Group contribution methodology based on the statistical associating fluidtheory for heteronuclear molecules formed from Mie segments

Vasileios Papaioannou,1 Thomas Lafitte,1 Carlos Avendano,1, 2 Claire S. Adjiman,1 George Jackson,1 Erich A.Muller,1 and Amparo Galindo1, a)1)Department of Chemical Engineering, Centre for Process Systems Engineering, Imperial College London,South Kensington Campus, London SW7 2AZ, United Kingdom2)School of Chemical Engineering and Analytical Sciences, University of Manchester, Sackville Street,Manchester M13 9PL, United Kingdom

(Dated: 2 December 2013)

A generalization of the recent statistical associating fluid theory for variable range Mie potentials [Lafitte et al.,J. Chem. Phys., 139, 154504, (2013)] is formulated within the framework of a group contribution approach(SAFT-γ Mie). Molecules are modeled as comprising distinct functional (chemical) groups based on a fusedheteronuclear molecular model, where the interactions between segments are described by the Mie (generalizedLennard-Jonesium) potential of variable attractive and repulsive range. A key feature of the presented theoryis the accurate description of the monomeric group-group interactions by application of a high-temperatureperturbation expansion up to third order. The capabilities of the SAFT-γ Mie approach are exemplifiedby studying the thermodynamic properties of two chemical families, the n-alkanes and the n-alkyl esters,by developing parameters for the methyl, methylene, and carboxylate functional groups (CH3, CH2, andCOO). The approach is shown to describe accurately the fluid phase behavior of the compounds consideredwith absolute average deviations of 1.20% and 0.42% for the vapor pressure and saturated liquid density,respectively, which shows a clear improvement over other existing SAFT-based group contribution approaches.The use of Mie potentials to describe the group-group interaction is shown to allow accurate simultaneousdescriptions of the fluid-phase behavior and second-order thermodynamic derivative properties of pure fluidsbased on a single set of group parameters. Furthermore, the application of a perturbation expansion tothird order for the description of the reference monomeric fluid improves the predictions of the theory forthe fluid-phase behavior of pure components in the near-critical region. The predictive capabilities of theapproach stem from its formulation within a group-contribution formalism: predictions of the fluid-phasebehavior and thermodynamic derivative properties of compounds not included in the development of groupparameters are demonstrated. The performance of the theory is also critically assessed with predictions ofthe fluid-phase behavior (vapor-liquid and liquid-liquid equilibria) and excess thermodynamic properties ofa variety of binary mixtures, including polymer solutions, where very good agreement with the experimentaldata is seen, without the need for adjustable mixture parameters.

Keywords: SAFT; Group Contribution Methods; Heteronuclear Models; Mie Potential; Derivative Properties

I. INTRODUCTION

Thermodynamic tools are being continually developedand improved in order to meet the need for accurateproperty prediction in many sectors of the chemical in-dustries. Predictive approaches have come to play animportant role in the design of processes and the ac-curacy of their output can significantly affect process-design decisions1,2. Despite the wide variety of methodsthat are available, industrial requirements in propertyprediction highlight the need for approaches that can beapplied in an ever-expanding range of applications, in-cluding, among others, polymer processing, biotechnol-ogy, and solvent screening3. Advances in thermodynamicmodeling have led to novel applications, such as the in-tegrated design of solvents and processes, where molec-ular characteristics of solvents are determined as part of

a)Author to whom correspondence should be addressed. Email:a.galindo@imperial.ac.uk

the optimization of the process4–7. An important aspectin the development of thermodynamic methodologies istheir predictive capability, which is commonly perceivedas the capability to provide predictions of phase behav-ior and other bulk properties without the need for ex-perimental data for the determination of the molecularmodel parameters.

An important class of thermodynamic methodologieswith high predictive power is that of group contribu-tion (GC) methods. GC methods are developed basedon the assumption that the properties of a given com-pound can be calculated as appropriate functions of thechemically distinct functional groups that the compoundcomprises. The contribution of each functional groupto the molecular properties is assumed to be indepen-dent of the molecular structure that the group appearsin. Such approaches have a long history of researchand application with the first developments focused onthe prediction of pure-component properties, such as theGC methods of Lydersen8, Joback and Reid9, and thelater works of Gani and co-workers10,11, to name but afew. Furthermore, GC approaches have found extensive

2

application to the study of the thermodynamic proper-ties of binary and multi-component mixtures, often withemphasis on fluid-phase behavior, initially within theframework of activity coefficient methods. An exampleof this type of methodology is the well-established uni-versal quasi-chemical functional group activity coefficient(UNIFAC) formalism12. The UNIFAC approach and itssubsequent modifications13 are widely considered to bethe state-of-the-art predictive methodologies for indus-trial applications due to their accuracy in predicting thephase behavior of a wide range of mixtures and the avail-ability of extensive parameter tables. Alongside UNI-FAC other predictive activity coefficient methods havebeen developed, a prominent example of which is thecomputational-chemistry based conductor-like screeningmodel for real solvents (COSMO-RS)14. Such methodsare however subject to a number of limitations: in theseapproaches one resorts to a treatment of the liquid phaseonly, neglecting pressure effects and only the subset ofthermodynamic properties that can be calculated fromactivity coefficients are obtained. With the aim of over-coming these limitations, GC methods have been cou-pled with equations of state (EoSs); prominent examplesinclude the PSRK EoS15, the GC16 and GCA17 EoSs,and more recently the VTPR EoS18–20. Equations ofstate have the advantage of treating the vapor and liq-uid phases on an equal footing, can account for pressureeffects, and can be applied to the study of a wide rangeof thermodynamic properties. Of particular relevance tothe work presented here are the applications of GC withinthe framework of the statistical associating fluid theory(SAFT)21,22. These are reviewed briefly in the followingdiscussion; for a more thorough discussion of the differ-ent applications of GC methods, the reader is redirectedto the reviews of references [23] and [24].

The various versions of SAFT EoS have been suc-cessfully applied to the study of the properties of ther-modynamically challenging systems, such as polymeric,hydrogen-bonding, and reactive systems. Details ofthe underlying theory and the different applications ofSAFT can be found in a number of comprehensive re-views25–28. The early work dedicated to developingSAFT-based GC approaches involved the application ofthe group contribution concept at the level of determin-ing the molecular parameters that describe a compound.In these homonuclear approaches the molecular modelemployed was the traditional SAFT representation ofmolecular chains formed from identical bonded segments,with all segments described by the same set of parame-ters, so that the underlying theory remained unchanged.Lora et al.29 were one of the first to combine a groupcontribution method with the original SAFT EoS21,22,determining the molecular parameters that describe sev-eral poly(acrylates) by examining compounds of lowermolecular weight. Subsequently, Vijande et al.30 devel-oped a SAFT-based GC approach with an emphasis onthe prediction of the properties of hydrofluoroethers. Inthe same spirit, Tamouza et al.31,32 have developed GC

methods based on SAFT approaches which employ bothfixed and variable-range interaction potentials for thestudy of the chemical families of the n-alkanes, alkylben-zenes, olefins, and alcohols. The methodology was thenextended to explicitly account for interactions due to po-larity based on the treatment of Gubbins and Twu33 forthe study of light and heavy esters34, and has since beenapplied to a wide range of systems including polycyclichydrocarbons35, ethers, aldehydes, ketones36, amines37,and long-chain multifunctional molecules including alka-nediols and alkanolamines38. Alongside the aforemen-tioned SAFT-based group contribution methods, othertechniques have been developed where molecular prop-erties are not calculated as functions of the occurrencesof functional groups alone, but also account for the posi-tioning of the groups in the molecule by defining super-structures. In the context of GC methods this invari-ably involves a consideration of second-order groups (asin the work of Constantinou and Gani10). The aim ofsuch approaches is to account for proximity effects (i.e.,how the presence of a functional group influences thephysicochemical properties of a neighboring group) andto distinguish the representation of isomers39–41.

As has been mentioned earlier, homonuclear SAFT-based GC can be used successfully to describe the ther-modynamic properties of a wide range of systems. Never-theless, the predictive capabilities of these methodologiesare typically limited to the prediction of the properties ofpure compounds as no information on the nature of theunlike interactions between groups and/or molecules canbe extracted from the pure-component data that is usedin the characterization of the functional groups. The de-termination of the unlike interaction parameters, whichcan also be calculated in a GC fashion as described byLe Thi et al.42, typically relies on the use of experimen-tal data for mixtures. Methods based on London’s the-ory can be used to predict the values of the unlike in-teraction parameters from the physicochemical proper-ties of the molecules within the SAFT formalism43: thistype of approach has been applied to mixtures of CO2,methane, and ethane with n-alkanes44,45, methanol withn-alkanes46, and aqueous solutions of alkanes, aromatichydrocarbons, and alcohols36.

Group contribution approaches have also been devel-oped within the framework of SAFT incorporating het-eronuclear molecular models (and the corresponding the-ory). In heteronuclear approaches molecules are rep-resented as chains of segments that are not necessarilyidentical. Within the Wertheim TPT1 methodology47–53

it is possible to formally develop, on the premise of adetailed heteronuclear molecular model, an equation ofstate maintaining a link between the nature of the dif-ferent segments making up the molecules and the corre-sponding parameters. The first such studies of bondedheteronuclear models formed from hard-sphere segmentsincluded heteronuclear dimers54, linear triatomic55, andarbitrary polyatomic molecules56,57. Within the contextof a SAFT treatment, several variants have been reformu-

3

lated based on a heteronuclear tangent molecular model.Examples include the works of Banaszak et al.58,59, Adid-harma and Radosz60,61, and McCabe and co-workers62,63

who implement a square-well (SW) potential to de-scribe the group-group interactions, the work of Blasand Vega64 based on molecules formed from Lennard-Jones (LJ) segments, and the work of Gross et al.65. Aswell as describing molecules with heteronuclear models,it is apparent that a model with fused (as opposed totangent) segments is needed to provide accurate ther-modynamic properties of fluids66,67. Fused models aretreated within the SAFT-γ GC approach by represent-ing the heteronuclear molecules as fused SW segments ofdifferent type with a shape factor to characterize the de-gree of overlap between the different groups. The methodwas shown be accurate in the description of the phasebehavior of a wide range of pure components includ-ing the chemical families of n-alkanes, branched alka-nes, alkylbenzenes, olefins, carboxylic acids, alcohols, ke-tones, and amines66,67, and was used to study aqueoussolutions of alkanes and alcohols68, and systems of ionicliquids69. Peng et al.70 have presented a closely relatedheteronuclear generalization of the SAFT-VR EoS basedon the use of SW potentials. As discussed in a recentreview28, the two theories, which were developed inde-pendently, are essentially identical and the only distin-guishing feature being the treatment of the contributiondue to the formation of chain-like molecules from distinctfunctional groups. Within the SAFT-γ EoS the contri-bution to the free energy due to the formation of theheteronuclear chain molecules is calculated based on amolecular average of the contact of the radial distribu-tion function; in the case of the GC-SAFT-VR EoS ofPeng et al. the contact values of the different monomericsegments are appropriately summed, in an attempt toretain the connectivity of the distinct groups that formthe molecule. As shown in Appendix A both approachesprovide an essentially equivalent description of the ther-modynamic properties of heteronuclear diatomic and tri-atomic molecules. We should also mention that a similartreatment to that of Peng et al. has been followed inthe recent work of Paduszynski and Domanska71. A keyadvantage of heteronuclear molecular approaches is thatthey can be used to predict the thermodynamic proper-ties of mixtures based on unlike interaction parametersobtained from pure component data alone.

In the context of group contribution approaches basedon heteronuclear molecular models only potentials com-prising hard-repulsive cores (e.g., hard-sphere or square-well potentials) or of fixed form (e.g., Lennard-Jones po-tential) have been considered thus far to describe the in-teractions between the molecular segments. Both the SWand the LJ form of interactions provide a successful rep-resentation of the fluid-phase equilibria of a wide varietyof systems, but are known to have limitations. Potentialswith infinitely repulsive cores are limited in the range ofproperties that can be accurately described (e.g., in thesimultaneous description of the fluid-phase behavior and

thermodynamic derivative properties), due to the simpli-fied representation of intermolecular forces72,73. On theother hand, potentials of fixed form are limited to thetype of compounds that can be accurately represented.As an example, the LJ potential cannot be applied to de-scribe the intermolecular interactions of perfluorocarbonswhich are known to be characterized by much steeper re-pulsive interactions than what can be represented withthe LJ form74. The development of a SAFT-based groupcontribution approach that incorporates the versatile Mieintermolecular potential of variable attractive and repul-sive range is a key objective of the current work. Wewill show that the resulting SAFT-γ Mie methodologyconstitutes a thermodynamic approach with a high pre-dictive capability that can be applied to a wide varietyof systems of different chemical nature providing an ac-curate description of thermodynamic properties, rang-ing from fluid-phase behavior to second-order derivativeproperties, such as the speed of sound, heat capacity, orisothermal compressibility.

Over the past decade there has been increased interestin the performance of EoSs in the description of second-order derivative properties. These derivative propertiesare of interest from a practical perspective (e.g., theJoule-Thomson inversion curve in the Linde technique asa standard process in the petrochemical industry), anda precise description of such properties, for which theperformance of traditional cubic EoSs has been shownto be relatively poor75–78, is highly challenging from atheoretical perspective75. An accurate representation ofderivative properties is a stringent test even with a moresophisticated thermodynamic treatment such as SAFT.One of the first detailed studies of the performance ofSAFT-type methods in the description of such proper-ties was carried out by Colina et al.79. These authorsexamined the Joule-Thomson inversion curve obtainedwith the LJ-based soft-SAFT EoS64,80 and found thatalthough good agreement could be achieved, the descrip-tion was very sensitive to the values of the molecularparameters. In a subsequent study different versions ofthe SAFT EoS were compared, and deviations of up to20% from the experimental values of the derivative prop-erties were found81. Llovell et al.82 have also presenteda detailed analysis on the performance of the soft-SAFTEoS for the derivative properties of pure components andmixtures83 concluding that the accuracy of the descrip-tion can range from 1 to 20% depending on the compoundand the property of interest, with the highest deviationstypically obtained for the speed of sound of long chainn-alkanes. In other work on smaller molecules whereSW or LJ forms of the potential are implemented, thereported errors for the description of the derivative prop-erties is found to be within 10%84. A number of stud-ies has been carried out with the aim of improving thedescription of the derivative properties76,77,85–87. How-ever, the simultaneous description of fluid-phase behav-ior and derivative properties is found to come at a cost:Maghari et al.86,87 have employed an empirical temper-

4

ature dependence of the dispersion energy by means ofa compound-specific constant, while Kontogeorgis andco-workers77 re-correlated the universal constants of thePC-SAFT EoS to include derivative property data.

Derivative properties have also been extensively stud-ied with renormalization-group crossover approaches astheir singular behavior at the critical point makes themespecially interesting. Examples of such studies that re-late to the general SAFT framework include the worksof Llovell et al.88, Dias et al.89, Vilaseca et al.90, andForte et al.91. Other crossover methods have also beenapplied to the study of derivative properties, such as thecrossover-cubic EoS92, and the crossover lattice equationof state93.

An aim our current work is to demonstrate the impactof the intermolecular potential model that is employedwithin a SAFT GC treatment on the accuracy that canbe achieved in the simultaneous description of phase be-havior and second-order derivative thermodynamic prop-erties. It is important to note that SAFT approachesbased on segment-segment interactions treated with Miepotentials of variable attractive and repulsive ranges havebeen found to provide such a capability72,94.

An improved version of the SAFT-VR Mie EoS hasbeen recently developed73, where several approximationsinherent in the underlying theory of the EoS presentedin 200672,95 are revisited and the perturbation expansion(in the description of the free energy due to monomericinteractions) has been taken to third order to incorpo-rate the effect of higher-order contributions. The SAFT-VR Mie approach has been shown to yield a very ac-curate methodology for the simultaneous description ofthe fluid-phase behavior and thermodynamic derivativeproperties of pure substances and mixtures. In view ofits success in describing the thermodynamic propertiesof real systems, we now reformulate SAFT-VR Mie asa group contribution approach based on a heteronuclearmodel of fused segments interacting via Mie potentialsof variable repulsive and attractive range: SAFT-γ Mie.The goal is to develop a predictive group-contributioncapability that provides accurate thermodynamic prop-erties, including both fluid-phase behavior and second-order derivatives of the free energy, such as the heat ca-pacity, compressibility, or speed of sound.

The remainder of the paper is organized in the follow-ing way. A brief discussion on the underlying molecularmodel and the intermolecular potential is first presentedin Section II, followed by a detailed description of theSAFT-γ Mie group contribution approach (Section III)and the procedure adopted for the estimation of groupparameters (Section IV). The parameters for the func-tional groups of two chemical families (n-alkanes and n-alkyl esters) are presented in Section VA and the per-formance of the theory in describing the fluid-phase be-havior of pure compounds is thoroughly discussed. Thepredictive capabilities of the approach in the prediction ofsecond-order thermodynamic derivatives of pure compo-nents and the fluid-phase behavior and excess properties

of binary mixtures are demonstrated in Section VB.

II. MOLECULAR MODEL AND INTERMOLECULARPOTENTIAL

As with other group contribution approaches the deter-mination of molecular properties with the SAFT-γ MieEoS is achieved by subdividing the molecules into dis-tinct functional groups chosen to represent the variouschemical moieties of a molecule, with appropriate sum-mations over the contributions of all of the functionalgroups. A fused heteronuclear model is employed, wherethe molecules are constructed from distinct monomericsegments which in our case are taken to interact throughMie potentials of variable attractive and repulsive range.An example of the fused heteronuclear molecular modelused to represent n-hexane is shown in figure 1, wherethe CH3 and CH2 functional groups that characterizen-hexane are highlighted. In line with physical expec-tations a fused heteronuclear united-atom representationhas been shown to be more appropriate than a tangen-tial heteronuclear model in the description of the phaseequilibria of real compounds within a SAFT-γ treat-ment66. A potential of the Mie form has been usedfor homonuclear models within the SAFT-VR frameworkby Davies et al.96, as well as in the original develop-ment of the SAFT-VR Mie EoS72,95. The main differ-ence between these two approaches is in the treatment ofthe radial distribution function (RDF) of the Mie fluid,which in the approach of Davies et al.96 was approxi-mated as the RDF of the Sutherland-6 potential. How-ever, it is known that the accuracy of the representationof the RDF of the monomer reference system is of greatimportance in the development of EoSs based on thefirst-order thermodynamic perturbation theory (TPT1)of Wertheim47–50. In the latest incarnation of the SAFT-VR Mie EoS73 an accurate second-order expansion RDFis developed; we use the latter form of the RDF in ourcurrent work.

The Mie intermolecular potential97 is a generalizedLennard-Jonesium potential98–102, where the attractiveand the repulsive exponents which characterize the soft-ness/hardness and the range of the interaction are al-lowed to vary freely. The form of the pair interactionenergy between two segments k and l as a function of theintersegment distance rkl is given by

ΦMiekl (rkl) = Cklϵkl

[(σkl

rkl

)λrkl

−(σkl

rkl

)λakl

], (1)

where σkl is the segment diameter, ϵkl is the depth ofthe potential well, and λr

kl and λakl are the repulsive and

attractive exponents of the intersegment interactions, re-spectively. The prefactor Ckl is a function of these expo-nents and ensures that the minimum of the interaction is−ϵkl:

5

Ckl =λrkl

λrkl − λa

kl

(λrkl

λakl

) λakl

λrkl

−λakl

. (2)

In common with other SAFT approaches, additionalshort-range association sites can be placed on the molec-ular segments to mimic the association interactions(hydrogen bonding and other short-range interactions)present in some polar compounds. More specifically theassociation interactions are modeled by means of square-well sites, so that the interaction between a site of type aplaced on a segment of type k and a site of type b placedon a segment of type l is given by

ΦHBkl,ab(rkl,ab) =

−ϵHB

kl,ab if rkl,ab ≤ rckl,ab ,

0 if rkl,ab > rckl,ab ,(3)

where rkl,ab is the center-center distance between sitesa and b, −ϵHB

kl,ab is the association energy, and rckl,ab thecut-off range of the interaction between sites a and b ongroups k and l, respectively. Each site is positioned at adistance rdkk,aa from the center of the segment on whichit is placed.We should also note that within the SAFT-γ treat-

ment a group may comprise several identical segments67.The number of identical segments forming a group is la-beled ν∗k . To summarize, the parameters that fully de-scribe a functional group k are the number ν∗k of identi-cal segments that the group comprises, the diameter σkk

of the segments of the group, the energy of interactionϵkk between the segments of the group, and the valuesλrkk and λa

kk of the repulsive and attractive exponents,respectively, that determine the form of the interactionpotential. The extent to which the segments of a givengroup k contribute to the overall molecular properties ischaracterized with a key parameter of the methodology,the shape factor Sk. The interactions between groups ofdifferent types k and l are specified through the corre-sponding unlike parameters σkl, ϵkl, λ

rkl, and λa

kl. In thecase of associating groups, the number NST,k of the dif-ferent site types, the number of sites of each type, e.g.,nk,a, nk,b, . . ., nk,NST,k

, together with the position rdkl,abof the site, and the energy ϵHB

kl,ab and range rckl,ab of theassociation between sites, of the same or different type,have to be determined.

III. SAFT-γ MIE

The total Helmholtz free energy A of a mixture of het-eronuclear associating molecules formed from Mie seg-ments is written in dimensionless form as the sum of fourseparate contributions,

A

NkBT=

Aideal

NkBT+

Amono.

NkBT+

Achain

NkBT+

Aassoc.

NkBT, (4)

where Aideal is the free energy of the ideal gas, Amono. isthe term accounting for interactions between monomericMie segments, Achain is the contribution to the free en-ergy for the formation of molecules from Mie segments,Aassoc. is the term accounting for the association interac-tions, N is the total number of molecules, kB the Boltz-mann constant, and T the absolute temperature. In thefollowing sections, the separate contributions to the freeenergy are discussed in detail.

A. Ideal Term

The ideal contribution to the free energy of the mixtureis given by103

Aideal

NkBT=

(NC∑i=1

xi ln(ρiΛ

3i

))− 1 , (5)

where xi is the mole fraction of component i in the mix-ture, ρi = Ni/V is the number density of component i,Ni being the number of molecules of component i. Thesummation in eq. (5) is over all of the components NC

present in the mixture (and N =

NC∑i

Ni). The ideal free

energy incorporates the effects of the translational, rota-tional and vibrational contributions to the kinetic energyimplicitly in the thermal de Broglie volume, Λ3

i .

B. Monomer Term

The monomer term Amono. describes the contributionof segment-segment interactions, taken to be of the Mieform in our case, to the total Helmholtz free energy ofthe system. This contribution is obtained with a Barkerand Henderson perturbation theory104,105 to third order.The intermolecular potential of eq. (1) is first written asthe sum of a reference repulsive contribution Φ0(rkl), anda perturbation attractive contribution Φ1(rkl):

ΦMiekl (rkl) = Φ0(rkl) + Φ1(rkl) , (6)

where

Φ0(rkl) =

ΦMie

kl (rkl) if rkl ≤ σkl ,

0 if rkl > σkl ,(7)

and

Φ1(rkl) =

0 if rkl < σkl ,

ΦMiekl (rkl) if rkl ≥ σkl .

(8)

6

Barker and Henderson105 have shown that the free en-ergy of the system can then be obtained as a perturbationexpansion in the inverse temperature relative to the ref-erence system. In the case of soft-core potentials, such asthe Mie potential, the reference system can be approxi-mated as an equivalent system of hard spheres of effective(temperature dependent) diameter dkk, since the proper-ties of the reference system described by the potential ineq. (7) are not generally known. The high-temperatureperturbation expansion can then be expressed as

Amono.

NkBT=

AHS

NkBT+

A1

NkBT+

A2

NkBT+

A3

NkBT, (9)

where AHS is the free energy of the hard-sphere referencesystem of diameter dkk. For a given group k, the effectivehard-sphere diameter is obtained from105

dkk =

∫ σkk

0

[1− exp

−ΦMie

kk (rkk)

kBT

]dr . (10)

The integral of eq. (10) is obtained by means of theGauss-Legendre quadrature, a technique previously em-ployed by Paricaud106 who showed that a 5-point Gauss-Legendre procedure is adequate for an accurate represen-tation of the effective hard-sphere diameter dkk.The hard-sphere Helmholtz free energy of the mixture

is given by66

AHS

NkBT=

(NC∑i=1

xi

NG∑k=1

νk,iν∗kSk

)aHS , (11)

where NG is the number of types of groups present, νk,ithe number of occurrences of a group of type k on com-ponent i, and aHS is the dimensionless contribution tothe hard-sphere free energy per segment, obtained usingthe expression of Boublık107 and Mansoori et al.108:

aHS =6

πρs

[(ζ32ζ23

− ζ0

)ln(1− ζ3)

+ 3ζ1ζ21− ζ3

+ζ32

ζ3(1− ζ3)2

]. (12)

In eq. (12), ρs is the segment number density which isrelated to the molecular density ρ through

ρs = ρ

(NC∑i=1

xi

NG∑k=1

νk,iν∗kSk

), (13)

and the moment densities ζm are expressed as

ζm =πρs6

NG∑k=1

xs,kdmkk , m = 0, 1, 2, 3 , (14)

where the effective hard-sphere diameter dkk of the ref-erence fluid (cf. eq. (10)) is used. The summation ofeq. (14) is expressed in terms of the fraction xs,k of seg-ments of a group of type k in the mixture, which is definedas

xs,k =

NC∑i=1

xiνk,iν∗kSk

NC∑i=1

xi

NG∑l=1

νl,iν∗l Sl

. (15)

After substituting the expression of the group fractionxs,k (eq. (15)) in the definition of the reduced densities(eq. (14)) and expressing the reference hard-sphere en-ergy per segment as a function of the molecular density,one obtains the following expression for the hard-sphereHelmholtz free energy per molecule:

AHS

NkBT=

6

πρ

[(ζ32ζ23

− ζ0

)ln(1− ζ3)

+ 3ζ1ζ21− ζ3

+ζ32

ζ3(1− ζ3)2

], (16)

which is identical to the form of the Helmholtz free energyof a hard-sphere mixture.

The first-order term A1 of the perturbation expansioncorresponds to the mean-attractive energy, and as for thehard-sphere term it is obtained as a summation of thecontributions to the mean-attractive energy per segmenta1:

A1

NkBT=

1

kBT

(NC∑i=1

xi

NG∑k=1

νk,iν∗kSk

)a1 . (17)

The mean-attractive energy per segment is obtained bysumming the pairwise interactions a1,kl between groupsk and l over all types of functional groups present in thesystem,

a1 =

NG∑k=1

NG∑l=1

xs,kxs,la1,kl , (18)

where it can be shown73 that

a1,kl = Ckl[xλakl

0,kl

(as1,kl(ρs;λ

akl) +Bkl(ρs;λ

akl))

− xλrkl

0,kl

(as1,kl(ρs;λ

rkl) +Bkl(ρs;λ

rkl))]

. (19)

Here, Ckl is the pre-factor of the potential (cf. eq. (2)),x0,kl is defined as x0,kl = σkl/dkl, and Bkl is given by

Bkl(ρs;λkl) = 2πρsd3klϵkl

[1− ζx/2

(1− ζx)3I(λkl)

− 9ζx(1 + ζx)

2(1− ζx)3J(λkl)

]. (20)

7

This expression is written using a generalized notationwhere the range λkl indicates that the expression can beevaluated for both the repulsive λr

kl and the attractiveλakl exponents. In eq. (20) ζx is the packing fraction of a

hypothetical pure fluid of diameter dx, obtained by ap-plying the van der Waals (vdW) one-fluid mixing rule

d3x =

NG∑k=1

NG∑l=1

xs,kxs,ld3kl , (21)

so that:

ζx =πρs6

NG∑k=1

NG∑l=1

xs,kxs,ld3kl . (22)

The unlike effective hard-sphere diameter dkl is obtainedwith an appropriate combining rule (cf. section III E).The quantities I(λkl) and J(λkl) are introduced in or-der to simplify the integration of the potential so as toobtain analytical expressions for the first-order perturba-tion term as discussed in detail in ref. [73]. Both I(λkl)and J(λkl) are functions of the parameters of the inter-molecular interaction potential alone and are calculated(for either λr

kl or λakl) as

73

I(λkl) =

∫ x0,kl

1

x2

xλkldx =

1− (x0,kl)(3−λkl)

(λkl − 3), (23)

and

J(λkl) =

∫ x0,kl

1

x3 − x2

xλkldx

=1− (x0,kl)

(4−λkl)(λkl − 3) + (x0,kl)(3−λkl)(λkl − 4)

(λkl − 3)(λkl − 4).

(24)

The free energy as1,kl(ρs;λkl) appearing in eq. (19) cor-responds to the first-order perturbation term of a Suther-land fluid characterized by a hard-core diameter dkl, aninteraction range exponent λkl, and energy well-depthϵkl. The exact evaluation of this term requires knowledgeof the radial distribution function of the hard-sphere sys-tem over a range of separations. In order to derive ananalytical expression in the same spirit as in the origi-nal SAFT-VR approach109,110, the mean-value theoremis applied in order to integrate over the radial distributionfunction by mapping it to its value at contact dkl at aneffective packing fraction ζeff.

x . This procedure is shownto provide a description which is in excellent agreementwith the evaluation of as1,kl(ρs;λkl) by full quadrature us-ing an integral equation theory for the radial distributionfunction73. The as1;kl(ρs;λkl) term is then evaluated as

as1,kl(ρs;λkl) = −2πρs

(ϵkld

3kl

λkl − 3

)1− ζeff.

x /2

(1− ζeff.x )3

. (25)

The effective packing fraction has been parameterized73

for ranges of the exponents of 5 < λkl ≤ 100 and canbe expressed as a function of the vdW one-fluid packingfraction ζx as

ζeff.kl = c1,klζx + c2,klζ

2x + c3,klζ

3x + c4,klζ

4x , (26)

where the coefficients (c1,kl, c2,kl, c3,kl, c4,kl) are obtainedas functions of the range λkl as

c1,klc2,klc3,klc4,kl

=

0.81096 1.7888 −37.578 92.2841.0205 −19.341 151.26 −463.50−1.9057 22.845 −228.14 973.921.0885 −6.1962 106.98 −677.64

×

11/λkl

1/λ2kl

1/λ3kl

. (27)

The second-order perturbation term of the high-temperature expansion (cf. eq. (9)) represents the fluc-tuation of the attractive energy in the system and is ob-tained as

A2

NkBT=

(1

kBT

)2(

NC∑i=1

xi

NG∑k=1

νk,iν∗kSk

)a2 , (28)

where the fluctuation term per segment a2 is obtainedfrom the appropriate sum of the contributions of the pair-wise interactions a2,kl between groups k and l as

a2 =

NG∑k=1

NG∑l=1

xs,kxs,la2,kl . (29)

The expression for a2 is obtained based on the improvedmacroscopic compressibility approximation (MCA) pro-posed by Zhang et al.111 combined with a correction inthe same spirit as that proposed by Paricaud106 for softpotentials. The final expression for a2;kl is given by

a2,kl =1

2KHS(1 + χkl)ϵklC2

kl

x2λa

kl

0,kl

[as1,kl(ρs; 2λ

akl)

+ Bkl(ρs; 2λakl)]

− 2x(λa

kl+λrkl)

0,kl

[as1,kl(ρs;λ

akl + λr

kl)

+ Bkl(ρs;λakl + λr

kl)]

+ x2λr

kl

0,kl

[as1,kl(ρs; 2λ

rkl) +Bkl(ρs; 2λ

rkl)]

, (30)

where KHS is the isothermal compressibility of the hypo-thetical vdW one-fluid (vdW-1) system (with the packingfraction ζx cf. eq. (22)) and is obtained from the Carna-han and Starling hard-sphere expression112 as

8

KHS =(1− ζx)

4

1 + 4ζx + 4ζ2x − 4ζ3x + ζ4x. (31)

The correction factor χkl is obtained from73

χkl = f1(αkl)ζ∗x + f2(αkl)(ζ

∗x)

5

+ f3(αkl)(ζ∗x)

8 , (32)

where the quantities fm, for m = 1, 2, 3, as shown ineq. (39), are functions of αkl, a dimensionless form of theintegrated van der Waals energy of the Mie potential:

αkl =1

ϵklσ3kl

∫ ∞

σ

ΦMiekl (r)r2dr

= Ckl(

1

λakl − 3

− 1

λrkl − 3

). (33)

Note that in eq. (32) the quantity ζ∗x is used (not to beconfused with ζx defined in eq. (22)) which is the packingfraction of a hypothetical pure fluid of diameter σx. Thediameter σx is obtained based on a vdW one-fluid mixingrule (cf. eq. (21)) as

σ3x =

NG∑k=1

NG∑l=1

xs,kxs,lσ3kl , (34)

so that the packing fraction ζ∗x is now obtained as:

ζ∗x =πρs6

NG∑k=1

NG∑l=1

xs,kxs,lσ3kl . (35)

The third-order perturbation term is obtained from thecontribution per segment a3 as

A3

NkBT=

(1

kBT

)3(

NC∑i=1

xi

NG∑k=1

νk,iν∗kSk

)a3 , (36)

where a3 is obtained by summing the pairwise segment-segment contributions a3,kl on groups k and l as previ-ously for the a1 and a2 terms:

a3 =

NG∑k=1

NG∑l=1

xs,kxs,la3,kl . (37)

The contribution a3,kl, is obtained using the followingempirical expression73

a3,kl = −ϵ3klf4(αkl)ζ∗xexp [f5(αkl)ζ

∗x

+ f6(αkl)(ζ∗x)

2]. (38)

A temperature dependence of the a3,kl term is avoidedby expressing it as a function of ζ∗x , as opposed to thepacking fraction ζx which has an implicit temperaturedependence through the effective diameter dkl.

The functions fm for (m = 1, . . . , 6) appearing ineqs. (32) and (38) are calculated as73

fm(αkl) =

3∑n=0

ϕm,nαnkl

1 +6∑

n=4

ϕm,nαn−3kl

, for m = 1, . . . , 6 . (39)

The values of the coefficients ϕm,n are listed in table I.The coefficients that appear in the modified MCA ex-pression for the a2,kl term, i.e., fm for m = 1, 2, 3, wereobtained from an analysis of the Monte Carlo simulationdata for the fluctuation term and pure-component vapor-liquid equilibrium data for selected Mie fluids (λr,λa)73.The coefficients employed for the calculation of the a3,klterm were obtained in order to best reproduce the sim-ulation data for vapor-liquid equilibrium data and crit-ical points of several Mie (λr,λa) fluids as explained inref. [73]. It is important to note that the coefficients in-cluded in the calculation of the a3,kl are obtained fromsimulation data of the fluid-phase behavior of monomers,and as a consequence the final expression for the free en-ergy incorporates higher-order terms of the perturbationexpansion of Barker and Henderson104 (in fact, for thecomplete series), rather than just the third-order term.A retrospective analysis of the third-order perturbationterm has shown that the calculations using the empiricalexpression of eq. (38) are in good agreement with eval-uations of the equivalent term obtained from molecularsimulation73, which indicates that the series is conver-gent and terms beyond the third-order make a reducedcontribution to the Helmholtz free energy.

C. Chain Term

The treatment of the contribution to the free energydue to the formation of molecular chains from fusedMie segments is undertaken following a formal TPT1treatment, where the contact value of the radial dis-tribution function at an effective average diameter isrequired66,67. An alternative treatment has been pro-posed70, and later employed in the work of Paduszynskiand Domanska71, where the chain contribution is calcu-lated using the segment-segment contact values of theRDF. In that way the connectivity of the molecule is re-tained in the theoretical description. However, as shownin Appendix A, both the RDF of an average diameterused in our current work and the segment-segment con-tact approach70,71 reproduce the thermodynamic behav-ior of tangentially bonded heteronuclear chains with sim-ilar accuracy in comparison with simulation data.

9

In order to apply the SAFT-γ treatment of the chaincontribution, average molecular parameters (σii, dii, ϵiiand λii) for each molecular species i in the mixture are in-troduced66,67; this allows for the evaluation of the radialdistribution function at a molecularly averaged effectivedistance. The averaging of the molecular size and en-ergy parameters is independent of the composition of themixture and makes use of the molecular fraction zk,i ofa given group k on a molecule i:

zk,i =νk,iν

∗kSk

NG∑l=1

νl,iν∗l Sl

. (40)

The quantity zk,i is not to be confused with the fractionxs,k of a given group k in the mixture, which is compo-sition dependent (cf. eq. (15)). The average molecularsegment size σii and the effective hard-sphere diameterdii are defined as

σ3ii =

NG∑k=1

NG∑l=1

zk,izl,iσ3kl , (41)

and

d3ii =

NG∑k=1

NG∑l=1

zk,izl,id3kl . (42)

The averaging rule for the effective hard-sphere diameterdii is chosen such that the value of the vdW one-fluidpacking fraction of a mixture of heteronuclear monomericsegments, as calculated in eq. (22), is the same whencalculated for monomeric segments of average molecularsize dii, i.e.,

ζx =πρs6

NG∑k=1

NG∑l=1

xs,kxs,ld3kl

=πρs6

NC∑i=1

NC∑j=1

xi(

NG∑k=1

ν∗kνk,iSk)xj(

NG∑k=1

ν∗kνk,jSk)d3ij .

(43)

Other effective molecular parameters are obtained inthe same way, so that the average interaction energy ϵiiand exponents which characterize the range of the poten-tial λii are obtained as

ϵii =

NG∑k=1

NG∑l=1

zk,izl,iϵkl , (44)

and

λii =

NG∑k=1

NG∑l=1

zk,izl,iλkl . (45)

Relation (45) holds for both the repulsive, λrii, and the

attractive, λaii, exponents.

The contribution to the free energy of the mixture dueto the formation of a chain of tangent or fused segmentsis based on the first-order perturbation theory (TPT1)of Wertheim47–50,52 but using the effective molecular pa-rameters:

Achain

NkBT= −

NC∑i=1

xi

NG∑k=1

(νk,iν∗kSk − 1) lngMie

ii (σii; ζx) ,

(46)where gMie

ii (σii; ζx) is the value of the radial distributionfunction (RDF) of the hypothetical one-fluid Mie systemat a packing fraction ζx evaluated at the effective diam-eter σii. An accurate estimate of the contact value ofthe RDF for a Mie fluid can be obtained by means of asecond-order expansion73:

gMieii (σii; ζx) = gHS

d (σii)exp[βϵiig1(σii)/gHSd (σii)

+ (βϵii)2g2(σii)/g

HSd (σii)] . (47)

The zeroth-order term of the expansion, gHSd (σii), is

the radial distribution function of a system of hardspheres of diameter dii evaluated at distance σii (andpacking fraction ζx). As shown in ref. [73], an expres-sion for this function can be obtained following Boublık’srecipe113,

gHSd (σii) = exp(k0 + k1x0,ii + k2x

20,ii + k3x

30,ii) , (48)

which is valid for 1 < x0,ii <√2 (with x0,ii = σii/dii).

The coefficients km in this expression are obtained asfunctions of the vdW one-fluid packing fraction ζx (cf.eq. (22)) of the hypothetical pure fluid as

k0 = −ln(1− ζx) +42ζx − 39ζ2x + 9ζ3x − 2ζ4x

6(1− ζx)3, (49)

k1 =ζ4x + 6ζ2x − 12ζx

2(1− ζx)3, (50)

k2 =−3ζ2x

8(1− ζx)2, (51)

and

k3 =−ζ4x + 3ζ2x + 3ζx

6(1− ζx)3. (52)

10

The first-order term g1(σii) of the expansion for thecontact value of the RDF (cf. eq. (47)) is approxi-mated by its value at the contact distance dii, obtainedby means of a self-consistent method for the calcula-tion of pressure from the virial and the free energyroutes73,109,110 so that

g1(σii) ≈ g1(dii) =1

2πεiid3ii

[3∂a1,ii∂ρs

− Ciiλaiix

λaii

0,ii

as1,ii(ρs; λaii) + Bii(ρs; λ

aii)

ρs

+ Ciiλriix

λrii

0,ii

as1,ii(ρs; λrii) + Bii(ρs; λ

rii)

ρs

]. (53)

In this expression Cii is the prefactor of the effec-tive molecular interaction potential of component i ob-tained using the values of the average molecular repul-sive and attractive exponents (λr

ii and λaii), i.e., Cii =

λrii

λrii−λa

ii

(λrii

λaii

) λaii

λrii

−λaii . The quantity Bii(ρs; λii) is obtained

using eq. (20) as

Bii(ρs; λii) = 2πρsd3iiϵii

[1− ζx/2

(1− ζx)3I(λii)

− 9ζx(1 + ζx)

2(1− ζx)3J(λii)

], (54)

but with I(λii) and J(λii) (cf. eq. (23) and (24)) nowbased on the effective exponents λr

ii and λaii and the ratio

x0,ii = σii/dii.The first-order term of the radial distribution functionfurther depends on the density derivative of the effectivefirst-order perturbation term a1,ii for the contribution ofthe monomeric interactions to the free energy per seg-ment, which is calculated following expression (19) as

a1,ii = Cii[xλaii

0,ii

(as1,ii(ρs; λ

aii) + Bii(ρs; λ

aii))

− xλrii

0,ii

(as1,ii(ρs; λ

rii) + Bii(ρs; λ

rii))]

. (55)

The integrated energy of the Sutherland fluid as1;ii(ρs; λii)calculated for the effective molecular parameters is ob-tained as

as1,ii(ρs; λii) = −2πρs

(ϵiid

3ii

λii − 3

)1− ζeff.

ii /2

(1− ζeff.ii )3

, (56)

where the effective packing fraction ζeff.ii used for the

mapping of the radial distribution function at contact,is now calculated as

ζeff.ii = c1,iiζx + c2,iiζ

2x + c3,klζ

3x + c4,iiζ

4x . (57)

The coefficients of eq. (57) are obtained based on theaveraged values of the exponents of the potential (λii):

c1,iic2,iic3,iic4,ii

=

0.81096 1.7888 −37.578 92.2841.0205 −19.341 151.26 −463.50−1.9057 22.845 −228.14 973.921.0885 −6.1962 106.98 −677.64

×

1

1/λii

1/λ2ii

1/λ3ii

. (58)

The second-order term g2(σii) of eq. (47) is also ap-proximated by its value at the effective diameter dii. Itis obtained based on the expression of the macroscopiccompressibility approximation and an empirical correc-tion73:

g2(σii) ≈ g2(dii) = (1 + γc,ii)gMCA2 (dii) . (59)

The correction factor γc,ii is given as a function of tem-perature, the one-fluid packing fraction ζ∗x (see eq. (35)),and the averaged values of the exponents of the potentialas

γc,ii = ϕ7,0 − tanh [ϕ7,1(ϕ7,2 − αii)]− 1× ζ∗xθ exp(ϕ7,3ζ

∗x + ϕ7,4(ζ

∗x)

2) , (60)

where θ = exp(βϵii) − 1, the values of the coefficientsϕ7,0, . . . , ϕ7,4 are given in table I, and αii is obtainedfrom the molecular averaged exponents of the potential(cf. eq. 33), as

αii = Cii(

1

λaii − 3

− 1

λrii − 3

). (61)

The second-order term from the macroscopic compress-ibility approximation gMCA

2 (dii) of eq. (59) is obtainedbased on the fluctuation term of the Sutherland poten-tial as

gMCA2 (dii) =

1

2πϵ2i d3ii

[3

∂

∂ρs

(a2,ii

1 + χii

)− ϵiiK

HSC2iiλ

riix

2λrii

0,ii

as1,ii(ρs; 2λrii) + B(ρs; 2λ

rii)

ρs

+ ϵiiKHSC2

ii(λrii + λa

ii)x(λr

ii+λaii)

0,ii

×as1,ii(ρs; λ

rii + λa

ii) + B(ρs; λrii + λa

ii)

ρs

− ϵiiKHSC2

iiλaiix

2λaii

0,ii

as1,ii(ρs; 2λaii) + B(ρs; 2λ

aii)

ρs

].

(62)

where all parameters and free energy terms are evaluatedusing the effective molecular parameters. Note that the

11

empirical correction to the MCA expression is based onthe molecular parameters:

χii = f1(αii)ζxx30,ii + f2(αii)(ζxx

30,ii)

5

+ f3(αii)(ζxx30,ii)

8 , (63)

where the coefficients f1, f2, f3 are obtained from eq. (39)by using αii instead of αkl.

D. Association Term

The contribution to the Helmholtz free energy arisingfrom the association of molecules due to the bonding sitesis obtained as47–50,52,53

Aassoc.

NkBT=

NC∑i=1

xi

NG∑k=1

νk,i

NST,k∑a=1

nk,a

×(lnXi,k,a +

1−Xi,k,a

2

), (64)

where NST,k is the total number of site types on a givengroup k, and nk,a the number of sites of type a on groupk. Xi,k,a represents the fraction of molecules of compo-nent i that are not bonded at a site of type a on groupk. It is obtained from the solution of the mass actionequation as50,53,66

Xi,k,a =1

1 +

NC∑j=1

NG∑l=1

NST,l∑b=1

ρxjnl,bXj,l,b∆ij,kl,ab

. (65)

Here, ∆ij,kl,ab characterizes the association strength be-tween a site of type a on a group of type k of componenti and a site of type b on a group of type l of componentj. By introducing the square-well bonding potential andcarrying out an angle average, the following expression isobtained53:

∆ij,kl,ab = σ3ijFkl,abIkl,ab , (66)

where Fkl,ab = exp(ϵHBkl,ab/kBT )−1, and Ikl,ab is a dimen-

sionless integral defined as73

Ikl,ab =π

6σ3ijr

dkl,ab

2

∫ (2rdkl,ab+rckl,ab)

(2rdkl,ab−rckl,ab)

gMie(r)

× (rckl,ab + 2rdkl,ab − r)2(2rckl,ab − 2rdkl,ab)rdr . (67)

The determination of the integral in eq. (67) requiresan expression for the RDF of the reference Mie fluid overa range of distances. For a detailed discussion of the var-ious options on how this can be calculated see ref. [73

and 114]. For example, the RDF can be approximatedbased on a Barker-Henderson zeroth-order perturbationapproach, so that gMie(r) ≃ gHS

d (r). After assuming thatr2gHS

d (r) ≃ d2gHSd (d), a compact analytical form for the

association contribution can be obtained, so that the in-tegral of eq. (67) can be expressed as

Ikl,ab ≈ gHSd (dij)Kij,kl,ab . (68)

The bonding volume Kij,kl,ab is obtained from73

Kij,kl,ab =πd2ij

18σ3ijr

dkl,ab

2

[ln((rckl,ab + 2rdkl,ab)/dij)

× (6rckl,ab3 + 18rckl,ab

2rdkl,ab − 24rdkl,ab3)

+ (rckl,ab + 2rdkl,ab − dij)(22rdkl,ab

2 − 5rckl,abrdkl,ab

− 7rdkl,abdij − 8rckl,ab2 + rckl,abdij + d2ij)

](69)

The value of the radial distribution function at con-tact, gHS

d (dij), is readily available from the expressionof Boublık107

gHSd (dij) =

1

1− ζ3+ 3

diidjjdii + djj

ζ2

(1− ζ3)2

+ 2

(diidjj

dii + djj

)2ζ22

(1− ζ3)3 . (70)

E. Combining Rules

Combining rules for the unlike intermolecular param-eters are commonly employed within equations of stateto facilitate the study of binary and multicomponent sys-tems. In the specific case of the methodology presentedhere, the interactions between groups of different kindalso contribute to the description of pure componentswhen represented with a heteronuclear molecular model.The unlike segment diameter is obtained from a simplearithmetic mean (Lorentz rule115) as

σkl =σkk + σll

2, (71)

and the same combining rule is applied for the calculationof the unlike effective hard-sphere diameter, so that dklis calculated as

dkl =dkk + dll

2. (72)

Bearing in mind the definition of the Barker and Hen-derson hard-sphere diameter (cf. eq. (10)), a more rigor-ous way to obtain dkl would be to integrate the potentialof the unlike interaction between groups k and l as

12

dkl =

∫ σkl

0

[1− exp

−ΦMie

kl (rkl)

kBT

]dr . (73)

However, such an approach would require extensive nu-merical calculations. Moreover, in agreement with ear-lier findings73, the approximation of dkl as an arithmeticmean is found to have minimal impact on the perfor-mance of the method, as judged by the quality of thedescription of the properties of real compounds.The unlike dispersion energy ϵkl between groups k and

l, can be obtained by applying an augmented geometricmean, which also accounts for the asymmetry in size:

ϵkl =

√σ3kkσ

3ll

σ3kl

√ϵkkϵll . (74)

The combining rule for the repulsive λrkl and the attrac-

tive λakl exponents of the unlike interaction is obtained

by invoking the geometric mean for the integrated vander Waals energy of a Sutherland fluid,

αsvdW;kl = 2πϵklσ

3kl

(1

λkl − 3

), (75)

and by imposing the Berthelot condition115, αsvdW;kl =√

αsvdW;kkα

svdW;ll, which results in

λkl = 3 +√(λkk − 3)(λll − 3) . (76)

The combining rule of eq. (74) provides a first esti-mate of the value of the unlike dispersion energy. It isknown, however, that real systems often exhibit large de-viations from simple or augmented combining rules, espe-cially when the molecules comprise chemically differentcomponents and groups. As will be discussed in the fol-lowing sections, in practice the unlike dispersion energyϵkl often has to be treated as an adjustable parameter.In the case of associating compounds, the unlike range

of the association site-site interaction is obtained as

rckl,ab =rckk,aa + rcll,bb

2. (77)

The unlike value of the association energy can be ob-tained by means of a simple geometric mean:

ϵHBkl,ab =

√ϵHBkk,aaϵ

HBll,bb . (78)

However, typically, ϵHBkl,ab is treated as an adjustable

parameter, and is obtained by regression to experimentaldata.A number of combining rules for the average molecular

parameters required for the calculation of the chain and

association contributions to the free energy also need tobe considered. The unlike values for the effective segmentsize, σij and dij , dispersion energy, ϵij , and repulsive andattractive exponents of the potential, λij , are obtainedfrom

σij =σii + σjj

2, (79)

dij =dii + djj

2, (80)

ϵij =

√σ3iiσ

3jj

σ3ij

√ϵiiϵjj , (81)

and

λij = 3 +√

(λii − 3)(λjj − 3) . (82)

Once the full functional form of the Helmholtz free en-ergy has been specified other properties, such as the pres-sure p, chemical potential (activity) of each componentµi, internal energy u, enthalpy h, entropy S, Gibbs freeenergy G, and second-order derivative properties includ-ing the isochoric cV and isobaric cp heat capacities, thespeed of sound u, isothermal compressibility kT, ther-mal expansion coefficient α, and Joule-Thomson coeffi-cient µJT can be obtained algebraically from the stan-dard thermodynamic relations116. In order to determinethe fluid-phase equilibria the conditions of thermal, me-chanical, and chemical equilibria are solved by ensuringthat the temperature, pressure, and chemical potentialof each component are equal in each phase and that thestate corresponds to a global minimum in the Gibbs freeenergy(e.g., see ref. [117]).

IV. ESTIMATION OF GROUP PARAMETERS

The parameters that describe the contribution of eachfunctional group to the molecular properties are typicallyestimated from appropriate experimental data. In ourSAFT-γ Mie EoS, molecules are modeled as heteronu-clear chains formed from distinct functional (chemical)groups, and each non-associating group is fully charac-terized by the following set of parameters (cf. section IIfor more details): the number of segments ν∗k , the shapefactor Sk, the segment diameter σkk, the energy of dis-persion ϵkk, and the repulsive λr

kk and attractive λakk ex-

ponents of the potential. The number of segments com-prising a group is determined by examining the differentrealistic possibilities with a trial-and-error approach, andthe rest of the group parameters are estimated from ap-propriate experimental data.

13

As discussed in the previous section, the unlikesegment diameter, σkl, and the values of the exponentsof the unlike interaction potential, λr

kl and λakl, are

calculated by means of the combining rules given ineqs. (71) and (76), respectively. The value of theunlike dispersion energy, ϵkl, is typically treated asan adjustable parameter and is therefore obtained byregression to experimental data. In many cases, thevalue of the unlike interaction energy can be estimatedfrom pure-component data as will be shown in thefollowing section. This is a unique characteristic ofheteronuclear models; it can lead to accurate predictionsfor properties of mixtures from pure component dataalone. Mixture data are also used where necessary toestimate the unlike energy, as was demonstrated inprevious work68.

The group parameters are estimated from experimen-tal data for a series of pure substances belonging to agiven chemical family. In most group contribution ap-proaches the parameter estimation procedure first con-siders the n-alkanes series, where the parameters for themethyl (CH3) and the methylene (CH2) groups are ob-tained. Once the parameters for these groups have beendetermined, they are transferred to the representation ofcompounds with additional functional groups based onexperimental data for the corresponding homologous se-ries, e.g., the n-alkyl esters for the carboxylate (COO)group. The parameters that describe each functionalgroup in the SAFT-γ Mie EoS are obtained by regres-sion to pure-component vapor-liquid equilibrium data(i.e., vapor pressures, pvap, and saturated liquid den-sities, ρsat), as well as single-phase densities, ρliq(T, p),at given temperatures and pressures. The temperaturerange commonly used for vapor-liquid equilibrium datais between the triple point and 0.9 T exp

crit , with T expcrit rep-

resenting the experimental critical temperature of thesubstance under study. Experimental data closer than0.9 T exp

crit are not included in the parameter estimation inour current work, despite the improved description of thenear-critical region with the novel methodology. Sincethe equation of state is a classical theory it is character-ized by mean-field critical exponents and does not allowone to reproduce the density fluctuations in the criticalregion; including data closer to the critical point wouldbias the parameters towards a more accurate represen-tation of the critical point and, therefore result in a lossof their physical significance. This would in turn makethe extrapolation to other thermodynamic conditions lessreliable. For the single-phase density, experimental dataat high temperatures and pressures in the liquid and su-percritical regions are typically used. These data are in-cluded when available as they provide information on thecompressibility of the fluid and can help achieve an accu-rate prediction of derivative thermodynamic properties.The objective function used in parameter estimation isgiven by

minΩ

fobj =w1

Npvap

Npvap∑u=1

[pexpvap(Tu)− pcalcvap (Tu;Ω)

pexpvap(Tu)

]2

+w2

Nρsat

Nρsat∑v=1

[ρexpsat (Tv)− ρcalcsat (Tv;Ω)

ρexpsat (Tv)

]2

+w3

Nρliq

Nρliq∑y=1

[ρexpliq (Ty, py)− ρcalcliq (Ty, py;Ω)

ρexpliq (Ty, py)

]2, (83)

where Ω denotes the vector of the parameters to be es-timated, the indices u, v, and y allow for the summationover all experimental (exp) points for each property, de-noted as Npvap , Nρsat and Nρliq

for the vapor pressure,saturated liquid density, and single-phase density, respec-tively. The desired level of accuracy for each calculated(calc) property can be adjusted by means of a weightingfactor: w1 for pvap, w2 for ρsat and w3 for ρliq. The es-timations are performed using the commercial softwarepackage gPROMS c⃝. Multiple starting points are used asinput to local optimizations, carried out using a gradient-based successive quadratic programming algorithm118.

V. RESULTS AND DISCUSSION

A. SAFT-γ Mie group parameters

In the first instance models for the characterization ofthe functional groups of two chemical families are devel-oped within the framework of SAFT-γ Mie: the n-alkanesand n-alkyl esters. In the following sections, the defini-tions of the functional groups for each chemical family,together with the detailed results of the regression to theexperimental data are presented. The metric used in ourwork to quantify the accuracy of the theoretical descrip-tion of the experimental data for a property R of a givencompound is the percentage average absolute deviation(%AAD) defined as:

%AADR =1

NR

NR∑i=1

∣∣∣∣Rexp.i −Rcalc.

i

Rexp.i

∣∣∣∣ , (84)

where NR is the number of data points of a property,Rexp.

i the experimental value, and Rcalc.i the calculated

value for the same property, at the conditions of the ith

experimental point.

1. n-Alkanes

The chemical family of the n-alkanes is considered firstin order to obtain the model parameters that describe themethyl (CH3) and methylene (CH2) functional groups.The parameters estimated for these functional groups,

14

summarized in tables II and III, are estimated from ex-perimental data for the vapor pressure, saturated liquiddensity, and single-phase density of linear alkanes fromethane to n-decane. The number of data points and thetemperature range (and pressure range, for the single-phase density) for each compound and property consid-ered are reported in table IV.

The quality of the description of the pure-componentvapor-liquid equilibria of the n-alkanes included in theregression is depicted for the coexistence densities in fig-ure 2(a) and for the vapor pressure in figure 2(b). Fromthe figures it can be seen that the SAFT-γ Mie groupcontribution approach allows for an excellent descriptionof the phase behavior of the correlated compounds, fromethane to n-decane. The deviations (%AAD) from theexperimental data for the vapor pressure and saturatedliquid density are summarized in table IV. The aver-age deviation for all correlated n-alkanes was found tobe 1.55% for pvap and 0.59% for ρsat and shows a signifi-cant improvement when compared to the results obtainedby with the SAFT-γ approach for models based on thesquare-well potential (3.98% for pvap and 0.57% for ρsat)for the vapor pressure, with a similar performance on sat-urated densities66. The level of accuracy of the SAFT-γ Mie approach constitutes a clear improvement in thedescription of the pure-component phase behavior of then-alkanes when compared to other group contribution ap-proaches within SAFT. Tamouza et al.31 have reportedaverage deviations of 2.63% for the vapor pressure and2.29% for the saturated liquid density with their homonu-clear GC approach based on the SAFT-0119, and 1.66%and 2.23% with that based on SAFT-VR SW109,110; inthe case of the hetero GC-SAFT-VR of Peng et al.70 de-viations of 5.95% and 3.07% are reported for the vaporpressure and saturated liquid density, respectively. TheSAFT-γ Mie predictions compare favorably with the hsPC-SAFT71 deviations of 0.96% for pvap and 0.56% forρsat, which are obtained based on a smaller temperaturerange, namely from 0.5 − 0.9 T exp

crit . It should be noted,however, that though a comparison of the deviations ofdifferent methods provides a measure of the accuracy ofeach approach, these are based on different experimentaldata and temperature ranges, as well as, in some cases,different sets of compounds.

The deviations reported here are based on data withina temperature range up to 0.9 T exp

crit and as such fail toexpress the improved performance of the SAFT-γ Mieapproach in the description of the near-critical region ofpure substances. The improvement can be appreciatedwhen comparing graphically the vapor-liquid equilibriaof selected n-alkanes obtained with our SAFT-γ Mie ap-proach and the SAFT-γ SW66 (cf. figure 3). From thecomparison, it can be clearly seen that the SAFT-γ MieEoS provides a significantly improved description of thenear-critical region of systems of varying chain length,decreasing the overshoot of the critical point whilst re-taining a highly accurate description of the fluid-phasebehavior at temperatures far from the critical point. We

reiterate that the approach, being an analytical mean-field type theory, fails to reproduce the critical scalingobserved experimentally. An accurate representation ofthese critical properties would require the application ofa renormalization group treatment, e.g., as applied to theSAFT-VR EoS91,120,121.

The description of the n-alkane single-phase densitiesρliq (cf. table IV) is also excellent, with an average devi-ation for all compounds of 0.59%. When examining eachcompound it can be seen that the highest deviations areobserved for ethane, as was also observed for the SAFT-γSW approach66. One may argue that ethane should notbe included in the parameter estimation procedure, sinceethane does not contain a CH3-CH2 interaction, whileall longer n-alkanes do. Furthermore, group contributiontechniques are generally not well suited to the study ofsmall molecules as proximity effects are neglected. Withthis in mind the procedure was repeated omitting thedata for ethane and the inclusion of these experimentaldata was not found to bias the parameters of the groupsin a way that would result in a sub-optimal descriptionfor the higher n-alkanes. When a more accurate model isneeded, ethane can be described as a separate functionalgroup (as would be the case for, e.g., water, methanol,etc.). This strategy has been followed within the het-ero GC-SAFT-VR of Peng et al.70 and the GC-SAFT ofTamouza et al.31 where n-propane was the first moleculeconsidered for the estimation of the parameters for then-alkane series.

As has already been mentioned, a particular advan-tage of the Mie potential is that the detailed form of thepair interaction potential between segments can be mod-ified by adjusting the values of the repulsive and attrac-tive exponents. This allows one to capture the finer fea-tures of the interaction which are important in providingan accurate description of the thermodynamic derivativeproperties. For the parameters that determine the formof the interaction potential between the CH3 and CH2

functional groups, the value of the attractive exponent isfixed to λa

kk = 6 for both groups. This choice is basedon a consideration of the chemical nature of the groups,which, given their non-polarity, are expected to interactvia London dispersion forces which are characterized byan attractive exponent of six43,122. The values of the re-pulsive exponents estimated from the experimental datafor the n-alkanes are found to be λr

CH3,CH3= 15.050

and λrCH2,CH2

= 19.871. Potoff and Bernard-Brunel123

have developed a force field for the methyl and methy-lene groups based on the Mie potential for molecular sim-ulation of the fluid-phase behavior of n-alkanes, findingthat a good description can be obtained with values ofthe repulsive exponent of 16 for both groups. The val-ues of the shape factors for the CH3 and CH2 groupsare also determined from the parameter estimation pro-cedure. The optimal values obtained are SCH3 = 0.5725and SCH2 = 0.2293, which are found to be quite differentfrom those used in previous work with the square-well po-tential66. In the previous study, the shape factors for the

15

CH3 and CH2 groups were fixed to 1/3 and 2/3, respec-tively, as these yield the molecular aspect ratio typicallyused for the n-alkanes in a homonuclear model57,124–126.The segment size of the CH3 group is found to be smallerthan the size of the CH2 group (4.077 A compared to4.880 A), which is in line with the smaller value of theshape factor for the CH2 group. Comparing the values ofthe segment diameters to the ones obtained with SAFT-γ SW66, one finds a significant increase; this will be dis-cussed in more detail when the predictions of the the-ory for longer n-alkanes and in the polyethylene polymerlimit are examined in section VB1.

2. n-Alkyl esters

Having determined the parameters for the CH3 andCH2 functional groups, these can be transferred in the de-scription of other homologous series of compounds to de-termine the parameters of additional functional groups.In the current work we develop parameters for the COOgroup, and its interactions with the CH3 and CH2 groups,based on experimental data for selected n-alkyl acetates.A number of different GC schemes have been suggestedto represent these compounds: in the modified UNI-FAC (Dortmund) approach an acetyl (CH3COO) func-tional group is employed127; in the case of SAFT-basedgroup contribution methods, a COO group is used inthe work of Thi et al.34 with the homonuclear GC-SAFTapproach, and Peng et al.70 employ a methoxyl (CH3O)group (in conjunction with a carbonyl (C=O) group usedfor the ketones) in the framework of the hetero GC-SAFT-VR approach. In our work the COO group istreated as comprising one segment (ν∗COO = 1) and ismodeled as non-associating, as esters are not expected toself-associate. Association sites will have to be includedin order to capture the unlike association in some polarmixtures containing esters, including, e.g., water or alka-nols (cf. the work of Kleiner and Sadowski128). Here, wetreat the polarity of the esters implicitly through the vari-able interaction range of the potential. A more explicittreatment of the polar nature of these molecules can befollowed by including an additional polar term in the freeenergy, as in the work of Thi et al.34, or alternatively bythe addition of two association sites.In our case the value of the attractive exponent is fixed

to the London λaCOO,COO = 6, as for the groups of the

n-alkanes, and the remaining parameters are estimatedbased on regression to a compilation of pure-componentand mixture experimental data. The pure-componentdata used include experimental data for the saturated liq-uid density, vapor pressure, and single-phase liquid den-sity for ethyl acetate to n-heptyl acetate; data for methylacetate was not included in the regression, as typicallywithin GC approaches the first member of a chemicalfamily is treated separately (e.g., methanol, methane,etc.). The specifics of the temperature and pressureranges, the number of points for each property, as well as

the deviations for each property and each compound aregiven in table V; the corresponding estimated parametersare presented in tables II and III. In addition to the pure-component data, experimental data for the enthalpies ofmixing of two binary n-alkyl acetate+n-alkane mixtureshave been included in the parameter estimation: ethylacetate+n-hexane, and n-butyl acetate+n-octane. Thecorresponding objective function in this case is as givenin eq. (83) with an additional term accounting for thedeviation between the experimental and the calculatedvalue of the enthalpy of mixing for a given compositionof each mixture:

minΩ

fobj =w1

Npvap

Npvap∑u=1

[pexpvap(Tu)− pcalcvap (Tu;Ω)

pexpvap(Tu)

]2

+w2

Nρsat

Nρsat∑v=1

[ρexpsat (Tv)− ρcalcsat (Tv;Ω)

ρexpsat (Tv)

]2

+w3

Nρliq

Nρliq∑y=1

[ρexpliq (Ty, py)− ρcalcliq (Ty, py;Ω)

ρexpliq (Ty, py)

]2

+w4

NhE

NhE∑z=1

[hE,exp(Tz, pz, xz)− hE,calc(Tz, pz, xz;Ω)

hE,exp(Tz, pz, xz)

]2.

(85)

Experimental data for selected mixtures are included inthe estimation procedure so as to have a more completecharacterization of the unlike interactions between thealkyl and carboxylate functional groups, to ensure an ac-curate description of the highly non-ideal phase behaviorthat these mixtures exhibit, and to increase the statisti-cal significance of the interaction parameters.

A comparison between the SAFT-γ Mie descriptionand the experimental data is given for the saturated liq-uid densities in figure 4(a) and for the vapor pressures infigure 4(b). The overall deviations (%AAD) for the corre-lated compounds (cf. table V) are 0.83% for pvap, 0.24%for ρsat, and 0.31% for ρliq. It is evident from the reporteddeviations and the figures that the SAFT-γ Mie approachleads to an excellent description of the coexistence andsingle-phase properties of the compounds considered inthe parameter estimation procedure. Thi et al.34 haveapplied three versions of SAFT (the SAFT-0119, SAFT-VR109,110, and PC-SAFT129) combined with a polar termand parameterized the COO functional group (for allesters other than formates, HCOOR). The parametersfor the COO group were obtained based on a selectionof acetates, propanoates, butanoates, and other esters.For the same compounds as the ones used in the re-gression of the SAFT-γ Mie parameters for the COOgroup presented in our current work, the reported de-viations by Thi et al. were 4.76%, 3.05%, and 4.20% forthe pvap and 1.62%, 1.72%, and 2.87% for ρsat with theGC-SAFT based on SAFT-0, SAFT-VR, and PC-SAFT,respectively. The performance of the hetero GC-SAFT-VR of Peng et al.70, where alkyl acetates are modeled bymeans of separate CH3O and C=O groups is somewhat

16

less accurate with average deviations (from n-propyl ton-heptyl acetate) of 7.29% for pvap and 2.14% for ρsat.Although a direct comparison with the other methods isdifficult due to the differences in the experimental dataused, the reported deviations for each methodology in-dicate that the presented SAFT-γ Mie EoS leads to asignificant improvement in the description of the pure-component fluid-phase behavior of n-alkyl acetates.

The limited mixture data included in the regressionare also well represented with SAFT-γ Mie approach, asshown in figure 5. The average deviation (%AAD) ofthe excess enthalpies for the binary systems included inthe estimation was found to be 7.37% (8.78% for ethylacetate+n-hexane, and 5.95% for n-butyl acetate+n-octane), and the level of agreement was deemed sat-isfactory based on the sensitivity of the selected prop-erties to the unlike interaction parameters between thefunctional groups and the small values of the excess en-thalpies exhibited by these mixtures (∼ 0.5-1.5 kJ/mol).In order to assess the level of accuracy, the descriptionof the SAFT-γ Mie approach was compared with thatobtained using modified UNIFAC (Dortmund)127, wherethe group parameters have been obtained by regressionto data for different mixture properties including phaseequilibria and excess enthalpies. From the comparisonshown in figure 5, it can be seen that the modified UNI-FAC (Dortmund) results in a slightly better descriptionof the experimental excess enthalpies with an averagedeviation of 3.91%. Bearing in mind that within modi-fied UNIFAC approach group parameters are exclusivelydeveloped to describe (limited) properties of binary andmulti-component mixtures, and that within the SAFT-γ Mie approach the primary target is a more completethermodynamic description of pure components and mix-tures, the performance of the latter methodology in thedescription of the excess enthalpies is very satisfactory.

The resulting form of the interaction potential of theCOO group is described by an optimal value of the repul-sive exponent of λr

COO,COO = 31.189, which accounts forthe nature of the electrostatic interactions of the group inan effective manner. No explicit treatment of the polar-ity of the esters is included in the model developed in thecurrent work; as mentioned earlier a term accounting forpolar interactions has been considered in other model-ing approaches for esters34, but in the expense of a morecomplicated description in terms of additional parame-ters. The estimated value of the segment size of the COOgroup is σCOO,COO = 3.994 A, which is slightly smaller

than the size of the CH3 group (4.077A), but combinedwith the higher value of the shape factor (SCOO = 0.6526compared to SCH3

= 0.5726) corresponds to an over-all larger size for the COO group, as expected from thechemical composition of the functional groups. The en-ergy of interaction characterizing the COO group is foundto be rather high, ϵCOO,COO/kB = 868.923 K, which ismainly attributed to the fact that there are two oxygencenters and that the polar nature of the COO group istreated implicitly through the dispersive interactions.

B. Predictions

1. Pure Components

An assessment of the adequacy of the parameters ob-tained for the n-aklyl CH3 and CH2 functional groupscan be made by examining the performance of the the-ory in describing the thermodynamic properties of com-pounds not included in the parameter estimation proce-dure. The application of the Mie intermolecular potentialwas previously shown to allow for an accurate simulta-neous description of fluid-phase equilibria and thermody-namic derivative properties72,73 when cast as a theory forhomonuclear models. The prediction of derivative prop-erties is a stringent test in any thermodynamic descrip-tion, and high deviations are to be expected with mostequations of state, be it with traditional cubic EoSs orthe SAFT variants not based on the Mie potential72,73.Here, we examine the performance of the SAFT-γ Mieapproach in the description of second-order thermody-namic derivative properties of representative n-alkanes.A comparison between the predictions of our current ap-proach and the results obtained with the SAFT-γ SWEoS66 in representing the experimental data of REF-PROP130 for the isobaric and isochoric heat capacities,the speed of sound, the isothermal compressibility, andthe thermal expansion coefficient of the linear n-alkanesfrom ethane to n-decane is provided in table VI. Forthe calculation of caloric and derivative properties, thefree energy of the ideal gas is calculated based on theJoback and Reid GC approach9 for the ideal gas heatcapacity, cp,0. Though these expressions are based oncorrelations for a temperature range between 273 and ap-proximately 1000 K, a reliable description is seen whenextrapolated to lower temperatures. From the table itcan be seen that the SAFT-γ Mie EoS provides a veryaccurate prediction of the second-order thermodynamicderivative properties of the n-alkanes examined, and con-stitutes a significant improvement in comparison to thedescription obtained with the SAFT-γ SW EoS basedon the square-well potential66. The level of agreementbetween the predictions of the SAFT-γ Mie theory andthe correlated experimental data from REFPROP130 forthe speed of sound, isothermal compressibility, isobaricheat capacity, and thermal expansion coefficient of se-lected n-alkanes at different thermodynamic conditionsis also shown in figure 6. In figure 7 the predictions ofour SAFT-γ Mie approach are compared to the exper-imental data for the isochoric heat capacity of selectedn-alkanes, at conditions both close to and far from thecritical point. From figure 7(a) it is evident that the the-ory provides an excellent representation of the tempera-ture dependence of the isochoric heat capacity of threeprototypical alkanes (n-butane, n-hexane, and n-octane,from bottom to top), for both the liquid and the vaporbranches, including the discontinuity at the liquid-vaportransition. The description of the pressure dependence ofthe isochoric heat capacity is depicted in figure 7(b) for

17

n-propane along both sub- and super-critical isotherms.The theory is shown to provide a satisfactory descriptionof the isochoric heat capacity at all conditions; slightdeviations are observed in the near-critical isotherm at atemperature of T=350 K (with T exp

crit,C3H8= 369.89 K130),

as well as for the super-critical isotherm of T=400 K. Aweak divergence is reproduced by the theory, howeveran accurate description of the singular behavior of theisochoric heat capacity along the critical isotherm wouldrequire a formal renormalization-group treatment.

It is important to reiterate that the calculations pre-sented in this section for the second-order thermody-namic derivative properties of n-alkanes are predictions,as no such data were included in obtaining the param-eters that characterize each functional group. The as-sessment of the performance of the theory for these chal-lenging thermodynamically properties provides a goodindication about the physical robustness of the param-eters obtained for the CH3 and CH2 functional groups.This is of crucial importance as these functional groupsconstitute the molecular backbone of a wide variety ofsubstances, and given that within the approach used heregroup parameters are obtained in a sequential manner, itis essential to ensure that high-fidelity parameters havebeen derived for the methyl and methylene groups.

A key advantage of a group contribution formulationlies in the transferability of the group parameters. Inorder to assess this feature, the parameters for the CH3

and the CH2 functional groups are applied to the predic-tion of pure-component properties of systems not usedto estimate the group parameters. Predictions with theSAFT-γ Mie EoS of the fluid-phase behavior for someheavier n-alkanes (namely n-pentadecane, n-eicosane, n-pentacosane, and n-triacontane) are compared with theexperimental data in figure 8, and the corresponding de-viations are summarized in table VIII. From this anal-ysis it is clear that the methyl and methylene group pa-rameters can be successfully transferred to the predictionof the pure-component fluid-phase behavior of longer n-alkanes; the description of the saturated liquid densityremains good, while higher deviations are seen in the va-por pressure. The latter could be related to the highuncertainty and low absolute values of the experimentaldata of the saturation pressure of the higher molecular-weight compounds at low temperatures. Apart from thefluid-phase behavior, the CH3 and CH2 group parame-ters result in a good description of derivative propertiesof long n-alkanes, as shown in figure 9(a) for the speed ofsound of n-pentadecane (n-C15H32) and figure 9(b) forthe isothermal compressibility of n-eicosane (n-C20H42).The results obtained with the SAFT-γ Mie approach andthe parameters developed for the CH3 and CH2 groupsshow a significantly improved performance in the pre-diction of the derivative properties of the long chain n-alkanes studied compared to the predictions with theSAFT-γ SW EoS, also included in figure 9.

The formulation of a SAFT-type approach within agroup contribution concept combines the wide range of

applicability of EoSs and the predictive nature of GCmethods. This can be illustrated in the description of thethermodynamic properties of polymers. The SAFT EoShas been successfully applied as a general methodology tothe study of polymeric systems (e.g., see refs. [119, 131–134]); the theory is particularly suited to polymers due tothe explicit consideration of the chain topology. Withinthe scope of the current work, and taking advantageof the group contribution formulation, the properties ofpolyethylene polymers can be predicted, using the pa-rameters estimated for the methyl and methylene groupsfrom the lower n-alkanes (cf. section VA1). The pre-dictions of the SAFT-γ Mie approach are comparedwith the experimental single-phase densities of liquid(molten) linear polyethylene in figure 10. The param-eters for the methyl and (predominantly) for the methy-lene group are seen to allow for an accurate descrip-tion of linear polyethylene with a molecular weight of126,000 g mol−1 over a broad range of temperatures andpressures. Achieving this level of agreement with theexperimental data is particularly challenging, especiallywhen one recalls that the parameters for the CH3 andCH2 groups were obtained based on the properties ofthe much shorter n-alkanes, ranging from ethane to n-decane. The description obtained with the SAFT-γ SWEoS using published parameters66 are also plotted in fig-ure 10. From the comparison it is apparent that the rep-resentation of the polymer with Mie segments provides amarked improvement over that obtained with square-wellsegments, and lends credence to the physical relevance ofthe parameters for the CH3 and CH2 groups (cf. table II).

A similar assessment is performed for the chemicalfamily of the n-alkyl acetates, where the performanceof the methodology in the prediction of second-orderderivative properties is examined. The deviations of theSAFT-γ Mie predictions from the experimental data forthe speed of sound, isobaric and isochoric heat capaci-ties, and isothermal and isentropic compressibilities forselected n-alkyl acetates are summarized in table VII.From the table, it can be seen that higher deviationsare observed in the description of the thermodynamicderivative properties considered for the lower molecular-weight compounds. This could be due to the dominanceof the polar nature of the COO functional group in themolecules with a short alkyl chain, which is not accu-rately reproduced by the theory in the case of derivativeproperties. It should be noted that this is somewhatexpected, as group contribution approaches are less suit-able for the description of small molecules. As the chainlength of the n-alkyl acetates is increased, the level ofagreement between the predictions of the theory and theexperimental data improves, and a very good represen-tation is seen for the case of n-decyl acetate. This is alsodepicted in figure 11, where the predictions of the the-ory for the speed of sound and the isobaric heat capacityof n-alkyl acetates of varying chain length are plottedtogether with the experimental data.

An assessment of the SAFT-γ Mie EoS in the pre-

18

diction of the pure-component fluid-phase behavior forlonger compounds not included in the parameter estima-tion procedure has also been performed for the n-alkylacetates. The parameters for the CH3, CH2, and COOgroups allow one to predict the fluid-phase behavior offive high-molecular-weight n-alkyl acetates, namely n-nonyl, n-decyl, n-undecyl, n-dodecyl, and n-tetradecylacetate and the deviations from the experimental data,available for the vapor pressure only, are summarized intable VIII. From the table it can be seen that the SAFT-γ Mie EoS can be extrapolated with good accuracy to theprediction of the pure component fluid phase behavior oflong-chain n-alkyl acetates. Aside from predictions forpure-component fluid-phase behavior the SAFT-γ Mieapproach can be successfully applied to the prediction ofsingle-phase densities and second-order thermodynamicderivative properties of long-chain esters. This is demon-strated in figure 12, where the predictions of the the-ory are compared with the experimental data for thesingle-phase density, the speed of sound and the isother-mal and isentropic compressibility of several long-chainethyl esters, namely ethyl caprylate (ethyl n-octanoate),ethyl caprate (ethyl n-decanoate), ethyl laurate (ethyl n-dodecanoate), ethyl myristate (ethyl n-tetradecanoate),and ethyl palmitate (ethyl n-hexadecanoate) fatty acidesters of key relevance in the production of biofuels. Itshould be noted that these compounds are not of thesame category as the ones used in the development ofthe COO functional group, where data for alkyl acetateswere used; however, since the functional group developedis the more versatile COO group (as opposed to mod-els where a CH3COO group is employed) the propertiesof these ethyl esters can be predicted in a reliable andstraightforward manner.

2. Binary Systems

The ability to obtain information about the nature ofthe unlike dispersion interaction between segments be-longing to the same molecule is an inherent advanta-geous feature of the SAFT-γ approach. For the specificcase of the chemical families considered in this work, thismeans that the values of the unlike dispersion energiesbetween the CH3 and CH2 groups are obtained by pure-component data of the n-alkanes alone (see table III).This allows for calculations of the fluid-phase behaviorand other properties of mixtures of n-alkanes and/or es-ters in a purely predictive manner.An example of the predictive capabilities of the SAFT-

γ Mie approach is shown in figure 13, where the theo-retical description is compared to experimental data forthe vapor-liquid phase behavior of the binary system n-butane+n-decane at different temperatures. It is clearthat the SAFT-γ Mie predictions are in very good agree-ment with the experimental data for the fluid-phase equi-libria of the mixture. The improvement in the descrip-tion of the near-critical fluid-phase behavior within the

SAFT-γ Mie approach allows for an accurate represen-tation of the system over a wide range of conditions, in-cluding the high-pressure critical region for isotherms attemperatures above the critical point of n-butane. Theovershoot of the critical points of the phase envelopes ofthis binary system is significantly reduced in comparisonto the results obtained with the SAFT-γ SW approach66.

The same parameter set can be applied to the pre-diction of the fluid-phase behavior of binary mixtures ofcompounds not included in the regression of the groupparameters. An example of this is shown in figure 14,where the predictions of the theory are compared to theexperimental data for asymmetric systems of a short anda polymer-like long-chain n-alkane. More specifically, infigure 14(a) the theoretical predictions are compared withthe available experimental data for the liquid-liquid equi-libria of the propane+n-hexacontane mixture at threedifferent temperatures. This system was chosen as aprototypical example of the type of fluid-phase equilib-ria exhibited in polymer solutions of hydrocarbons135.The n-C60H122 molecule is the highest molecular-weightmonodisperse n-alkane commercially available, so themixture can be modeled without the need to accountfor polydispersity136. Polydispersity generally has to beconsidered in representing polymer+solvent systems andit can have a significant effect on the phase boundaries,particularly in the description of liquid-liquid equilibria(e.g., see ref. [134]). From figure 14(a) it can be seenthat the SAFT-γ Mie predictions are in very good agree-ment with the experimentally determined phase behaviorof the mixture, accurately reproducing the width of thecoexistence region at higher pressures, with only a minorovershoot of the liquid-liquid critical point. The pro-posed methodology provides a significantly improved de-scription of the mixture in comparison to the correspond-ing results obtained with the SAFT-γ SW EoS based onthe SW potential66,67. A comparison of the predictionsof the theory with the vapor-liquid equilibria that poly-mer+solvent systems can exhibit is shown in figure 14(b),for the mixture of n-pentane and polyethylene of molec-ular weight 76,000 g mol−1 at two temperatures. Thepolymer in question has quite a high degree of polydis-persity characterized by a polydispersity index of 6.91137;this degree of polydispersity is not expected to signifi-cantly affect the vapor-liquid phase behavior of the sys-tem. From the plot it can be seen that both the SAFT-γMie and the SAFT-γ SW methods provide a descriptionof similar accuracy for the vapor-liquid equilibrium ofthe system. As a final point it is important to empha-size that the same set of group parameters is employedfor the prediction of different types of phase behavior:vapor-liquid equilibria for the n-butane+n-decane andthe n-pentane+polyethylene, and liquid-liquid equilibriafor the propane+n-hexacontane mixture.

The performance of SAFT-γ Mie can also be assessedin the prediction of excess thermodynamic properties ofbinary mixtures of n-alkanes. Properties such as the ex-cess volume and enthalpy have been the subject of numer-

19

ous studies, including among others the seminal work ofFlory and co-workers138–140 and more recently the workof Blas and co-workers141–143 within the SAFT frame-work. An example of the performance of the SAFT-γMie approach is illustrated in figure 15, where the predic-tions of the theory are compared to experimental data forthe excess speed of sound (uE) of selected n-hexane+n-alkane binary mixtures144 (cf. figure 15(a)), and theexcess molar volumes (V E) of the binary mixture of n-hexane+n-hexadecane at different temperatures145 (cf.figure 15(b)). As for the fluid-phase behavior, the pre-dictions of the theory are in very good agreement withthe experimental data. For the excess speed of sound,the predictions of the theory are seen to be in quanti-tative agreement with the experimental data; the trendto lower values of the excess speed of sound as the differ-ence in the chain length of the components of the mixturedecreases is correctly reproduced. Quantitative agree-ment for the excess molar volume is seen with the dataat higher temperatures, and the trend of the data to lowervalues with decreasing temperatures is reproduced by thetheory. Given the small magnitude of the data (espe-cially at lower temperatures), and the fact that one ofthe components of the mixture (n-hexadecane) is not in-cluded in the parameter estimation procedure, good over-all agreement of the theory with the experimental datais seen. Comparing the predictions of the SAFT-γ Mieapproach to the corresponding description obtained withthe SAFT-γ SW EoS, a clear improvement is seen for theexcess speed of sound of the mixtures in figure 15. In thecase of the predictions of the excess volumes, both the-ories provide a description of equivalent accuracy, withthe SAFT-γ SW in better agreement with the experi-mental data at low temperatures, and SAFT-γ Mie athigher temperatures. Despite an accurate overall perfor-mance, it has to be noted that the theory cannot repro-duce the fine features of the excess enthalpies of binarysystems of asymmetric n-alkanes, as these arise from con-formational effects that the theory does not account for.Such effects can be incorporated in the general theoreti-cal framework by including a term that accounts for thespecific intramolecular interactions as has been shown bydos Ramos and Blas143.

The group parameters obtained for the COO groupand the unlike interaction values shown in tables IIand III allow for the prediction of properties of binarymixtures of n-alkanes and n-alkyl esters. A comparisonbetween the predictions of the SAFT-γ Mie and the mod-ified UNIFAC (Dortmund)13 approaches for the fluid-phase behavior of selected binary mixtures are presentedin figures 16 and 17. In figure 16, the fluid phase behav-ior of mixtures of ethyl acetate with n-alkanes of vary-ing chain length (n-pentane, n-octane, and n-decane) ispresented as an isobaric temperature-composition repre-sentation. The theory accurately describes the vapor-liquid phase behavior of the mixtures studied, charac-terized by a minimum boiling azeotrope, and the shiftof the azeotropic composition to higher compositions of

ethyl acetate with increasing chain length of the n-alkaneis reproduced. In figure 17 the vapor-liquid phase behav-ior of a selection of mixtures of n-hexane with n-alkylacetates of varying chain length is shown. In this case,a more distinct azeotrope is observed for the mixture n-hexane+ethyl acetate, the composition of which is accu-rately reproduced by the theory. Regarding the perfor-mance of the SAFT-γ Mie EoS in comparison to the well-established modified UNIFAC (Dortmund) approach, itis apparent from figures 16 and 17 that in the case of thesystems studied both approaches provide the same levelof agreement with the experimental data.

The adequacy of the theory in describing excess prop-erties of mixing of binary systems of n-alkanes+n-alkylacetates is also examined. The excess enthalpy of mixingpredicted with SAFT-γ Mie for three binary mixturesof n-heptane with n-alkyl acetates of varying size(n-propyl, n-butyl, and n-pentyl acetate) is comparedto the experimental data in figure 18(a). It can beseen that the enthalpy of mixing of the three selectedmixtures is described well, and the correct dependenceof the magnitude of the excess enthalpy with varyingchain length of the n-alkyl acetate is also predicted.The SAFT-γ Mie predictions are seen to be of similaraccuracy to those with modified UNIFAC (Dortmund),as shown in figure 18(a). This is very pleasing, giventhe small absolute values of the excess enthalpy forthis mixtures (hE ≤ 1.5 kJ mol−1). One has to bearin mind that the SAFT-γ Mie parameters are obtainedpredominantly from pure-component experimental dataand data for selected mixtures, while the UNIFACparameters result from extensive regression to binarymixture (fluid-phase behavior and other properties)data. In figure 18(b), the predictions of the SAFT-γMie theory for the excess volumes of the same mixturesare compared to experimental data. In this case,comparisons with the modified UNIFAC (Dortmund)approach are not possible, as UNIFAC was developedon the basis of a lattice-fluid model and thus cannot beused directly for density calculations. The performanceof the SAFT-γ Mie approach is very satisfactory, andexcellent agreement between the calculations and theexperimental data is seen.

Finally, we examine the performance of the theory forthe prediction of second-order thermodynamic derivativeproperties of binary mixtures of n-hexane with n-alkylacetates of different chain length. In figure 19 a com-parison of the experimental data and the SAFT-γ Miepredictions for the speed of sound (figure 19(a)) and theexcess isentropic compressibility (figure 19(b)) is shownfor the binary mixtures of n-hexane with ethyl, n-pentyl,and n-decyl acetate. In the case of the speed of sound,and in line with the deviations reported in table VII, itcan be seen that the theory provides a qualitative de-scription of the experimental data; the deviation in theprediction of the speed of sound in the limit of pure ethylacetate is, however, quite significant. As the chain length

20

of the n-alkyl acetate is increased, the agreement of thepredictions with experiments improves, and the resultsfor the mixture of n-hexane+n-decyl acetate are seen tobe in excellent agreement with the experimental data.Regarding the predictions for the excess isentropic com-pressibility of the same mixtures (cf. figure 19(a)), theSAFT-γ Mie predictions agree with the change of sign ob-served for mixtures of n-hexane with ethyl and n-pentylacetate.

VI. CONCLUDING REMARKS

We have presented the development of the SAFT-γ Mieapproach as a reformulation of the SAFT-VR Mie EoS73

within a group contribution formalism, where the inter-actions between segments are represented by means ofthe Mie pair potential of variable attractive and repulsiveranges. The molecular model employed for the descrip-tion of pure substances and mixtures is a fused heteronu-clear model, an idea following from previous work66. To-gether with the implementation of the new intermolec-ular potential, a key novelty of the theory lies in thetreatment of the monomeric segment contribution to thefree energy, where a third-order perturbation expansionis employed. The performance of the theory in the de-scription of real systems is examined in the study of twochemical families (the n-alkanes and the esters), whereparameters for the methyl, methylene, and carboxylatefunctional groups (CH3, CH2, and COO) are obtainedfrom fluid-phase behavior data and single-phase densi-ties of the pure components. In the development of thecarboxylate group parameters (and the unlike parame-ters with the n-alkane groups) some data for the excessenthalpies of selected binary mixtures are used, in orderto better characterize the nature of the unlike interac-tions.The SAFT-γ Mie approach is shown to provide an ex-

cellent description of the pure-component properties ofthe correlated compounds, with average relative errorsof 1.19% for the vapor pressure, 0.42% for the saturatedliquid density, and 0.45% for the single-phase liquid den-sity. It is also shown that the treatment of the monomerterm proposed leads to a significant improvement in thedescription of the near-critical region of fluids, comparedto the SAFT-γ SW formulation which is based on thesquare-well potential66. Apart from fluid-phase behav-ior, second-order thermodynamic derivative propertiesare also considered. The Mie intermolecular potentialmodel employed in the proposed methodology allows foran accurate representation of properties such as the speedof sound, the isochoric and isobaric heat capacities, andthe thermal expansion coefficient as shown in detail forthe n-alkanes, while slightly higher deviations are foundfor the n-alkyl esters.The predictive capability of the method is examined

in the description of fluid-phase equilibria and, whenavailable, derivative properties of high-molecular-weight

compounds not included in the regression of the groupparameters, where the performance of the SAFT-γ Mieapproach is found to be very satisfactory. A predic-tive study in the limit of high-molecular-weight poly-mers confirms the robustness of the presented method-ology and the corresponding group parameters obtainedin this work. Finally, the performance of SAFT-γ Mieis examined in the prediction of the fluid-phase behav-ior and excess properties of binary mixtures of n-alkanesand n-alkane and n-alkyl esters. The theory is seen toaccurately describe the fluid-phase behavior of a vari-ety of systems, with different types of phase equilibria(vapor-liquid and liquid-liquid), over a wide range of con-ditions, including the high-pressure critical points of themixtures. The excess and second-order thermodynamicderivative properties of binary mixtures are also very welldescribed and, for the cases considered, the predictionsof SAFT-γ Mie compare favorably to the calculationsusing the modified UNIFAC (Dortmund) GC approach.The description of such properties is a challenging taskas these are known to be very sensitive to the specificmolecular details, and the performance of the SAFT-γMie demonstrates the future potential of the theory forthe study of other chemical families.

ACKNOWLEDGMENTS

V.P. is very grateful to the Engineering and Phys-ical Sciences Research Council (EPSRC) of the UKfor the award of a PhD studentship. T.L. and C.A.thank the Engineering and Physical Sciences ResearchCouncil (EPSRC) of the UK for the award of postdoc-toral fellowships. Additional funding to the MolecularSystems Engineering Group from the EPSRC (grantsGR/T17595, GR/N35991, EP/E016340, EP/J003840and EP/J014958), the Joint Research Equipment Ini-tiative (JREI) (GR/M94426) and the Royal Society-Wolfson Foundation refurbishment scheme is also grate-fully acknowledged.

Appendix A: Comparison between the SAFT-γ and heteroGC-SAFT-VR treatment of the chain term

Within a SAFT treatment the contribution of the for-mation of molecular chains from monomeric segmentsis determined from a knowledge of the radial distribu-tion function of the reference monomeric fluid at con-tact, based on Wertheim’s first-order thermodynamicperturbation theory (TPT1)47–50. Such a treatment israther straightforward for homonuclear models formedfrom tangent segments, but, in the case of heteronuclearmolecular models formed from fused segments, an alter-native treatment is required since the monomeric seg-ments forming the molecular chains are bonded at differ-ent distances.

Two different approaches have been suggested for the

21

formulation of the chain contribution within a SAFT-type theory based on a fused heteronuclear molecularmodel. In SAFT-γ an effective TPT1 treatment is em-ployed, where the following relation is used for the cal-culation of the chain term:

Achain

NkBT= −

NC∑i=1

xi

NG∑k=1

(νk,iν∗kSk − 1) ln gii(σii) , (A1)

The radial distribution function of the reference systemof monomers at contact, gii(σii), is evaluated at an av-erage molecular distance obtained as a weighted averageover the segment sizes σkl of the constituent monomericsegments of the molecular chain66,67. A different for-mulation was proposed in the development of the heteroGC-SAFT-VR63,70, and was subsequently employed inthe heterosegmented PC-SAFT71, where the Helmholtzfree energy change due to chain formation is obtainedbased on an appropriate summation of the monomericradial distribution function at the different intersegmentcontacts σkl as

Achain

NkBT= −

NC∑i=1

xi

NG∑k=1

[νk,i (ν∗kSk − 1) ln gkk(σkk)

+

NG∑l=k

bikl ln gkl(σkl)

], (A2)

where bikl denotes the number of bonds between seg-ments k and l on molecule i. It is important to notethat both expressions revert to the original treatmentof the chain term in the case of a homonuclear chain.Eqs. (A1) and (A2) are written in a general form in termsof the contact value of the radial distribution function;for a given intermolecular potential the RDF can be de-termined from Barker-Henderson perturbation theory asshown for the square-well potential in ref. [66, 70] and forthe Mie potential in section III C.Here, we examine the performance of the two method-

ologies presented for the chain term based on compari-son with simulation data for square-well and hard-sphereheteronuclear diatomic and triatomic molecules. Thoughour main interest is in models based on the Mie potential;to our knowledge there are no suitable data for heteronu-clear molecules formed from Mie segments. Comparisonsof the theoretical treatment with simulation data for het-eronuclear SW molecules should however provide a goodindication of the adequacy of the approximations madein the generic chain term. A first comparison is shown infigure 20, where calculations with the SAFT-γ66,67 andthe hetero GC-SAFT-VR63,70 treatments of the chainterm are compared to the corresponding simulation datafor several pressure-density isotherms of a heteronuclearsquare-well dimer with a fixed segment size ratio of σ11

= 0.5σ22. It should be noted that the only differencebetween the SAFT-γ SW and the hetero GC-SAFT-VR

calculations shown in the plot is in the treatment of thechain term; for the monomer contribution the SAFT-γtheory66,67 is used, neglecting the minor differences be-tween the two theories, as discussed in detail in refs. [24and 28]. In the remainder of the paper, this “hybrid” the-ory is referred to as SAFT-γ∗ for brevity. From the figureit is apparent that both approaches accurately capturethe effect of the heteronuclear nature of the molecule.

It is of further interest to examine how the two ap-proaches perform for heteronuclear dimers for differentsegment size ratios. Simulation data for pressure-densityisotherms for purely repulsive heteronuclear hard-spheredimers and compared with the theoretical descriptionin figure 21. The calculations are performed using theSAFT-γ approach and discarding the contributions aris-ing from attractions between monomeric segments. Fromthe comparison, it appears that the predictions of thetwo methodologies in the description of the thermody-namic properties of the heteronuclear hard-sphere dimerfor similar segment sizes (i.e., high values of the σ11/σ22

ratio) are almost indistinguishable. As the segments be-come more different in size, the SAFT-γ∗ description is inbetter agreement with the simulation data, compared tothe SAFT-γ calculations. However, this is of little conse-quence in practice as such differences in the segment size(σ11/σ22 ≈ 0.25) would be rather unrealistic when repre-senting real compounds, especially given the extension ofSAFT-γ to treat large chemical groups by means of multi-ple identical segments67. The smallest segment size ratioreported in the published SAFT-γ parameter table67 isseen between the hydroxyl OH and carbonyl C=O func-tional groups with a value of σC=O/σOH ≈ 0.59.

As a final example we consider heteronuclear hard-sphere trimers. A comparison between the SAFT-γ andSAFT-γ∗ calculations and the corresponding simulationdata for the pressure-density isotherms of trimers withdifferent segment ratios and/or connectivity is shown infigure 22. It should be noted that in this case, the chaincontribution in SAFT-γ∗ is calculated by explicitly ac-counting for the connectivity of the monomeric segmentsto form the molecular chain, while in SAFT-γ the onlystructural information is the number of occurrences ofeach monomeric segment. This difference in the descrip-tion of the molecular chain term, however, is not found toaffect the agreement of the theoretical description withthe simulation data to a great extent. From this analysisit can be concluded that both theories provide a sat-isfactory description of the thermodynamic behavior ofthe heteronuclear trimers considered, with the SAFT-γ∗

approach performing marginally better. The calculationof the chain contribution within SAFT-γ successfully re-produces the thermodynamic properties of heteronuclearmolecules, and in that way the heteronuclear nature ofthe molecular model is retained within the theory at thelevel of the formation of the molecular chain from dis-tinct monomers. The SAFT-γ treatment is shown tobe almost indistinguishable from a SAFT-γ∗ treatment,apart from cases of extreme differences in size of the seg-

22

ments forming the chain. However, such extreme valuesare of purely theoretical interest, as highly asymmetricmolecular models are not expected to be appropriate inthe modeling of real fluids. Nevertheless, it has to beacknowledged that some isomers are treated differentlywhen eq. (A2) is used to calculate the chain Helmholtzfree-energy contribution, whereas their properties wouldbe the same when eq. (A1) is used.The comparison made thus far between SAFT-γ and

SAFT-γ∗ is based on simulation data for tangentiallybonded heteronuclear models. A full comparison forfused segments is more difficult as a relation betweenthe degree of overlap and the bond length has to be es-tablished. However, an assessment of the two approachescan be examining the description of the properties of realcompounds. This is undertaken here for the chemicalfamily of n-alkanes, where the description of the SAFT-γSW approach with the parameters from ref [66] is com-pared against the results obtained with SAFT-γ∗. Theparameters for the CH3 and CH2 groups with SAFT-γ∗

are obtained here using the same approach as in the workof Lymperiadis et al.66,67, with the same experimentaldata. Regarding the parameters, the shape factors forboth the CH3 and the CH2 groups are assigned valuesof 1/3 and 2/3, respectively, while the only unlike pa-rameter regressed is the energy of the unlike interaction,ϵCH3−CH2 , as opposed to the original work with the het-ero GC-SAFT-VR70 where the range of the unlike inter-action λCH3−CH2 is treated as an additional adjustableparameter. The parameters obtained with SAFT-γ∗ arelisted in table IX, and the deviations for the vapor pres-sure and saturated liquid density are summarized in ta-ble X. From these results it is clear that the SAFT-γtreatment of the chain term leads to a better descriptionof the thermodynamic properties of real compounds witha fused heteronuclear molecular model compared to thedescription obtained within the same theory, but follow-ing the hetero GC-SAFT-VR treatment of the chain term(cf. eq (A2)).

1R. Dohrn and O. Pfohl, Fluid Phase Equilib. 194-197, 15(2002).

2W. B. Whiting, J. Chem. Eng. Data 41, 935 (1996).3S. Gupta and J. D. Olson, Ind. Eng. Chem. Res. 42, 6359 (2003).4L. E. K. Achenie, R. Gani, and V. Venkatasubramanian, Com-puter Aided Molecular Design: Theory and Practice, 1st ed.(Elsevier Science B.V., Amsterdam, The Netherlands, 2003).

5F. E. Pereira, E. Keskes, A. Galindo, G. Jackson, and C. S. Ad-jiman, “Chapter 8: Integrated design of CO2 capture processesfrom natural gas,” in Process Systems Engineering Vol. 5: En-ergy Systems Engineering, edited by M. C. Georgiadis, E. S.Kikkinides, and E. N. Pistikopoulos (Wiley-VCH, Weinheim,2008) pp. 231–248.

6A. Bardow, K. Steur, and J. Gross, Ind. Eng. Chem. Res. 49,2834 (2010).

7F. E. Pereira, E. Keskes, A. Galindo, G. Jackson, and C. S.Adjiman, Comput. Chem. Eng. 35, 474 (2011).

8A. L. Lydersen, University of Wisconsin College Engineering,Eng. Exp. Stn. Rep. 3, Madison, WI (1955).

9K. G. Joback and R. C. Reid, Chem. Eng. Commun. 57, 233(1987).

10L. Constantinou and R. Gani, AIChE J. 40, 1697 (1994).

11J. Marrero and R. Gani, Fluid Phase Equilib. 183-184, 183(2001).

12A. Fredenslund, R. L. Jones, and J. M. Prausnitz, AIChE J.21, 1086 (1975).

13U. Weidlich and J. Gmehling, Ind. Eng. Chem. Res. 26, 1372(1987).

14A. Klamt and F. Eckert, Fluid Phase Equilib. 172, 43 (2000).15T. Holderbaum and J. Gmehling, Fluid Phase Equilib. 70, 251(1991).

16S. Skjold-Jørgensen, Fluid Phase Equilib. 16, 317 (1984).17H. P. Gros, S. Bottini, and E. A. Brignole, Fluid Phase Equilib.116, 537 (1996).

18J. Ahlers and J. Gmehling, Fluid Phase Equilib. 191, 177(2001).

19J. Ahlers and J. Gmehling, Ind. Eng. Chem. Res. 41, 3489(2002).

20J. Ahlers and J. Gmehling, Ind. Eng. Chem. Res. 41, 5890(2002).

21W. G. Chapman, K. E. Gubbins, G. Jackson, and M. Radosz,Fluid Phase Equilib. 52, 31 (1989).

22W. G. Chapman, K. E. Gubbins, G. Jackson, and M. Radosz,Ind. Eng. Chem. Res. 29, 1709 (1990).

23B. E. Poling, J. M. Prausnitz, and J. P. O’Connell, The prop-erties of gases and liquids, 5th ed. (Mc Graw Hill, 2007).

24V. Papaioannou, C. S. Adjiman, A. Galindo, and G. Jack-son, “Chapter 4: Group contribution methodologies for theprediction of thermodynamic properties and phase behavior inmixtures,” in Process Systems Engineering Vol. 6: MolecularSystems Engineering, edited by C. S. Adjiman and A. Galindo(Wiley-VCH, Weinheim, 2010) pp. 135–173.

25E. A. Muller and K. E. Gubbins, Ind. Eng. Chem. Res. 40, 2193(2001).

26I. G. Economou, Ind. Eng. Chem. Res. 41, 953 (2002).27S. P. Tan, H. Adidharma, and M. Radosz, Ind. Eng. Chem.Res. 47, 8063 (2008).

28C. McCabe and A. Galindo, “Chapter 8: SAFT Associating flu-ids and fluid mixtures,” in Applied Thermodynamics of Fluids,edited by A. R. H. Goodwin, J. V. Sengers, and C. J. Peters(Royal Society of Chemistry, UK, 2010) pp. 215–279.

29M. Lora, F. Rindfleisch, and M. A. McHugh, J. Appl. Polym.Sci. 73, 1979 (1999).

30J. Vijande, M. M. Pineiro, D. Bessieres, H. Saint-Guirons, andJ. L. Legido, Phys. Chem. Chem. Phys. 6, 766 (2004).

31S. Tamouza, J. P. Passarello, P. Tobaly, and J. C.de Hemptinne, Fluid Phase Equilib. 222-223, 67 (2004).

32S. Tamouza, J. P. Passarello, P. Tobaly, and J. C.de Hemptinne, Fluid Phase Equilib. 228-229, 409 (2005).

33K. E. Gubbins and C. H. Twu, Chem. Eng. Sci. 33, 863 (1978).34T. X. N. Thi, S. Tamouza, P. Tobaly, J. P. Passarello, and J. C.de Hemptinne, Fluid Phase Equilib. 238, 254 (2005).

35D. Nguyen-Huynh, M. Benamira, J. P. Passarello, P. Tobaly,and J. C. de Hemptinne, Fluid Phase Equilib. 254, 60 (2007).

36D. Nguyen-Huynh, J. C. de Hemptinne, R. Lugo, J. P. Pas-sarello, and P. Tobaly, Ind. Eng. Chem. Res. 50, 7467 (2011).

37J. Rozmus, J. C. de Hemptinne, and P. Mougin, Fluid PhaseEquilib. 303, 15 (2011).

38J. Rozmus, J. C. de Hemptinne, N. Ferrando, and P. Mougin,Fluid Phase Equilib. 329, 78 (2012).

39J. Vijande, M. M. Pineiro, J. L. Legido, and D. Bessieres, Ind.Eng. Chem. Res. 49, 9394 (2010).

40A. Tihic, G. M. Kontogeorgis, N. von Solms, M. L. Michelsen,and L. Constantinou, Ind. Eng. Chem. Res. 47, 5092 (2008).

41A. Tihic, N. von Solms, M. L. Michelsen, G. M. Kontogeorgis,and L. Constantinou, Fluid Phase Equilib. 281, 60 (2009).

42C. Le Thi, S. Tamouza, J. P. Passarello, P. Tobaly, and J. C.de Hemptinne, Ind. Eng. Chem. Res. 45, 6803 (2006).

43A. J. Haslam, A. Galindo, and G. Jackson, Fluid Phase Equilib.266, 105 (2008).

44D. Nguyen-Huynh, J. P. Passarello, P. Tobaly, and J. C.de Hemptinne, Ind. Eng. Chem. Res. 47, 8847 (2008).

23

45D. Nguyen-Huynh, T. K. S. Tran, S. Tamouza, J. P. Passarello,P. Tobaly, and J. C. de Hemptinne, Ind. Eng. Chem. Res. 47,8859 (2008).

46M. Mourah, D. Nguyen-Huyhn, J. P. Passarello, J. C.de Hemptinne, and P. Tobaly, Fluid Phase Equilib. 298, 154(2010).

47M. S. Wertheim, J. Stat. Phys. 35, 19 (1984).48M. S. Wertheim, J. Stat. Phys. 35, 35 (1984).49M. S. Wertheim, J. Stat. Phys. 42, 459 (1986).50M. S. Wertheim, J. Stat. Phys. 42, 477 (1986).51M. S. Wertheim, J. Chem. Phys. 87, 7323 (1987).52W. G. Chapman, G. Jackson, and K. E. Gubbins, Mol. Phys.65, 1057 (1988).

53G. Jackson, W. G. Chapman, and K. E. Gubbins, Mol. Phys.65, 1 (1988).

54A. L. Archer and G. Jackson, Mol. Phys. 73, 881 (1991).55M. D. Amos and G. Jackson, Mol. Phys. 74, 191 (1991).56R. P. Sear, M. D. Amos, and G. Jackson, Mol. Phys. 80, 777(1993).

57A. L. Archer, M. D. Amos, G. Jackson, and I. A. McLure, Int.J. Thermophys. 17, 201 (1996).

58M. Banaszak, C. K. Chen, and M. Radosz, Macromolecules 29,6481 (1996).

59M. Banaszak and M. Radosz, Fluid Phase Equilib. 193, 179(2002).

60H. Adidharma and M. Radosz, Ind. Eng. Chem. Res. 37, 4453(1998).

61H. Adidharma and M. Radosz, Fluid Phase Equilib. 158-160,165 (1999).

62C. McCabe, A. Gil-Villegas, G. Jackson, and F. del Rıo, Mol.Phys. 97, 551 (1999).

63Y. Peng, H. Zhao, and C. McCabe, Mol. Phys. 104, 571 (2006).64F. J. Blas and L. F. Vega, Mol. Phys. 92, 135 (1997).65J. Gross, O. Spuhl, F. Tumakaka, and G. Sadowski, Ind. Eng.Chem. Res. 42, 1266 (2003).

66A. Lymperiadis, C. S. Adjiman, A. Galindo, and G. Jackson,J. Chem. Phys. 127, 234903 (2007).

67A. Lymperiadis, C. S. Adjiman, G. Jackson, and A. Galindo,Fluid Phase Equilib. 274, 85 (2008).

68V. Papaioannou, C. S. Adjiman, G. Jackson, and A. Galindo,Fluid Phase Equilib. 306, 82 (2011).

69S. S. Ashrafmansouri and S. Raeissi, J. Supercrit. Fluids 63, 81(2012).

70Y. Peng, K. D. Goff, M. C. dos Ramos, and C. McCabe, FluidPhase Equilib. 277, 131 (2009).

71K. Paduszynski and U. Domanska, Ind. Eng. Chem. Res. 51,12967 (2012).

72T. Lafitte, D. Bessieres, M. M. Pineiro, and J. L. Daridon, J.Chem. Phys. 124, 024509 (2006).

73T. Lafitte, A. Apostolakou, C. Avendano, A. Galindo, C. S.Adjiman, E. A. Muller, and G. Jackson, J. Chem. Phys. 139,154504 (2013).

74G. C. Maitland, M. Rigby, and E. B. Smith, Intermolecularforces: their origin and determination (Clarendon Press Ox-ford, 1981).

75J. Gregorowicz, J. P. O’Connell, and C. J. Peters, Fluid PhaseEquilib. 116, 94 (1996).

76I. Polishuk, Ind. Eng. Chem. Res. 50, 4183 (2011).77X. Liang, B. Maribo-Mogensen, K. Thomsen, W. Yan, andG. M. Kontogeorgis, Ind. Eng. Chem. Res. 51, 14906 (2012).

78M. R. Faradonbeh, J. Abedi, and T. G. Harding, Can. J. Chem.Eng. 91, 101 (2013).

79C. M. Colina, L. F. Turrens, K. E. Gubbins, C. Oliveira-Fuentes,and L. F. Vega, Ind. Eng. Chem. Res. 41, 1069 (2002).

80F. J. Blas and L. F. Vega, Ind. Eng. Chem. Res. 37, 660 (1998).81F. Castro-Marcano, C. G. Olivera-Fuentes, and C. M. Colina,Ind. Eng. Chem. Res. 47, 8894 (2008).

82F. Llovell and L. F. Vega, J. Phys. Chem. B 110, 11427 (2006).83F. Llovell, C. J. Peters, and L. F. Vega, Fluid Phase Equilib.248, 115 (2006).

84N. I. Diamantonis and I. G. Economou, Energy Fuels 25, 3334(2011).

85J. Chen and J.-G. Mi, Fluid Phase Equilib. 186, 165 (2001).86A. Maghari and M. S. Sadeghi, Fluid Phase Equilib. 252, 152(2007).

87A. Maghari and M. Hamzehloo, Fluid Phase Equilib. 302, 195(2011).

88F. Llovell and L. F. Vega, J. Supercrit. Fluids 41, 204 (2007).89A. M. A. Dias, F. Llovell, J. A. P. Coutinho, I. M. Marrucho,and L. F. Vega, Fluid Phase Equilib. 286, 134 (2009).

90O. Vilaseca, F. Llovell, J. Yustos, R. M. Marcos, and L. F.Vega, J. Supercrit. Fluids 55, 755 (2010).

91E. Forte, F. Llovell, J. P. M. Trusler, and A. Galindo, FluidPhase Equilib. 337, 274 (2013).

92M. S. Shin, Y. Lee, and H. Kim, J. Chem. Thermodyn. 40, 688(2008).

93Y. Lee, M. S. Shin, and H. Kim, J. Chem. Thermodyn. 40,1580 (2008).

94A. J. de Villiers, C. E. Schwarz, A. J. Burger, and G. M. Kon-togeorgis, Fluid Phase Equilib. 338, 1 (2013).

95T. Lafitte, M. M. Pineiro, J. L. Daridon, and D. Bessieres, J.Phys. Chem. B 111, 3447 (2007).

96L. A. Davies, A. Gil-Villegas, and G. Jackson, Int. J. Thermo-phys. 19, 675 (1998).

97G. Mie, Ann. Phys. 316, 657 (1903).98E. A. Gruneisen, Z. Elektrochem. Angew. Phys. Chem. 17, 737(1911).

99E. A. Gruneisen, Ann. Phys. (Berlin) 344, 257 (1912).100J. E. Jones, Proc. Roy. Soc. London A 106, 441 (1924).101J. E. Jones, Proc. Roy. Soc. London A 106, 463 (1924).102J. E. Lennard-Jones, Proc. Phys. Soc. 43, 461 (1931).103L. L. Lee, Molecular thermodynamics of nonideal fluids (But-

terworths, Boston, 1988).104J. A. Barker and D. Henderson, Rev. Mod. Phys. 48, 587 (1976).105J. A. Barker and D. Henderson, J. Chem. Phys. 47, 4714 (1967).106P. Paricaud, J. Chem. Phys. 124, 154505 (2006).107T. Boublık, J. Chem. Phys. 53, 471 (1970).108G. A. Mansoori, N. F. Carnahan, K. E. Starling, and T. W.

Leland, J. Chem. Phys. 54, 1523 (1971).109A. Gil-Villegas, A. Galindo, P. J. Whitehead, S. J. Mills,

G. Jackson, and A. N. Burgess, J. Chem. Phys. 106, 4168(1997).

110A. Galindo, L. A. Davies, A. Gil-Villegas, and G. Jackson, Mol.Phys. 93, 241 (1998).

111B.-J. Zhang, Fluid Phase Equilib. 154, 1 (1999).112N. F. Carnahan and K. E. Starling, J. Chem. Phys. 51, 635

(1969).113T. Boublık, Mol. Phys. 59, 775 (1986).114S. Dufal, T. Lafitte, A. J. Haslam, A. Galindo, and G. Jackson,

Mol. Phys. (2013), (submitted).115J. S. Rowlinson and F. L. Swinton, Liquids and liquid mixtures,

3rd ed. (Butterworths, 1982).116K. E. Bett, J. S. Rowlinson, and G. Saville, Thermodynamics

for chemical engineers (MIT Press, 1975).117F. E. Pereira, G. Jackson, A. Galindo, and C. S. Adjiman, Fluid

Phase Equilib. 299, 1 (2010).118PSE Ltd., gPROMS v. 3.4.0 (2011),

http://www.psenterprise.com/.119S. H. Huang and M. Radosz, Ind. Eng. Chem. Res. 29, 2284

(1990).120C. McCabe and S. B. Kiselev, Ind. Eng. Chem. Res. 43, 2839

(2004).121E. Forte, F. Llovell, L. F. Vega, J. P. M. Trusler, and

A. Galindo, J. Chem. Phys. 134, 154102 (2011).122F. London, Trans. Faraday Soc. 33, 8 (1937).123J. J. Potoff and D. A. Bernard-Brunel, J. Phys. Chem. B 113,

14725 (2009).124G. Jackson and K. E. Gubbins, Pure Appl. Chem. 61, 1021

(1989).125C. McCabe and G. Jackson, Phys. Chem. Chem. Phys. 1, 2057

24

(1999).126P. Paricaud, A. Galindo, and G. Jackson, Fluid Phase Equilib.

194-197, 87 (2002).127J. Gmehling, J. Li, and M. Schiller, Ind. Eng. Chem. Res. 32,

178 (1993).128M. Kleiner and G. Sadowski, J. Phys. Chem. C 111, 15544

(2007).129J. Gross and G. Sadowski, Ind. Eng. Chem. Res. 40, 1244

(2001).130E. W. Lemmon, M. L. Hubert, and M. O. McLinden, “NIST

standard reference database 23: Reference fluid thermodynamicand transport properties - REFPROP, version 9.0,” NationalInstitute of Standards and Technology, Standard ReferenceData Program, Gaithersburg (2010).

131C.-S. Wu and Y.-P. Chen, Fluid Phase Equilib. 100, 103 (1994).132S. J. Han, C. J. Cregg, and M. Radosz, Ind. Eng. Chem. Res.

36, 5520 (1997).133J. Gross and G. Sadowski, Ind. Eng. Chem. Res. 41, 1084

(2002).134P. Paricaud, A. Galindo, and G. Jackson, Ind. Eng. Chem. Res.

43, 6871 (2004).135A. R. Shultz and P. J. Flory, J. Am. Chem. Soc. 74, 4760 (1952).136C. E. Schwarz and I. Nieuwoudt, J. Supercrit. Fluids 27, 145

(2003).137R. K. Surana, R. P. Danner, A. B. de Haan, and N. Beckers,

Fluid Phase Equilib. 139, 361 (1997).138P. J. Flory, R. A. Orwoll, and A. Vrij, J. Am. Chem. Soc. 86,

3507 (1964).139P. J. Flory, R. A. Orwoll, and A. Vrij, J. Am. Chem. Soc. 86,

3515 (1964).140R. A. Orwoll and P. J. Flory, J. Am. Chem. Soc. 89, 6822 (1967).141F. J. Blas and I. Fujihara, Mol. Phys. 100, 2823 (2002).142F. J. Blas, Mol. Phys. 100, 2221 (2002).143M. C. dos Ramos and F. J. Blas, J. Phys. Chem. B 109, 12145

(2005).144A. Tourino, M. Hervello, V. Moreno, M. Iglesias, and

G. Marino, Phys. Chem. Liq. 42, 37 (2004).145M. F. Bolotnikov, Y. A. Neruchev, Y. F. Melikhov, V. N. Ver-

veyko, and M. V. Verveyko, J. Chem. Eng. Data 50, 1095(2005).

146E. P. J. Linstrom and W. G. Mallard, NIST Chemistry Web-Book, NIST Standard Reference Database Number 69 (NationalInstitutes of Standards and Technology WebBook, GaithersburgMD, 20899) http://webbook.nist.gov/, (retrieved March 2013).

147N. B. Vargaftik, Handbook of thermophysical properties of gasesand fluids (Nauka, Moscow, 1972) (cited in DETHERM).

148E. A. Macedo and P. Rasmussen, J. Chem. Eng. Data 27, 463(1982).

149R. L. Gardas, I. Johnson, D. M. D. Vaz, I. M. A. Fonseca, andA. G. M. Ferreira, J. Chem. Eng. Data 52, 737 (2007).

150S. Young and G. L. Thomas, J. Chem. Soc., Trans. 63, 1191(1893).

151E. Lladosa, J. B. Monton, M. C. Burguet, and R. Munoz, J.Chem. Eng. Data 53, 108 (2008).

152G. Liessmann, W. Schmidt, and S. Reiffarth, Data compilationof the Saechsische Olefinwerke Boehlen Germany (1995), (citedin DETHERM).

153W. V. Steele, R. D. Chirico, S. E. Knipmeyer, and A. Nguyen,J. Chem. Eng. Data 41, 1255 (1996).

154E. L. Krasnykh, S. P. Verevkin, B. Koutek, and J. Doubsky, J.Chem. Thermodynamics 38, 717 (2006).

155E. F. Meyer, M. J. Awe, and R. E. Wagner, J. Chem. Eng.Data 25, 371 (1980).

156S. L. Oswal, P. Oswal, P. S. Modi, J. P. Dave, and R. L. Gardas,Thermochim. Acta 410, 1 (2004).

157S. Salerno, M. Cascella, D. May, P. Watson, and D. Tassios,Fluid Phase Equilib. 27, 15 (1986).

158C. Viton, M. Chavret, E. Behar, and J. Jose, Int. Electron. J.Phys. Chem. Data 2, 215 (1996), (cited in DETHERM).

159R. D. Chirico, A. Nguyen, W. V. Steele, M. M. Strube, and

C. Tsonopoulos, J. Chem. Eng. Data 34, 149 (1989).160T. Sawaya, I. Mokbel, N. Ainous, E. Rauzy, C. Berro, and

J. Jose, J. Chem. Eng. Data 51, 854 (2006).161B. D. Smith and R. Srivastava, Thermodynamic data for pure

compounds part A: Hydrocarbons and ketones (Elsevier, Ams-terdam, 1986).

162L. Pias, M. I. Paz-Andrade, F. Sarmiento, E. Rodriguez-Nunez,and J. Fernandez, Fluid phase Equilib. 28, 183 (1986).

163L. Fernandez, E. Perez, J. Ortega, J. Canosa, and J. Wisniak,J. Chem. Eng. Data 55, 5519 (2010).

164J.-L. Daridon, H. Carrier, and B. Lagourette, Int. J. Thermo-phys. 33, 697 (2002).

165S. Dutour, J.-L. Daridon, and B. Lagourette, High Temp. HighPressures 23, 371 (2001).

166P. Zoller and D. J. Walsh, Standard pressure-volume-temperature data for polymers (Technomic, Lancaster-Basel,1995).

167M. J. Pratas, S. Freitas, M. B. Oliveira, S. C. Monteiro, A. S.Lima, and J. A. P. Coutinho, J. Chem. Eng. Data 55, 3983(2010).

168E. H. I. Ndiaye, D. Nasri, and J. L. Daridon, J. Chem. Eng.Data 57, 2667 (2012).

169E. H. I. Ndiaye, M. Habrioux, J. A. P. Coutinho, M. L. L. Pare-des, and J. L. Daridon, J. Chem. Eng. Data 58, 1371 (2013).

170H. H. Reamer and B. H. Sage, J. Chem. Eng. Data 9, 24 (1964).171L.-C. Feng, C.-H. Chou, M. Tang, and Y.-P. Chen, J. Chem.

Eng. Data 43, 658 (1998).172J. Ortega, F. Espiau, and F. J. Toledo, J. Chem. Thermody-

namics 36, 193 (2004).173S. L. Oswal, P. Oswal, and J. P. Dave, J. Mol. Liq. 94, 203

(2001).

25

TABLE I. Coefficients ϕm,n for the a2,kl term (eq. (32)), thea3,kl term (eq. (38)), and γc,ii of the g2 term (eq. (59)). N/Adenotes that no value is needed.

n ϕ1,n ϕ2,n ϕ3,n ϕ4,n ϕ5,n ϕ6,n ϕ7,n

0 7.536 555 7 −359.44 1550.9 −1.199 32 −1911.28 9236.9 101 −37.604 63 1825.6 −5070.1 9.063 632 21 390.175 −129 430 102 71.745 953 −3168.0 6534.6 −17.9482 −51 320.7 357 230 0.573 −46.835 52 1884.2 −3288.7 11.340 27 37 064.54 −315 530 −6.74 −2.467 982 −0.823 76 −2.7171 20.521 42 1103.742 1390.2 −85 −0.502 72 −3.1935 2.0883 −56.6377 −3264.61 −4518.2 N/A6 8.095 688 3 3.7090 0 40.536 83 2556.181 4241.6 N/A

TABLE II. Group parameters for the methyl and methylenefunctional groups for the n-alkanes (CH3 and CH2) and thecarboxylate functional group for the n-alkyl esters (COO)within the SAFT-γ Mie group contribution approach.

Functional group k ν∗k Sk λr

kk λakk σkk [A] (ϵkk/kB) [K]

CH3 1 0.5726 15.050 6 4.077 256.766CH2 1 0.2293 19.871 6 4.880 473.389COO 1 0.6526 31.189 6 3.994 868.923

TABLE III. Unlike dispersion interaction energies ϵkl for themethyl and methylene functional groups for the n-alkanes(CH3 and CH2) and the carboxylate functional group for then-alkyl esters (COO) within the SAFT-γ Mie group contri-bution approach.

ϵkl/kB [K] CH3 CH2 COO

CH3 256.766 350.772 402.751CH2 350.772 473.389 498.856COO 402.751 498.856 868.923

26

TABLE

IV.Percentageav

erageabsolute

deviations(%

AAD)fortheva

porpressurespvap(T

),thesaturatedliquid

den

sities

ρsa

t(T

),andthesingle-phase

den

sities

ρliq(T

,p)forthen-alkanes

obtained

withtheSAFT-γ

Mie

groupcontributionapproach

withresp

ectto

thecorrelatedexperim

entaldata

from

NIST

146,wherenis

the

number

ofdata

points.

Compound

Trange[K

]n

%AAD

pvap(T

)T

range[K

]n

%AAD

ρsa

t(T

)T

range[K

]prange[M

Pa]

n%AAD

ρliq(T

,p)

C2H

6125-275

31

2.24

125-275

31

1.48

150-550

10-50

123

0.96

n-C

3H

8147-332

38

2.22

147-332

38

0.74

150-500

10-50

108

0.49

n-C

4H

10

170-385

44

1.27

170-385

44

0.37

150-550

10-50

123

0.50

n-C

5H

12

187-422

48

1.90

187-422

48

0.36

150-550

10-50

120

0.60

n-C

6H

14

201-456

52

1.68

201-456

52

0.27

188-548

10-50

108

0.52

n-C

7H

16

216-486

55

1.01

216-486

55

0.46

193-553

10-50

108

0.62

n-C

8H

18

227-512

58

1.22

227-512

58

0.54

226-546

10-50

95

0.64

n-C

9H

20

237-532

60

0.69

237-532

60

0.59

230-550

10-50

95

0.50

n-C

10H

22

245-555

63

1.75

245-555

63

0.52

253-553

10-50

89

0.47

Average

--

1.55

--

0.59

--

-0.59

27

TABLE

V.Percentageav

erageabsolute

deviations(%

AAD)fortheva

porpressurespvap(T

),thesaturated

liquid

den

sities

ρsa

t(T

),and

thesingle-phase

den

sities

ρliq(T

,p)forthen-alkylesters

obtained

withtheSAFT-γ

Mie

groupcontributionapproach

withresp

ectto

theexperim

entaldata,wheren

isthenumber

ofdata

points.

Trange

%AAD

Trange

%AAD

Trange

prange

%AAD

Compound

[K]

npvap

Ref.

[K]

nρsa

tRef.

[K]

[MPa]

nRef.ρliq(T

,p)

CH

3COOCH

2CH

3307.98-473.15

29

0.37

[147,148]273.15-473.15

21

0.25

[147]

298.15-393.15

1-20

80

[149]

0.43

CH

3COO(C

H2) 2CH

3303.15-493.15

20

0.49

[150]

303.15-493.15

20

0.17

[150]

298.15-393.15

1-20

80

[149]

0.43

CH

3COO(C

H2) 3CH

3334.36-399.05

27

0.73

[151]

298.15-523.15

25

0.42

[149,152]298.15-393.15

1-20

80

[149]

0.19

CH

3COO(C

H2) 4CH

3321.369-462.33

23

0.85

[153]

298.15-393.15

20

0.18

[149]

298.15-393.15

1-20

80

[149]

0.19

CH

3COO(C

H2) 5CH

3274.5

-459.14

33

1.86

[154]

273.27-431.07

18

0.24

[155]

--

--

-CH

3COO(C

H2) 6CH

3274.5

-478.14

30

0.67

[154,155]273.27-428.15

19

0.20

[155]

--

--

-

Average

--

0.83

--

-0.24

--

--

-0.31

28

TABLE

VI.

Comparisonofthepercentageav

erageabsolute

deviations(%

AAD)fortheisobaricheatcapacity

c p(T

,p),

theisoch

oricheatcapacity

c V(T

,p),

speed

ofsoundu(T

,p),

isothermalcompressibilitykT(T

,p),

and(volume)

thermalexpansionco

efficien

tαV(T

,p)forthen-alkanes

obtained

withtheSAFT-γ

Mie

andthe

SAFT-γ

SW

66groupcontributionapproach

esforthecorrelatedexperim

entaldata

from

REFPROP

130,wherenis

thenumber

ofdata

points.

%AAD

SAFT-γ

Mie

%AAD

SAFT-γ

SW

Compound

Trange[K

]prange[M

Pa]

nc p(T

,p)c V

(T,p)u(T

,p)kT(T

,p)αV(T

,p)c p(T

,p)c V

(T,p)u(T

,p)kT(T

,p)αV(T

,p)

C2H

6150-600

10-50

138

3.18

3.76

2.49

3.53

3.50

7.86

10.3

7.42

8.27

9.78

n-C

3H

8150-600

10-50

138

1.76

0.79

1.31

2.70

3.31

6.98

8.43

4.16

9.65

9.32

n-C

4H

10

150-570

10-50

129

1.98

1.92

0.78

1.84

3.70

7.96

11.06

4.48

12.4

10.45

n-C

5H

12

150-600

10-50

138

1.83

1.40

1.21

3.16

5.20

7.74

10.3

5.37

15.71

11.21

n-C

6H

14

190-600

10-50

124

0.96

1.31

0.82

3.36

5.91

5.33

7.48

4.62

11.55

10.12

n-C

7H

16

190-600

10-50

122

0.96

1.54

1.47

4.07

6.44

5.75

8.11

5.93

14.18

10.23

n-C

8H

18

230-600

10-50

111

0.59

2.19

0.92

3.45

7.33

4.73

7.51

5.29

12.07

10.61

n-C

9H

20

230-600

10-50

110

0.47

1.23

2.08

5.47

6.32

4.83

6.57

6.15

14.56

9.22

n-C

10H

22

260-670

10-50

123

0.42

1.73

2.26

6.25

7.74

3.89

5.56

5.64

12.81

10.57

Average

--

-1.35

1.76

1.48

3.76

5.49

6.12

8.37

5.45

12.36

10.17

29

TABLE VII. Percentage average absolute deviations (%AAD)for the isobaric heat capacity cp(T, p), the isochoric heat ca-pacity cV(T, p), speed of sound u(T, p), isothermal compress-ibility kT(T, p), and isentropic compressibility kS(T, p) for se-lected n-alkyl acetates obtained with the SAFT-γ Mie groupcontribution approach for the experimental data at ambientpressure156, where n is the number of data points.

%AAD SAFT-γ MieCompound T range [K] n cp(T, p) cV(T, p) u(T, p) kT(T, p) kS(T, p)

CH3COOCH2CH3 298.15 - 328.15 7 5.62 2.21 9.89 14.56 17.32CH3COO(CH2)2CH3 298.15 - 328.15 7 4.53 1.19 7.12 10.11 13.08CH3COO(CH2)3CH3 298.15 - 328.15 7 2.18 1.80 4.96 9.16 9.52CH3COO(CH2)4CH3 298.15 - 328.15 7 0.76 0.68 3.88 6.34 7.54CH3COO(CH2)9CH3 298.15 - 328.15 7 1.23 3.32 1.01 0.38 1.78

Average - - 2.86 1.84 5.37 8.11 9.85

TABLE VIII. Percentage average absolute deviations(%AAD) for the vapor pressures pvap(T ) the and saturatedliquid densities ρsat(T ) of the predictions with the SAFT-γ Mie group contribution approach from the experimentaldata (where n is the number of data points) for long-chainn-alkanes and n-alkyl acetates not included in the estimationof the group parameters.

Compound T range [K] n %AAD pvap(T ) Ref. T range [K] n %AAD ρsat(T ) Ref.

n-C15H32 293-576 35 7.19 [157,158] 273-633 11 0.71 [152]n-C20H42 388-625 29 15.66 [159] 293-683 11 0.98 [152]n-C25H52 381-461 13 29.87 [160] 293-695 11 3.10 [152]n-C30H62 432-452 5 37.50 [160] 293-727 11 4.01 [152]CH3COO(CH2)8CH3 276.7 - 309.2 12 1.37 [154] - - - -CH3COO(CH2)9CH3 284.4 - 530.0 32 2.94 [154,155] - - - -CH3COO(CH2)10CH3 288.9 - 329.4 14 2.31 [154] - - - -CH3COO(CH2)11CH3 288.9 - 333.4 15 4.46 [154] - - - -CH3COO(CH2)13CH3 303.1-340.1 13 2.95 [154] - - - -

Average - - 11.58 - - - 2.20 -

TABLE IX. Group parameters for the methyl and methylenefunctional groups (CH3 and CH2) of the n-alkanes within theSAFT-γ∗ SW approach. The optimal value of the energy ofthe unlike interaction was found to be ϵCH3−CH2/kB = 240.615K.

Functional group k ν∗k Sk λkk σkk [A] (ϵkk/kB) [K]

CH3 1 0.6667 1.433 3.806 244.628CH2 1 0.3333 1.714 3.996 217.741

30

TABLE X. Percentage average absolute deviations (%AAD)for the vapor pressures pvap(T ) and the saturated liquid densi-ties ρsat(T ) for the n-alkanes obtained with the SAFT-γ∗ SWapproach from the experimental data161, compared to the de-scription with the SAFT-γ SW approach66. n is the numberof experimental data points for each property.

SAFT-γ SW SAFT-γ∗ SW SAFT-γ SW SAFT-γ∗ SWCompound T range [K] n %AAD pvap(T ) %AAD pvap(T ) %AAD ρsat(T ) %AAD ρsat(T )

C2H6 91-270 38 3.93 2.89 1.02 1.89n-C3H8 97-331 38 4.09 6.46 0.65 1.80n-C4H10 139-379 38 3.44 4.81 0.55 2.06n-C5H12 144-417 38 5.39 7.08 0.54 1.56n-C6H14 177-455 38 3.60 4.05 0.47 1.46n-C7H16 191-483 37 4.09 5.08 0.51 1.46n-C8H18 216-512 38 3.69 4.68 0.45 1.47n-C9H20 344-424 15 3.01 6.19 0.40 0.97n-C10H22 243-553 63 4.63 5.22 0.52 1.38

Average - - 3.98 5.16 0.57 1.56

31

LIST OF FIGURES

FIG. 1. Representation of the fused heteronuclear molecu-lar model employed within the SAFT-γ Mie approach. Theexample depicted is for n-hexane, comprising two instancesof the methyl CH3 group (highlighted in grey), and four in-stances of the methylene CH2 group (highlighted in red).

0 5 10 15 20

ρsat

/ mol dm-3

100

200

300

400

500

600

700

T /

K100 200 300 400 500 600 700

T / K

10-7

10-5

10-3

10-1

101

p vap /

MP

a

(a)

(b)

FIG. 2. The description with the SAFT-γ Mie group contri-bution approach of the n-alkanes included in the estimationof the methyl CH3 and methylene CH2 parameters: (a) thecoexistence densities ρsat (n-ethane to n-decane from bottomto top); and (b) the vapor pressures pvap in a logarithmicrepresentation (n-ethane to n-decane from left to right). Theopen symbols represent the experimental data, the filled sym-bols the corresponding experimental critical points from theNIST146 database, and the continuous curves the calculationswith the theory.

32

0 5 10 15

ρsat

/ mol dm-3

200

300

400

500

600T

/ K

200 300 400 500 600T / K

0

1

2

3

4

5

6

p vap /

MP

a

(a)

(b)

FIG. 3. Comparison of the description with the SAFT-γSW66 (dashed curves) and SAFT-γ Mie (solid curves) groupcontribution approaches for the pure-component vapor-liquidequilibria of n-butane, n-hexane, and n-octane: (a) coexis-tence densities ρsat (from bottom to top); and (b) vapor pres-sures pvap (from left to right). The open symbols representthe experimental data and the filled symbols the correspond-ing experimental critical points from the NIST146 database.

0 5 10 15

ρsat

/ mol dm-3

200

300

400

500

600

700

T /

K200 300 400 500 600 700

T / K

10-7

10-5

10-3

10-1

101

p vap /

MP

a

(a)

(b)

FIG. 4. The description with the SAFT-γ Mie group con-tribution approach of the n-alkyl esters included in the esti-mation of the carboxylate COO group parameters: (a) thecoexistence densities ρsat (ethyl acetate to n-heptyl acetatefrom bottom to top); and (b) the vapor pressures pvap in alogarithmic representation (ethyl acetate to n-heptyl acetatefrom left to right) . The symbols represent experimental dataand the continuous curves the calculations with the theory.

33

0.00 0.20 0.40 0.60 0.80 1.00xacetate

0

500

1000

1500

hE / J

mol

-1

ethyl acetate + n-hexanen-butyl acetate + n-octaneSAFT-γ Miemod (Do) UNIFAC

FIG. 5. Excess enthalpies of mixing hE as a function of themole fraction x of the n-alkyl acetate of the binary systemsincluded in the estimation of the parameters for the carboxy-late COO group. The triangles are experimental data for thesystem of n-butyl acetate+n-octane at a temperature of T =298.15 K and pressure of p = 0.1 MPa162, and the circles forethyl acetate+n-hexane at T = 318.15 K and p = 0.101325MPa163. The continuous curves are the calculations of theSAFT-γ Mie EoS and the dashed curves are the correspond-ing calculations of the modified UNIFAC (Dortmund)127 ap-proach.

34

300 400 500 600 700T / K

100

400

700

1000

u /

m s

-1

p = 5.0 MPap = 7.5 MPap = 10.0 MPa

500 550 600 650 700T /K

250

750

1250

1750

c p / J

mol

-1 K

-1

p = 2.5 MPap = 3.0 MPap = 5.0 MPa

300 400 500 600 700T / K

0

300

600

900

kT /

GP

a-1

p = 5.0 MPap = 7.5 MPap = 10.0 MPa

300 400 500 600 700T /K

10-3

10-2

10-1

100

α V /

K-1

p = 3.0 MPap = 4.0 MPap = 5.0 MPa

(a)

(c) (d)

(b)

FIG. 6. Prediction of second-order thermodynamic derivative properties of selected n-alkanes with the SAFT-γ Mie groupcontribution approach at various pressures: (a) speed of sound u of n-pentane; (b) isothermal compressibility kT of n-butane;(c) isobaric heat capacity cp of n-decane; and (d) (volumetric) thermal expansion coefficient αV of n-octane, where the symbolsrepresent correlated experimental data from REFPROP130 and the continuous curves the theoretical predictions. The hightemperature data for n-octane shown as filled points in part (d) are based on extrapolations provided by REFPROP that lieoutside the temperature range of the correlations.

35

200 300 400 500 600T / K

50

100

150

200

250

300

350c V

/ J

mol

-1 K

-1

0 2 4 6 8 10

p / MPa

50

60

70

80

90

100

c V /

J m

ol-1

K-1

(a)

(b)

FIG. 7. Comparison of the predictions of the SAFT-γ Miegroup contribution approach and the experimental data forthe isochoric heat capacity of: (a) n-butane (diamonds), n-hexane (squares), and n-octane (circles) as a function of tem-perature at a pressure of p = 0.1 MPa; and (b) n-propane as afunction of pressure at a temperature of T = 200 K (squares),250 K (circles), 300 K (diamonds), 350 K (triangles), and400 K (asterisks). All symbols are correlated experimentaldata130 and the continuous curves are the predictions of thetheory.

36

0 1 2 3 4

ρsat

/ mol dm-3

200

400

600

800

1000T

/ K

n-pentadecanen-eicosanen-pentacosanen-triacontane

200 400 600 800 1000T / K

10-8

10-6

10-4

10-2

100

p vap /

MP

a(a)

(b)

FIG. 8. Comparison of the predictions of the SAFT-γ Miegroup contribution approach and the experimental data forthe pure-component vapor-liquid equilibria of long-chain n-alkanes not included in the estimation of the methyl andmethylene group parameters: (a) saturated liquid densi-ties ρsat and (b) vapor pressures pvap of n-pentadecane (n-C15H32), n-eicosane (n-C20H42), n-pentacosane (n-C25H52),and n-triacontane (n-C30H62).

0 50 100 150 200

p / MPa

0.0

0.5

1.0

1.5

2.0

k T /

GP

a-1

T = 393.15 KT = 373.15 KT = 353.15 KT = 333.15 K

0 50 100 150 200p / MPa

1000

1200

1400

1600

1800

2000

u / m

s-1

T = 313.15 KT = 333.15 KT = 353.15 KT = 373.15 K

(a)

(b)

FIG. 9. Comparison of the SAFT-γ Mie (continuous curves)and the SAFT-γ SW66 (dashed curves) group contributionapproaches in the description of second-order thermodynamicderivative properties of long-chain n-alkanes not included inthe regression of the group parameters: (a) speed of soundu of n-pentadecane at temperatures of T = 313.15 K (cir-cles), 333.15 K (triangles), 353.15 K (squares), 373.15 K(diamonds)164; and (b) isothermal compressibility kT of n-eicosane at 333.15 K (circles), 353.15 K (triangles), 373.15 K(squares), and 393.15 K (diamonds)165.

37

350 400 450 500 550T / K

0.7

0.8

0.9

1.0

ρ liq /

g cm

-3p = 140 MPap = 100 MPap = 60 MPap = 20 MPa

FIG. 10. Comparison of the predictions of the SAFT-γ Mie(continuous curves) and the SAFT-γ SW66 (dashed curves)group contribution approaches with the experimental data166

of the single-phase liquid densities ρliq of linear polyethylene(MW = 126,000 g mol−1) at pressures of p = 20 MPa (cir-cles), 60 MPa (diamonds), 100 MPa (squares), and 140 MPa(triangles).

275 300 325 350 375

T / K

100

200

300

400

500

c p / J

mol

-1 K

n-decyl acetaten-pentyl acetaten-butyl acetaten-propyl acetateethyl acetate

275 300 325 350 375T / K

800

1000

1200

1400

u / m

s-1

n-decyl acetaten-pentyl acetaten-butyl acetaten-propyl acetateethyl acetate

(a)

(b)

FIG. 11. Prediction of the second-order thermodynamicderivative properties of selected n-alkyl acetates obtainedwith the SAFT-γ Mie group contribution approach: (a) speedof sound u, and (b) isobaric heat capacity cp of ethyl acetate(circles), n-propyl acetate (squares), n-butyl acetate (trian-gles), n-pentyl acetate (crosses), and n-decyl acetate (aster-isks) at a pressure of p = 0.1 MPa. All symbols are experi-mental data156 and the continuous curves are the predictionsof the theory.

38

0 50 100 150 200 250 p / MPa

0.0

0.5

1.0

1.5

k T, k

S / G

Pa-1

0 50 100 150 200 250p / MPa

1000

1250

1500

1750

2000

u / m

s-1

250 300 350 400T / K

2

3

4

5

6

ρ liq /

mol

dm

-3

(a)

(c)

(b)

FIG. 12. Predictions of the single phase densities and second-order thermodynamic derivative properties of long-chain ethylesters with the SAFT-γ Mie group contribution approach: (a)single phase densities ρliq of ethyl caprylate (diamonds), ethylcaprate (circles), ethyl laurate (squares), ethyl myristate (tri-angles), and ethyl palmitate (asterisks) at a pressure of p =0.10132 MPa167 (b) speed of sound u of ethyl caprate at tem-peratures of T = 303.15 K (circles), 383.15 K (triangles)168

and of ethyl myristate at temperatures of T = 293.15 K (aster-isks), and 363.15 K (crosses)169; and (c) isothermal compress-ibility kT of ethyl caprate at temperatures of T = 303.15 K(circles), 383.15 K (triangles)168 and isentropic compressibil-ity kS of ethyl myristate at temperatures of T = 293.15 K(asterisks), and 363.15 K (crosses)169. The continuous curvesare the predictions of the theory.

0.0 0.2 0.4 0.6 0.8 1.0xC4H10

, yC4H10

0

2

4

6

8

p / M

Pa

T = 510.93 KT = 477.59 KT = 444.26 KT = 377.59 K

FIG. 13. Pressure-composition (p-x) isothermal slices ofthe fluid-phase behavior (vapor-liquid equilibria) of the n-butane+n-decane binary mixture. The continuous curves rep-resent the predictions with the SAFT-γ Mie group contribu-tion approach, the dashed curves the corresponding predic-tions with SAFT-γ SW66, and the symbols the experimentaldata at temperatures of T = 377.59 K (circles), 444.26 K(diamonds), 477.59 K (squares), and 510.93 K (triangles)170.

39

0.0 0.2 0.4 0.6 0.8 1.0wC60H122

2.5

5.0

7.5

10.0

12.5

15.0

17.5p

/ MP

aT = 408.15 KT = 393.15 KT = 378.15 K

0.00 0.15 0.30 0.45 0.60wC5H12

0

1

2

3

4

5

p / M

Pa

T = 474.15 KT = 423.65 K

(a)

(b)

FIG. 14. Pressure-weight fraction (p-x) isothermal slices of:(a) the liquid-liquid equilibria of propane + n-hexacontane (n-C60H122); and (b) the vapor-liquid equilibria of n-pentane +low-density polyethylene (LDPE) of molecular weight 76,000g mol−1. The continuous curves represent the predictionswith the SAFT-γ Mie group contribution approach, thedashed curves the corresponding predictions with SAFT-γSW66, the dashed-dotted line the location of three-phase equi-libria, and the symbols the experimental data136,137 at differ-ent temperatures.

0.0 0.2 0.4 0.6 0.8 1.0xC6H14

0

5

10

15

20

25

30

u E /

m s

-1

0.0 0.2 0.4 0.6 0.8 1.0xC6H14

-1.5

-1.0

-0.5

0.0

V E

/ cm

3 mol

-1

(a)

(b)

FIG. 15. Predictions of selected excess thermodynamic prop-erties of binary mixtures of n-alkanes with the SAFT-γ Miegroup contribution approach: (a) excess speed of sound uE forn-hexane+n-dodecane (squares), n-hexane+n-decane (trian-gles) and n-hexane+n-octane (circles) at 298.15 K144; and(b) excess molar volumes V E for n-hexane+n-hexadecaneat 293.15 K (circles), 303.15 K (triangles-up), 313.15 K(squares), 323.15 K (diamonds) and 333.15 K (triangles-down)145 at a pressure of p = 0.10132 MPa. The continuouscurves represent the predictions with SAFT-γ Mie, and thedashed curves the corresponding description with the SAFT-γSW theory66,67.

40

0.00 0.25 0.50 0.75 1.00xCH3COOCH2CH3

, yCH3COOCH2CH3

300

325

350

375

400

425

450T

/ K

ethyl acetate + n-decaneethyl acetate + n-octaneethyl acetate + n-pentane

FIG. 16. Predictions of the fluid-phase behavior of bi-nary mixtures of ethyl acetate+n-alkane as temperature-composition (T -x) isobaric slices at p = 0.10132 MPa ob-tained with the SAFT-γ Mie group contribution approach.The symbols represent the experimental data163 for the vapor-liquid equilibria for ethyl acetate+n-pentane (circles), ethylacetate+n-octane (asterisks), and ethyl acetate+n-decane(squares). The continuous curves are the predictions ofSAFT-γ Mie, and the dashed curves the corresponding predic-tions with modified UNIFAC (Dortmund)13 using parametersfrom Ref. [127].

0.00 0.25 0.50 0.75 1.00xC6H14

, yC6H14

330

340

350

360

370

380

390

400

T /

K

n-hexane + n-butyl acetaten-hexane + n-propyl acetaten-hexane + ethyl acetate

FIG. 17. Predictions of the fluid-phase behavior of bi-nary mixtures of n-hexane+n-alkyl acetate as temperature-composition (T -x) isobaric slices at p= 0.10132 MPa obtainedwith the SAFT-γ Mie group contribution approach. The sym-bols represent the experimental data for the vapor-liquid equi-libria for n-hexane+ethyl acetate (circles)163, n-hexane+n-propyl acetate (asterisks)163, and n-hexane+n-butyl acetate(triangles)171. The continuous curves are the predictions ofSAFT-γ Mie and the dashed curves the corresponding pre-dictions with modified UNIFAC (Dortmund)13 using fromRef. [127].

0.0 0.2 0.4 0.6 0.8 1.0xC7H16

0

500

1000

1500

hE /

J m

ol-1

0.0 0.2 0.4 0.6 0.8 1.0xC7H16

0.0

0.2

0.4

0.6

0.8

1.0

V E

/ cm

3 mol

-1

(a)

(b)

FIG. 18. Predictions for selected excess properties of bi-nary systems of n-heptane+n-alkyl acetate obtained with theSAFT-γ Mie group contribution approach: (a) excess en-thalpy hE, and (b) excess volume V E for n-heptane+n-propylacetate (circles), n-heptane+n-butyl acetate (asterisks), andn-heptane+n-pentyl acetate (triangles) at a temperature ofT = 318.15 K and a pressure of p = 0.10132 MPa172. Thecontinuous curves represent the predictions with SAFT-γ Mieand the dashed curves (for the excess enthalpy) the calcula-tions with modified UNIFAC (Dortmund)13 using parametersfrom Ref. [127].

41

0.0 0.2 0.4 0.6 0.8 1.0xC6H14

1000

1100

1200

1300u

/ m s

-1n-hexane + n-decyl acetaten-hexane + n-pentyl acetaten-hexane + ethyl acetate

0.0 0.2 0.4 0.6 0.8 1.0xC6H14

-0.2

-0.1

0.0

0.1

0.2

k SE /

GP

a-1

n-hexane + ethyl acetaten-hexane + n-pentyl acetaten-hexane + n-decyl acetate

(a)

(b)

FIG. 19. Predictions of selected second-order thermodynamicderivative properties of binary mixtures of n-hexane+n-alkylacetate obtained with the SAFT-γ Mie group contribu-tion approach: (a) speed of sound u, and (b) excess isen-tropic compressibility of mixing kE

S for n-hexane+ethyl ac-etate (circles), n-hexane+n-pentyl acetate (asterisks), and n-hexane+n-decyl acetate (triangles) at a temperature of T =303.15 K and a pressure of p = 0.10132 MPa173. The contin-uous curves represent the predictions of the theory.

0.0 0.2 0.4 0.6 0.8

ρ ∗

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

p∗

T∗ = 1.0

T∗ = 1.5

T∗ = 2.0

T∗ = 3.0

FIG. 20. Comparison of the predictions of the SAFT-γ SWand SAFT-γ∗ SW approaches with isothermal-isobaric MonteCarlo simulation data for the density ρ∗ = ρσ∗

11 dependenceof the pressure p∗ = pπσ∗

11/(6kT ) at different temperaturesT ∗ = kT/ϵ11 square-well tangentially bonded dimers62 with:σ22 = 0.5σ11; ϵ11 = ϵ12 = ϵ22; λ11 = λ12 = λ22= 1.5. Thecontinuous curves represent the calculations with a SAFT-γtreatment of the chain term, the dashed curves are the resultswith the SAFT-γ∗.

0.0 0.2 0.4 0.6 0.8

ρ ∗

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

p∗

σ11/σ22 = 0.25σ11/σ22 = 0.50σ11/σ22 = 0.75

FIG. 21. Comparison of the predictions of the SAFT-γ SWand SAFT-γ∗ SW approaches with Monte Carlo simulationdata for the density ρ∗ = ρσ∗

11 dependence of the pressurep∗ = pπσ∗

11/(6kT ) of heteronuclear diatomic hard spheres atdifferent segment size ratios: σ11/σ22 = 0.25 (circles), σ11/σ22

= 0.50 (squares), and σ11/σ22 = 0.75 (triangles)54. Thecontinuous curves represent the calculations with a SAFT-γtreatment of the chain term, the dashed curves are the resultswith the SAFT-γ∗.

42

0.00 0.01 0.02

ρ ∗

0.00

0.05

0.10

0.15

0.20

p*

0.0 0.1 0.2 0.3 0.4 0.5

ρ ∗

0

2

4

6

p*

0.0 0.1 0.2 0.3 0.4 0.5

ρ ∗

0

2

4

6

p*

0.00 0.25 0.50 0.75 1.00

ρ ∗

0

2

4

6

8

10

p*

(a) (b)

(c) (d)

FIG. 22. Comparison of the predictions of the SAFT-γ SW and SAFT-γ∗ SW approaches with Monte Carlo simulation datafor the density ρ∗ = ρσ∗

11 dependence of the pressure p∗ = pπσ∗11/(6kT ) of heteronuclear triatomic hard spheres at different

segment size ratios and connectivity: (a) σ11 = 0.25σ22 = σ33, (b) σ11 = 4σ22 = σ33, (c) σ11 = σ22 = 2σ33, and (d) σ11 = 2σ22

= 4σ3355. The continuous curves represent the calculations with a SAFT-γ treatment of the chain term, the dashed curves are

the results with the SAFT-γ∗.