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From realistic to simple models of fluids. III. Primitivemodels of carbon dioxide, hydrogen sulphide andacetone, and their propertiesLukáš Vlček a & Ivo Nezbeda a b

a E. Hála Laboratory of Thermodynamics , ICPF, Academy of Sciences , 165 02 Prague, CzechRepublicb Department of Physics , J.E. Purkyně University, 400 96 Ústí n. Lab. , Czech Republicc E. Hála Laboratory of Thermodynamics , ICPF, Academy of Sciences , 165 02 Prague, CzechRepublic E-mail:Published online: 21 Feb 2007.

To cite this article: Lukáš Vlček & Ivo Nezbeda (2005) From realistic to simple models of fluids. III. Primitive models ofcarbon dioxide, hydrogen sulphide and acetone, and their properties, Molecular Physics: An International Journal at theInterface Between Chemistry and Physics, 103:14, 1905-1915, DOI: 10.1080/00268970500083630

To link to this article: http://dx.doi.org/10.1080/00268970500083630

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From realistic to simple models of fluids. III. Primitivemodels of carbon dioxide, hydrogen sulphide and acetone,

and their properties

LUKAS VLCEKy and IVO NEZBEDA*yz

yE. Hala Laboratory of Thermodynamics, ICPF, Academy of Sciences, 165 02 Prague, Czech RepubliczDepartment of Physics, J.E. Purkyne University, 400 96 Ustı n. Lab., Czech Republic

(Received 16 December 2004; in final form 18 February 2005)

Recently developed methodology to construct primitive models of associating fluids as directdescendants of complex realistic intermolecular potential functions [L. Vlcek, I. Nezbeda.Molec. Phys., 102, 485 (2004).] is extended to polar fluids and applied to three substances ofpractical importance: quadrupolar carbon dioxide, and dipolar hydrogen sulphide andacetone. It is shown that the structural properties (in terms of the site–site correlationfunctions) of the primitive models of polar fluids reproduce very well those of their parentrealistic models but, nonetheless, they perform worse than in the case of associating fluids.A number of thermodynamic properties of the developed models obtained by computersimulations is also reported (for their later use in theoretical investigation) and discussed.

1. Introduction

Extensive computer simulations performed over the lastdecade on realistic models of polar and associating fluids(for a review see [1]) have shown that their propertiesare governed primarily by short-range interactionswhich, unlike simple fluids, may be both repulsive andattractive. Specifically, the structure of these fluids,given in terms of the complete set of the site–sitecorrelation functions, is very similar (nearly identical) tothat of suitably defined short-range models. It is thuslegitimate to write properties of polar and associatingfluids in a perturbed form with the short-range modelsas a reference (short-range reference, SRR). To imple-ment the perturbation expansion, the properties of theSRR fluid must be well known and available, preferably,in an analytic form. For this purpose the so-calledprimitive models (PMs) may be used: the properties ofthe SRR fluid are estimated by means of an appropriatePM in the same way as the properties of soft repulsivesimple fluids are approximated by those of a fluid ofhard spheres [2, 3]. The conditions imposed on the PMsthus are: (i) they should reproduce to a high degree ofaccuracy the structure of the SRR fluids and (ii) theyshould be simple enough to be tractable by a theory.In addition to their use as a reference system in apotential perturbation theory, they have merit of their

own for their potential direct use in studies of molec-ular mechanisms of various processes in the same way asfluids of hard spheres.

Recently we have considered realistic models of anumber of associating fluids and developed a method-ology to construct the associated PMs as directdescendants of the parent realistic models [4, 5]. In theapproach adopted, the molecules are pictured as fused-hard-sphere bodies which are rigorously defined bythe parent SRR model. When approximating theCoulombic interactions at short separations, generalphysical criteria have been used to minimize thenecessity to resort to the known structural data.Despite the complexity of associating fluids, theproblem to develop their PMs is considerably simplifiedby predominance of the strong and short-range hydro-gen bonding (H-bonding). This, however, is not the caseof non-associating polar fluids in which the Coulombicinteractions do not create any specific short-rangeinteraction and any physical criterion (as, e.g. saturationof H-bonding) is also missing.

In this paper we modify the developed approach andan attempt is made to extend it to non-associating polarfluids and develop PMs for compounds of practicalindustrial importance: quadrupolar carbon dioxide, anddipolar acetone and hydrogen sulphide. In addition tothe structural data which are a prerequisite for devel-oping the models, we also report computer simulationdata both for the internal energy and pressure alonga number of isotherms for their later use and discuss,*Corresponding author. Email: ivonez@icpf.cas.cz

Molecular Physics, Vol. 103, No. 14, 20 July 2005, 1905–1915

Molecular PhysicsISSN 0026–8976 print/ISSN 1362–3028 online # 2005 Taylor & Francis Group Ltd

http://www.tandf.co.uk/journalsDOI: 10.1080/00268970500083630

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in general terms, the thermodynamic properties of themodels.

2. Basic definitions and technical details

Realistic pair potential models uð1, 2Þ assume that themolecule contains interaction sites which may, but neednot necessarily, coincide with the location of theindividual atoms, and for relatively small moleculesthe geometrical arrangement of the sites is fixed withinthe molecules (rigid monomer). The interaction sitesare the seat of two types of interactions: (1) neutral(non-electrostatic) interaction generating a strongrepulsion at short separations and a weak attractionat medium separations, represented commonly by theLennard–Jones (LJ) potential,

unon–elð1, 2Þ � uLJðrijÞ ¼ 4�ij�ijrij

� �12

��ijrij

� �6" #

, ð1Þ

and (2) the long-range Coulombic charge–chargeinteraction. A common realistic pair potential thus hasthe form

uð1, 2Þ ¼ unon–elð1, 2Þ þ uCoulð1, 2Þ

¼Xi2f1g

Xj2f2g

unon–el rð1Þi � r

ð2Þj

��� ���� �þ

qð1Þi q

ð2Þj

rð1Þi � r

ð2Þj

��� ���8<:

9=;,

ð2Þ

where (1, 2) stands for the separation and orientationof molecules 1 and 2, r

ðkÞi is the position vector of site

i on molecule k, rij ¼ jrð1Þi � r

ð2Þj j and q

ðkÞi is the partial

charge of site i of molecule k.In this paper we consider carbon dioxide defined by

the EPM model used recently by Harris and Yung [6] intheir vapour–liquid equilibria calculations, hydrogensulphide as given by the simple three-site model ofJorgensen [7], and for acetone we use the model studiedby Jedlovszky and Palinkas [8]. For acetone, the methylgroups are treated as united sites and the molecule thushas four interaction sites. The molecules of carbondioxide and hydrogen sulphide are much simpler andhave three interaction sites only. The geometricalarrangement of the sites of these models is depicted infigure 1 and the charges, qi, and the LJ parameters �ijand �ij for i¼ j are given in table 1. For the cross-interactions between the LJ sites we use the samecombining rules as in the original papers, i.e. thegeometric mean for the energy for all three compounds,

�ij ¼ ð�ii�jjÞ1=2, ð3Þ

and the arithmetic mean for �ij of acetone,

�ij ¼�ii þ �jj

2, ð4Þ

and the geometric mean for carbon dioxide,

�ij ¼ ð�ii�jjÞ1=2: ð5Þ

The short-range models uSRR result from theabove general models by gradually switching off their

Table 1. Potential parameters of the realistic potentialmodels, equation (2), and the switching ranges used.

Model Site �=kB=K �= �A q=e R0 R00

CO2 [6] C 28.999 2.785 0.6645 4.0 6.0

O 82.997 3.064 �0.33225

Acetone [8] C 52.84 3.75 0.566 6.6 8.0

O 105.68 2.96 �0.502

Me 85.00 3.88 �0.032

H2S [7] S 125.89 3.70 �0.470 5.5 7.5

H 0.0 0.0 0.235

OO

1.163Å

C

CMeMe

121.35˚

O

1.22

3Å

1.572Å

92.0˚

1.34ÅHH

S

CO O

C

MeMe

O

X

H H

X X

S

CARBON DIOXIDE

ACETONE

HYDROGEN SULPHIDE

Figure 1. Geometry of the considered realistic models(left column) and the descending primitive models(right column).

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long-range Coulombic part [9–11],

uSRRð1, 2Þ ¼ uð1, 2Þ � SðrOO;R0,R00ÞuCoulð1, 2Þ, ð6Þ

where Sðr;R0,R00Þ is the switch function defined by

Sðr;R0,R00Þ ¼

0, for r < R0,

ðr�R0Þ2

ð3R00 �R0 � 2rÞ=ðR00 �R0Þ3, for R0 < r < R00,

1, for r > R00:

8>>><>>>:

ð7Þ

It has been shown [1] that the structure of the SRRfluid, described in terms of the complete set of site–sitecorrelation functions, is practically identical to that ofthe full models (2).Carbon dioxide and acetone have already been

considered in studies on the effect of the long-rangeforces [11] and the recommended values for theirswitching range ðR0,R00Þ are given in table 1.Concerning hydrogen sulphide, for the purpose of thispaper we carried out the standard MC simulations in anNVT ensemble (for technical details see subsection 2.3below) [12] to examine the effect of the range ofinteractions and find the most appropriate switchingrange. The values obtained for ðR0,R00Þ are also given intable 1.The above defined SRR model (6) serves as the parent

model for a PM to be constructed. To maintain thedirect link between the parent and primitive models,geometry of the PM must copy that of its realisticparent model, that is the arrangement of sites and theirseparations. This arrangement for the compoundsconsidered is also shown in figure 1. To approximatethe force field of the parent model at short separations,the following general approach is adopted [4, 5].

(a) The non-electrostatic repulsive site–site interactionsare represented by a hard-sphere (HS) interactionwhich means that the molecule is made up of HSs(fused-hard-sphere body, FHS) of diameters d

ðSÞii .

(b) The effect of the Coulombic interactions at closeseparations is represented as follows:

(i) the repulsive interaction between the likecharges is represented by a hard-sphereinteraction, and

(ii) the attractive interaction between the unlikecharges is represented by a square-wellinteraction.

Denoting the sites which bear charges as ‘þ’ and ‘–’, andthe sites with non-electrostatic interactions as S, thecomplete intermolecular interaction energy of the

PM is given by

uPMð1, 2Þ ¼Xi, j2fSg

uHS jrð1Þi � r

ð2Þj j; d

ðSÞij

� �

þX

i, j2fþ,�g

i¼j

uHS jrð1Þi � r

ð2Þj j; dij

� �

þX

i, j2fþ,�g

i 6¼j

uSW jrð1Þi � r

ð2Þj j; �

� �, ð8Þ

where

uHSðr12; �Þ ¼ þ1, for r12 < �,

¼ 0, for r12 > �,ð9Þ

and

uSWðr12; �Þ ¼ ��HB, for r12 < �,

¼ 0, for r12 > �:ð10Þ

The first term in equation (8) defines a FHS body (hardcore) and the first two repulsive terms in (8) togetherdefine a pseudo-hard-body (PHB) [13]. Provided thatthe attractive interaction, the third term in (8), may betreated as a perturbation, the PHB fluid may serveas a suitable natural reference in such a perturbationexpansion.

Both the structural and thermodynamic propertiesof both the SRR and PM fluids were obtained from thestandard Metropolis Monte Carlo (MC) simulationsin an NVT ensemble with N¼ 216 particles in thesimulation box [12]. Although some data for the realisticSRR model are readily available from the literature[11, 14], for the purpose of this paper we carried out ourown additional simulations to get all properties ofinterest at the state points considered and used literaturedata to check correctness of the developed code.The simulations were arranged in cycles, with onecycle consisting of N trial steps, and about 4� 105 cycleswere generated to evaluate the desired quantities.The convergence of the simulations was controlled bythe histogram of the internal energy, and statisticalerrors were assessed by the block method [15].

In addition to the standard thermodynamicquantities (the internal energy, E, and pressure, P) andthe usual site–site correlation functions, gijðrijÞ, foracetone and carbon dioxide we also measured thenormalized multipole–multipole correlation functions,�l, defined as

�lðrÞ ¼h� r� jr1,C � r2,Cj� �

Pl cos �12ð Þi

h�ðr� jr1,C � r2,CjÞi, ð11Þ

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where Pl is the normalized Legendre polynomial, �12 isthe angle formed by the axis of molecules 1 and 2, � isthe Dirac delta distribution, rk,C is the position of thereference carbon site on molecule k and h�i denotes anensemble average. We also computed the second virialcoefficient, B2, by means of the standard MC integrationmethod.

3. Primitive model methodology

In realistic models of associating fluids the hydrogen siteengaged in H-bonding bears usually only a chargebut no non-electrostatic interaction. Thus, lacking arepulsive core, such a hydrogen site may get quite closeto the sites of another molecule to strongly influencethe entire force field. For this reason the core d

ðSÞii of

the descending PM has to be determined from thecomposite effect of all site–site interactions via anaverage Boltzmann factor defining an effective site–siteinteraction:

ussðrijÞ ¼ �kBT ln

Zrij¼const

exp�uð1, 2Þ

kBT

� dð1Þ dð2Þ

* +�

,

ð12Þ

where kB is the Boltzmann constant, T is the absolutetemperature and h�iO denotes an unweighted angleaverage. This is also the case of non-associatinghydrogen sulphide. For carbon dioxide and acetone,whose all sites bear the LJ non-electrostatic interaction,the problem of determining d

ðSÞij is considerably simpler

and dðSÞij can be determined only from the individual LJ

site–site interactions. In both cases we use the common

hybrid Barker–Henderson recipe: the site–site potential(either an effective one given by equation (12) or directlythe LJ potential) is decomposed at its minimum Rmin

ii

into the repulsive part, urep, and the attractive part, therepulsive part is shifted (see figure 2) and the hard corediameter is then computed from [2]

dðSÞij ¼

Z Rminij

0

1� exp �urepij ðrÞ=kBTh in o

dr: ð13Þ

With the FHS body determining the molecule havingnow been defined, we have to cope with the problem ofapproximating the electric field at short intermolecularseparations. A straightforward way would be to takedirectly the Coulombic sites and define in some way thecorresponding HS and SW interactions. However,preliminary test computations along with simple physi-cal considerations show that this method may work onlyfor quadrupolar carbon dioxide but not in the case ofstrong dipolar fluids. The problem is that someCoulombic sites (typically the hydrogen sites) may bedeeply buried within the LJ core and their replacementby the short-range SW attraction (to mimic theinteraction with an unlike site of the other molecule)may lead immediately to the loss of the considerabledirectional effect (for details see [5]). This problem issimilar to that encountered when constructing realisticmodels for which it has been bypassed by using(unphysical) auxiliary sites (cf. for instance, the three-site model of hydrogen fluoride [16] or the four-siteTIP4P model of water). Thus the same approach weadopt for developing PMs and use auxiliary sites in thesame way as for recently developed PMs for water andalcohols [4, 5].

uss

rmin0

εmin

urep

0rmin

uss−εmin

Figure 2. Schematic representation of the procedure defining the repulsive part of interactions used to obtain the hard spherediameters.

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The question now stands how to set some values tothe parameters for the Coulombic-like SW and HSinteractions. There are actually only two parameters tobe determined: (1) the size of the HS repulsion betweenthe like charges, and (2) the range of the SW attraction;the depth �HB just scales the temperature, T ! T[dimensionless]¼T=ð�HB=kBÞ � 1=�, and a link to realsystems may thus be made through this expression, seecaptions to figures 3, 5 and 7 in section 4. Nonetheless, itis worth mentioning here that the structural propertiesare only slightly temperature-dependent over a widerange of temperatures and such a link may provide onlya rough estimate for the range of potential values of �HB.A more appropriate (and physical) way is via thermo-dynamic properties which, however, require a perturba-tion term to be added to the primitive model interactionsto make it meaningful.Since the SW attraction is to approximate the slowly

decaying Coulombic interaction it would be natural totry to make its range as long as possible. This isstraightforward for associating fluids because there arecertain general criteria which H-bonding must satisfyand which thus set automatically upper limits on the SWrange [4, 5]. However, we are not aware of any criterionwhich could be applied for non-associating fluids and wemust therefore proceed using trial and error, andcommon sense. Similarly, for the HS repulsion betweenthe like Coulombic sites, we use the known computersimulation data on the structural properties of theparent SRR model to set their optimal values. Thecriteria we will use for both � and dii are (i) the positionof extremes of the site–site correlation functions andtheir overall shape, and (ii) the multipole–multipolecorrelation functions.

4. Results and discussion

4.1. Primitive models and their structural properties

Using the methods described in the previous section weare now going to determine the potential parameters forthe primitive models of the considered compounds.To keep the number of required potential parameters ata minimum we add auxiliary sites only when it turns outto be necessary to obtain as good as possible agreementfor the structural properties of the primitive model withits realistic parent. With respect to its industrialapplications we develop a PM for carbon dioxide forits primary use in the supercritical region and thereforeuse the temperature T¼ 320K and the critical density(d ¼ 0:4586 g cm�3) as its benchmark conditions.Acetone is considered at ambient conditions, andhydrogen sulphide at its boiling point at 1 atm(T ¼ 212:81K and density 0.948 g cm�3).

4.1.1. Carbon dioxide. The attractive SW interactionsare acting directly between the C and O sites.Furthermore, to preserve simplicity of the model, wedo not consider any Coulomb interaction-based HSrepulsion between the like sites and do not introduceany auxiliary sites. There is thus only the non-electro-static HS interaction defining the FHS body; theresulting parameters are given in table 2. Sufficiency(and convenience) of this choice is demonstrated infigures 3 and 4, where the site–site and quadrupole–quadrupole correlation functions of the PM arecompared with those of the SRR fluid. The cusps area direct consequence of the discontinuous characterof the interactions. Otherwise, the comparisonsuggests that this PM forms very similar molecularconfigurations as its parent model.

4.1.2. Acetone. The realistic parent potential considerscharges on all sites. However, as we see from table 1,the charge on the methyl groups is negligibly smalland when constructing a PM of acetone we thereforeneglect it and consider charges only on C and O sites.Regardless of this simplification, when identifying the‘þ’ and ‘–’ sites of the PM directly with the Coulombicsites of the parent model, a discrepancy in the structurebetween such a PM and the parent SRR is observedregardless of the values (within reasonable range) of theparameters. Thus, to obtain the desired directionaleffect, we use an auxiliary site placed on the Me–C–Mebisector a distance lX away from the C site, see figure 1.This site bears the ‘charge’ instead of the C site but wewill try to do without any additional X–X repulsion(which may be considered to be buried inside the FHSbody). The correlation functions of the PM/acetonemodel developed in this way are shown in figures 5 and 6.We see that the site–site correlation functions related to

Table 2. Parameters of the developed primitive models.

Model Parameter A dref

PM/CO2 dðSÞOO 2.949 1.00

dðSÞCC 2.536 0.86

�CO 4.718 1.60

PM/acetone dðSÞOO 2.881 1.00

dðSÞCC 3.529 1.225

dðSÞMeMe 3.739 1.298

‘X 1.008 0.350

�OX 4.177 1.45

PM/H2S dðSÞSS 3.443 1.00

dXX 2.410 0.70

�SX 2.927 0.85

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the central site are in very good agreement withits realistic parent and the same applies also to themultipole–multipole correlation functions. Slightly worseagreement found for gOO should in fact be expectedbecause this is a typical feature of off-centre sites.

4.1.3. Hydrogen sulphide. From the point of ability toform hydrogen bonds and from its liquid structure,

hydrogen sulphide belongs to non-associating fluids.However, due to a similarity of its molecule to those ofassociating liquids (water) with just one LJ atom andembedded hydrogens, we treat hydrogen sulphide in thesame way as associating fluids. Thus, as mentionedabove, the diameter of the central sphere we determinefrom the average Boltzmann factor of its parent model,equation (12). Furthermore, the ‘þ’ hydrogen-like sitesare placed as auxiliary X sites on the surface of thecentral sphere in order to obtain the desired directionaleffect (cf. the same arrangement of sites for the primitivemodel of SPC/E water [5]). The site–site correlationfunctions of the resulting PM/H2S are compared withthose of its liquid realistic counterpart in figure 7. We seeagain that the agreement is very good and comparablewith that for carbon dioxide and acetone. Nonetheless,it is worth mentioning that this agreement is not as goodas for true associating fluids.

4.2. Thermodynamics of primitive models

Although, unlike PMs for associating fluids, no theoryfor PMs of polar fluids is readily available, the form ofpotential (8) makes it tempting to write the propertiesof PM of polar fluids also in a similar composite way,XPM ¼ XPHB þ�X . For this reason we also split thediscussion of the properties of the PMs and discuss thePHB fluid separately.

4.2.1. Reference pseudo-hard body fluids. Two com-mon routes to the thermodynamic properties of PHBfluids are (1) the improved scaled particle theory (ISPT)and (2) the perturbed virial expansion [17]. The first

4 6 8 10 12

4 6 8 10 12

4 6 8 10 12

gCC

gCO

gOO

0

1

2

0

1

2

r[Å]

0

1

Figure 3. Comparison of the site–site correlation functionsof the realistic model of carbon dioxide at 320K and densityd ¼ 0:4586 g cm�3 and the descending primitive model at� ¼ 0:36 and the same density.

4 6 8 10

-0.4

-0.2

0.0

0.2

Π2

r[Å]

Figure 4. The same as in figure 3 for the normalizedquadrupole–quadrupole correlation function.

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route uses a parameter � to characterize non-sphericityof the body and evaluates pressure, P, from

zISPT ¼1þ ð3�� 2Þ�þ ð3�2 � 3�þ 1Þ�2 � �2�3

ð1� �Þ3, ð14Þ

where z � �P=� is the compressibility factor, � isthe number density and � is the packing fraction,

� ¼ �VPHB, where VPHB is the ‘volume’ of the PHB,defined by the interactions of the first twoterms in equation (8). Since VPHB is an ill-definedproperty, we approximate it by the volume of theunderlying FHS body, VPHB � VFHS. Following theconvex body fluid results, the parameter alpha isobtained from the second virial coefficient using therelation [18]

B2=VPHB ¼ 1þ 3�: ð15Þ

4 6 8 10 12

4 6 8 10 12

4 6 8 10 12

gCC

0

1

2

gCO

0

1

r[Å]

gCMe

0

1

Figure 5. Comparison of the site–site correlation functionsof the realistic model of acetone at ambient conditions and thedescending primitive model at � ¼ 2:5 and the same densityd ¼ 0:790 g cm�3.

4 6 8 10 12

4 6 8 10 12

gOO

0

1

gOMe

0

1

r[Å]

gMeMe

0

1

2

Figure 5. Continued.

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The other route evaluates the compressibility factorfrom a perturbed virial expansion,

zPVEð�Þ ¼ zref ð�ref Þ þXMi¼2

Bi�i�1 � Bref, i�

i�1ref

� �, ð16Þ

where quantities with subscript ‘ref’, refer to a suitablereference system, typically the system of HS at the samepacking fraction as the fluid of PHBs:

zHS ¼1þ �þ �2 � �3

ð1� �Þ3: ð17Þ

To implement the above equations, we used MCintegration to compute the lowest three virialcoefficients of the repulsive reference systems of theconsidered PMs. The virial coefficients and parameter� are listed in table 3. The compressibility factorspredicted by the above two equations are compared infigure 8 with simulation data. As can be seen, theperformance of the ISPT equation is in all cases betterthan that of the PVE and is also in quite good agreement

with the simulation results. As already mentioned,although there is no theory readily available for PMsof polar fluids, in a likely perturbation treatment,the fluid of PHB will be used as a reference andit is thus desirable to determine its properties asaccurately as possible. We have therefore followed theroute used for PMs of associating fluids and fittedthe experimental compressibility factors obtained from

4 6 8 10

gSS

0

1

2

3

2 4 6 8 10

gSH

0

1

r[Å]

2 4 6 8

gHH

0

1

Figure 7. Comparison of the site–site correlation functionsof the realistic model of hydrogen sulphide at 212.82K anddensity d ¼ 0:948 g cm�3, and the descending primitive modelat � ¼ 0:5 and the same density.

4 6 8 10 12

4 6 8 10 12

Π1

−1

0

Π2

0

1

r[Å]

Figure 6. The same as in figure 5 for the normalizedmultipole–multipole correlation functions.

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simulations by the function of the same functional formas in (14),

z ¼1þ a1�þ a2�

2 � a3�3

ð1� �Þ3: ð18Þ

This equation with coefficients ai listed in table 4provides an excellent fit to the simulation data for thepacking fractions at least up to � ¼ 0:5 and is used in thefollowing for the precise description of the consideredsystems.

4.2.2. Full primitive models. It is common to begin thediscussion of the thermodynamic properties with thoseat low density. These properties are readily availablefrom the truncated virial expansion. For the purpose ofthis paper we computed only the second virial coeffi-cients of the full primitive models, which can beexpressed analytically in the form

B2,PM ¼ B2,PHB þ b1½ðexp �Þ � 1�, ð19Þ

where b1 is a numerical constant given by an appropriatefraction of the configurational space with the SWattraction (for details see [19]). From (19) one candetermine the Boyle temperatures, which are given intable 5; knowledge of the Boyle temperature is usefulwhen the virial coefficients of model systems are to becompared to experimental data.

It has been recently shown [20] that the virialcoefficients can also be used for a rough estimate ofthe critical temperature. For this purpose the virialexpansion is written in a perturbed form (16), heretruncated after the first order term and with therepulsive PHB fluid described by equation (18) as a

Table 3. Virial coefficients, Bi, volumes VPHB, the parameter of non-sphericity, �, for the pseudo-hard body fluids underlyingthe given primitive models, and parameter b1 determining the second virial coefficient of the full primitive model.

Model B2=VPHB B3=V2PHB B4=V

3PHB VPHB=d

3ref � b1=VPHB

PM/CO2 4.6986 13.0411 24.2891 1.0476 1.2329 �7.8514

PM/acetone 4.6681 13.1181 25.7408 2.631 1.2227 �0.4316

PM/H2S 5.5268 18.1045 39.8774 0.524 1.5089 �1.1126

Table 4. Coefficients of the equation of state, equation (18),of the reference PHB fluids.

Model a1 a2 a3

PM/CO2 1.5886 2.9345 3.9036

PM/acetone 1.4830 3.3925 3.5254

PM/H2S 2.1927 8.8472 13.1815

0.0 0.1 0.2 0.3 0.4 0.5 0.6

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10

15

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25

30

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z

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60

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z

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(a)

(c)

(b)

Figure 8. Compressibility factors of the PHB fluids of(a) carbon dioxide, (b) acetone and (c) hydrogen sulphide.Comparison of simulation data (filled circles) with predictionsof the ISPT (solid line) and PVE (dotted line) equations ofstate.

From realistic to simple models of fluids 1913

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reference. The location of the critical point resultsthen from the solution of the set of the well knowntwo equations, @P=@� ¼ @2P=@�2 ¼ 0. The estimatedcritical temperatures are given in table 5. The trendfor acetone and hydrogen sulphide corresponds toexperimental data with the ratio of their criticaltemperatures being within 20% of the experimental

values. An unusually high (and unrealistic) criticaltemperature for carbon dioxide results from theway the PM/CO2 has been developed and has avery simple reason. From the requirement to obtainas good agreement as possible for the structure,the SW attraction results as being relatively longranged and shallow. Since the critical temperatureis inversely proportional to the depth of thisattraction we must then get a very high criticaltemperature.

The standard thermodynamic properties, the internalenergy and pressure, of the primitive models obtainedfrom simulations for a wide range of thermodynamicconditions are shown in figures 9 through 11. Bothproperties exhibit familiar dependence on densityalong isotherms; cf. similar results for associatingfluids presented in [4]. Without availability of any first-principal theoretical way to estimate these properties wemay hardly comment on these simulation results which

Table 5. The inverse critical temperatures obtained from thefirst-order perturbed virial expansion, and the inverse Boyle

temperatures.

Model �c �B

PM/CO2 0.7937 0.469

PM/acetone 2.9525 2.469

PM/H2S 3.1112 1.786

ρ

0.0 0.1 0.2 0.3 0.4 0.5 0.6

U/N

−20

−15

−10

−5

0

ρ

0.0 0.1 0.2 0.3 0.4 0.5 0.6

z

−10

0

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20

30

Figure 9. Density dependence of the excess internal energy(top; measured in �HB=kBT) and compressibility factor(bellow) for PM/CO2 at various temperatures(� ¼ 0, 0:1, 0:2, . . . , 1:5 from the top below).

ρ

0.00 0.05 0.10 0.15 0.20 0.25

U/N

−4

−3

−2

−1

0

ρ

0.00 0.05 0.10 0.15 0.20 0.25

z

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20

30

40

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Figure 10. The same as figure 9 for PM/acetone (for� ¼ 0, 0:5, . . . , 5:0).

1914 L. Vlcek and I. Nezbeda

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should be used primarily to test any theoreticalapproach to be developed.

5. Conclusions

In this paper we have attempted to develop simple short-range models of three important polar fluids along thesame lines that have worked for associating fluids.Surprisingly, due to the lack of the strong directionalshort-range attraction present in associating fluids, thisproblem turns out to be more complex. Consequently,a larger body of advance knowledge of structural datais required. Despite this more empirical approach, theresults for the structure are slightly worse than those forassociating fluids but yet in quite good agreement when

compared to the results for realistic models in order toclaim that they satisfy the criteria for their use as areference system in a perturbation expansion.

Another problem with polar fluids is the lack of anytheory readily applicable to their primitive models.Attempts to use, at least formally, the thermodynamicperturbation theory have failed completely (the modelsdo not satisfy the conditions of saturation neitherapproximately) and another perturbation theory musttherefore be developed. The developed models may thusbe used at present primarily in simulation studiesinvestigating molecular mechanisms governing variousprocesses, preferably in those exhibiting a sort of ananomalous behaviour.

Acknowledgement

This work was supported by the Grant Agency ofThe Academy of Sciences of the Czech Republic(Grant No. IAA4072303).

References

[1] I. Nezbeda. Molec. Phys., 103, 59 (2005).[2] T. Boublık, I. Nezbeda, K. Hlavaty. Statistical

Thermodynamics of Simple Liquids and Their Mixtures,Elsevier, Amsterdam (1983).

[3] J.P. Hansen, I.R. McDonald. Theory of Simple Liquids,Academic Press, London (1976).

[4] L. Vlcek, I. Nezbeda. Molec. Phys., 101, 2987 (2003).[5] L. Vlcek, I. Nezbeda. Molec. Phys., 102, 485 (2004).[6] J.G. Harris, K.H. Yung. J. phys. Chem., 99, 12021 (1995).[7] W.L. Jorgensen. Phys.Chem., 90, 6379 (1986).[8] P. Jedlovszky, G. Palinkas. Molec. Phys., 84, 217

(1995).[9] I. Nezbeda, J. Kolafa. Czech. J. phys. B, 40, 138 (1990).[10] J. Kolafa, I. Nezbeda. Molec. Phys., 98, 1505 (2000).[11] J. Kolafa, I. Nezbeda. Molec. Phys., 99, 1751 (2001).[12] M.P. Allen, D.J. Tildesley. Computer Simulation of

Liquids, Clarendon Press, Oxford (1987).[13] I. Nezbeda. Molec. Phys., 90, 661 (1997).[14] M. Kettler, I. Nezbeda, A.A. Chialvo, P.T. Cummings.

J. phys. Chem. B, 106, 7537 (2002).[15] H. Flyvbjerg, H.G. Petersen. J. chem. Phys., 91, 461

(1989).[16] M.E. Cournoyer, W.L. Jogensen. Molec. Phys., 51, 119

(1984).[17] M. Prdota, I. Nezbeda, Y.V. Kalyuznyi. Molec. Phys., 94,

937 (1998).[18] T. Boublik, I. Nezbeda. Coll. Czech. Chem. Commun., 51,

2301 (1986).[19] I. Nezbeda, J. Slovak. Molec. Phys., 90, 353 (1997).[20] I. Nezbeda, W.R. Smith. Fluid Phase Equil., 216, 183

(2004).

ρ

0.0 0.2 0.4 0.6 0.8

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ρ

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15

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Figure 11. The same as figure 9 for PM/H2S (for� ¼ 0, 0:5, . . . , 2:5).

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