential inter-action used [25]. The results are in overall good agreement with thesimulation data and better than those obtained from some other

0

1

2

3

4

5

6

7

0 0,2 0,4 0,6 0,8 1ρ∗

Ζ

Fig. 4. Compressibility factor Z for the Mie(14, 7) fluid. Points: analytic EOS fitted onthe MC simulation data of Ref. [31] for T*=0.7 (*); T*=1 (♦); T*=2 (▲); T*=4 (□)and T*=6 (○) and the critical point: Tc*=0.9941, ρc*=0.3278, Zc=0.276 (•) from[31]. The continuous lines are the results from this work. For clarity, the curve andthe corresponding data points for T*=6 (○) have been shifted upward by a unitywith respect to those immediately below (T*=4).

analytic EOS. In particular the present EOS are applicable to a largerrange of potentials than other EOS and in particular than the SAFT-VR theory, which cannot be applied to the Mie(α,β) for βN6 andαN12 and the HCMie(α, β). This is clearly of interest since the accu-rate modeling of the repulsion (parameter α) and the van der Waalsattractions (parameter β) between the monomer segments is crucialfor an extension of the SAFT-VR theory of chain fluids to potentialforms with a variable repulsive (and/or attractive) interactions.

This interplay between the repulsive and attractive parts of thepotential is also found for the hard-core n-Yukawa (HCnY) chainfluids composed of m spherical freely-jointed tangent monomersat the hard core, for which the SAFT-VR theory was successfullyused for n=2 and m=2, 4, 8, 16 in conjunction with the full meanspherical approximation (MSA) for multi-Yukawa interactions byKalyuzhnyi et al. [42]. However, as the full MSA entails rather com-plex expressions for the monomer thermodynamic and structuralproperties, which requires the solution of a set of n non-linear alge-braic equations, it was further simplified with a high-temperature ex-pansion (HTE) [43–46]. This is the first-order mean sphericalapproximation (FMSA) (also called sometimes MSA-HTE), whichwas applied to monomers (m=1) with n=1 [43], n=2 [44] andn=1, 2, 3, 4 [45], as well as to HCnY chain fluids for n=1 withm=2, 4, 16 [43], n=2 with m=2, 4 [44] and m=2, 4, 8, 16 [46]with two different HC2Y potentials. The SAFT-VR theory combinedto the FMSA instead of the full MSA is then completely analytical inthe sense that it bypasses the numerical solution of the set of n alge-braic equations mentioned above and this leads to simplified, tracta-ble and accurate expressions of the thermodynamic properties ofHCnY chain fluids. Hence, depending on the system considered, onecan either use the HC2Y potential, which combines a repulsive andan attractive Yukawa tail or the Mie(α, β) potential to model analyt-ically the variable attractive and repulsive parts of the monomerintermolecular potential. On the 20th anniversary of the SAFT theory,Vega and Jackson considered this flexible modeling as a potentialhope for the development of the theory in their presentation of its ad-vances and challenges [47].

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