from realistic to simple models of associating fluids. ii. primitive models of ammonia, ethanol and...

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From realistic to simple models of associating fluids. II.Primitive models of ammonia, ethanol and models ofwater revisitedLukáš Vl[cbreve]ek a & Ivo Nezbeda a ba E. Hála Laboratory of Thermodynamics , Institute of Chemical Process Fundamentals,Academy of Sciences , 165 02 Prague, Czech Republicb Physics Department , J. E. Purkynĕ University , 400 96 Ústí n. Lab., Czech RepublicPublished online: 20 Feb 2007.

To cite this article: Lukáš Vl[cbreve]ek & Ivo Nezbeda (2004) From realistic to simple models of associating fluids. II.Primitive models of ammonia, ethanol and models of water revisited, Molecular Physics: An International Journal at theInterface Between Chemistry and Physics, 102:5, 485-497, DOI: 10.1080/00268970410001668417

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From realistic to simple models of associating fluids. II.Primitive models of ammonia, ethanol and models

of water revisited

LUKAS VLCEK1 and IVO NEZBEDA1,2*1E. Hala Laboratory of Thermodynamics, Institute of Chemical ProcessFundamentals, Academy of Sciences, 165 02 Prague, Czech Republic

2Physics Department, J. E. Purkyne University, 400 96 Ustı n. Lab., Czech Republic

(Received 24 October 2003; revised version accepted 5 January 2004)

Recently developed methodology to construct primitive models of associating fluids directlyfrom realistic intermolecular potential functions is applied to ammonia, ethanol and severalmodels of water. Hard cores of the molecules are pictured as fused hard-sphere bodies definedby composite short-range repulsions, and the Coulombic repulsions and attractions areapproximated by hard-sphere and square-well potentials, respectively. Hard-sphere diametersare determined directly from the parent potential using a theoretical route and the range ofthe square-well attraction is adjusted using constraints imposed on hydrogen bonding. It isshown that the developed primitive models, despite their simplicity and lack of any long-rangeinteractions, are able to reproduce the structural properties (the set of the site–site correlationfunctions) of the parent realistic models and may thus serve well as a reference in theperturbation theory.

1. Introduction

Perturbation theories of fluids are based on therecognition of different impacts of different parts ofthe intermolecular potential on the properties of fluids.For normal fluids (i.e. the fluids without significantlong-range interactions) it was already well establishedmore than thirty years ago that their structure isdetermined by short-range (repulsive) interactions [1].This finding gave rise to the development of a varietyof theories and semi-empirical methods based on theknown properties of the fluid of hard spheres, whichis the simplest model of such interactions and serves asa suitable reference [2, 3].Over the last decade a sufficient body of pseudo-

experimental evidence has been gathered [4–8] to makea similar claim also for pure polar and associatingfluids, namely that their structure, defined in terms of acomplete set of the site–site correlation functions, isdetermined primarily by the short-range interactions(which may however be both repulsive and attractive)and that the long-range Coulombic interactions, nomatter how strong, may be treated as a perturbationonly. Thus, similarly as for normal fluids, this findingmakes it possible to set up a fast converging pertur-bation expansion about a suitable short-range reference

(SRR). The problem is that to accomplish such aperturbation scheme, properties of the SRR must bedetermined, preferably in an analytic form, and thismay be a formidable task. In fact, the same problem wasencountered also by perturbation theories of normalfluids and it may be useful therefore to recall how it wassolved: the properties of the soft-sphere reference fluidwere estimated by those of the fluid of hard spheres[2, 3]. A similar way may also be conveniently usedfor associating fluids. It means we would need a simplemodel which captures the essence of physics of the SRRand which would also be simultaneously amenable to atheoretical treatment.

There are a number of simple models for associatingfluids available which were developed during the 1980sand early 1990s [9–13] using intuition and speculationto mimic hydrogen bonding (H-bonding). Althoughthey capture the essence of association (and weretherefore used, for example, as the basis of a semi-empirical SAFT (statistical association fluid theory)approach [14]), they all suffer from a common draw-back, namely, that (i) they yield the structure only inqualitative agreement with observations on real fluidsand (ii) there does not seem to be a way to improvetheir performance. These defects are a consequence ofthe fact that the models are not linked to any realisticintermolecular potential and may therefore hardly be*Author for correspondence. e-mail: [email protected]

MOLECULAR PHYSICS, 10 MARCH 2004, VOL. 102, NO. 5, 485–497

Molecular Physics ISSN 0026–8976 print/ISSN 1362–3028 online # 2004 Taylor & Francis Ltdhttp://www.tandf.co.uk/journals

DOI: 10.1080/00268970410001668417

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used to develop a molecular theory, i.e. to predictproperties of a system defined by a Hamiltonian (cf. vander Waals equation which is also based, but again onlyintuitively, on a physically sound hard sphere–fluidreference system).With the set ultimate goal—to develop a theory for

any associating fluid defined by an appropriateHamiltonian—we have recently considered the OPLSmodel of methanol [15] and briefly also the SPC modelof water [16], and developed a methodology to con-struct such simple models as direct descendants of thechosen realistic parent ones. These models, that repro-duce quantitatively the structure (in terms of thecomplete set of the site–site correlation functions) ofthe parent SRR fluid, have been called primitive models(PMs). In the approach adopted, the molecules arepictured as fused hard-sphere bodies which are rigor-ously defined by the short-range repulsions of the parentSRR model. Additional hard-sphere (HS) repulsion andsquare-well (SW) attraction then mimic Coulombicinteractions at short separations. When approximatingthe Coulombic interactions, general physical criteriahave been used to minimize the necessity to resort to theknown structural data. In this paper we further refinethis approach and apply it to other associating fluids,ethanol and ammonia, and reconsider three qualitativelydifferent models of water and construct their associatedPMs.

2. Theoretical background and technical details

2.1. The parent modelsThere seems to be now a general consensus con-

cerning the functional form of realistic pair potentialmodels u(1, 2). It is assumed that the molecule containsinteraction sites which may, but need not necessarily,coincide with the location of the individual atoms. It isfurther assumed that for relatively small molecules thegeometrical arrangement of the sites is fixed withinthe molecules (rigid monomer). The interaction sites arethe seat of two types of interactions: (1) non-electro-static interaction generating a strong repulsion at shortseparations and a weak attraction at medium separa-tions, represented commonly by the Lennard-Jones (LJ)potential,

uLJðrijÞ ¼ 4�ij�ijrij

� �12

��ijrij

� �6" #

; ð1Þ

and (2) long-range Coulombic charge–charge interac-tion. A common realistic pair potential thus has the form

uð1; 2Þ ¼ unon–elð1; 2Þ þ uCoulð1; 2Þ

¼Xi2f1g

Xj2f2g

uLJðjrðiÞ1 � r

ð jÞ2 jÞ þ

qð1Þi qð2Þj

jrðiÞ1 � r

ð jÞ2 j

( ); ð2Þ

where (1, 2) stands for the separation and orientation ofmolecules 1 and 2, r

ðiÞk is the position vector of site i on

molecule k, rij ¼ jrðiÞ1 � r

ð jÞ2 j and q

ðiÞk is the partial charge

of site i of molecule k.The parent short-range models uSR (short-range

reference, SRR) result from the above general modelsby gradually switching off their long-range Coulombicpart [5, 17],

uSRRð1; 2Þ ¼ uð1; 2Þ � SðrOO;R0;R00ÞuCoulð1; 2Þ; ð3Þ

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

Sðr;R0;R00Þ

¼

0; for r<R0

ðr�R0Þ2ð3R00�R0�2rÞ=ðR00�R0Þ

3 ; for R0<r<R00,

1; for r>R00.

8><>:

ð4Þ

In this paper we consider the OPLS (optimizedpotentials for liquid simulations) models of Jorgensenand co-workers for ammonia [18] and ethanol [19],and three models of water: three-site SPC/E model [20],four-site TIP4P model [21] and five-site TIP5P model[22]. The geometrical arrangement of the sites of thesemodels is depicted in figure 1, and the charges, qi, alongwith the LJ parameters �ij and �ij for i ¼ j are given intable 1; for all these potentials the geometric mean isused to define the cross-interactions:

�ij ¼ ð�ii�jjÞ1=2; �ij ¼ ð�ii�jjÞ

1=2: ð5Þ

The full TIP5P and TIP4P models of water havealready been the subject of investigations and theirswitching range was determined [6, 8]; the recommendedvalues of (R0, R00) are given in table 1. For the remainingcompounds (models) we carried out the standardMetropolis MC simulations in an NVT ensemble (fortechnical details see subsection 2.3 below) [23] toexamine the effect of the range of interactions on thestructural properties and to find the most appropriateswitching range. The obtained values of (R0, R00) arealso given in table 1. Furthermore, for the purpose ofconstructing primitive models, for ammonia we fixed thebond lengths and angles at the most probable values(energy minima) and, similarly, fixed also the dihedralangle in the model of ethanol at 180� (see figure 1).

2.2. Primitive model methodologyThe above SRR model (3) serves as the parent model

for a PM to be constructed. To maintain the direct linkbetween the parent and primitive models, geometry ofthe PM must copy that of its realistic parent model,that is the arrangement of the sites and their separations.

486 L. Vlcek and I. Nezbeda

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This arrangement for the compounds considered isshown in figure 1. To approximate the force field ofthe parent model at short separations, the followingapproach is adopted [24, 25].

(1) The non-electrostatic repulsive interactions arerepresented by an HS interaction which meansthat the molecule is made up of hard spheres(fused hard-sphere body, FHS) of diameters dii.

(2) The Coulombic interactions are represented asfollows:

(i) the repulsive interaction between the likecharges is represented by an HS interaction,and

(ii) the attractive interaction between the unlikecharges is represented by an SW interaction.

Denoting the sites which bear charges as P (positivecharge) and N (negative charge), and the sites withnon-electrostatic interactions as S, the complete inter-molecular interaction energy of the PM is given by

uPMð1;2Þ

¼Xi;j2fSg

uHSðjrð1Þi � r

ð2Þj j;dijÞþ

Xði;jÞ2fPgði;jÞ2fNg

uHSðjrð1Þi � r

ð2Þj j;dijÞ

þX

i;j2fP;Ng

i6¼j

uSWðjrð1Þi � r

ð2Þj j;�Þ ð6Þ

� uFHSð1;2ÞþX

ði;jÞ2fPgði;jÞ2fNg

uHSðjrð1Þi � r

ð2Þj j;dijÞ

þX

i;j2fP;Ng

i6¼j

uSWðjrð1Þi � r

ð2Þj j;�Þ ð7Þ

Table 1. Potential parameters of the realistic potential models, the used switching range and the energeticcriterion for the formation of the H-bond.

Model Site �=kB=K �= �A q=e R0 R00 uHB=kJmol�1

NH3 N 85.548 3.42 �1.02 5.0 7.0 –

H 0.0 0.0 0.34

EtOH O 85.548 3.070 �0.700 5.7 7.7 12.2

H 0.0 0.0 0.435

CH2 70.451 3.905 0.265

Me 104.17 88.064 0.0

SPC/E O 78.205 3.166 �0.8476 5.0 7.0 13.4

H 0.0 0.0 0.4238

TIP4P O 78.08 3.154 0.00 4.0 6.0 33.5

H 0.0 0.0 0.52

M 0.0 0.0 �1.04

TIP5P O 80.57 3.12 0.0 5.0 7.0 16.8

H 0.0 0.0 0.241

M 0.0 0.0 �0.241

Figure 1. Geometry of the considered realistic models ofassociating fluids (left column) and of their associatedprimitive models (right column).

From realistic to simple models of associating fluids. II 487

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where the summation in the second term of equation (6)runs over the pairs of like charges, in the third term overthe pairs of unlike charges, and

uHSðr12; �Þ ¼ þ1; for r12 < �

¼ 0; for r12 > �;ð8Þ

and

uSWðr12; �Þ ¼ ��HB; for r12 < �;

¼ 0; for r12 > �:ð9Þ

For normal fluids, the FHS body may be deriveddirectly from the repulsive part of the individual site–siteinteractions [26]. However, in the case of associatingliquids, the short-range part of the Coulombic inter-actions is known to be indispensable and no site–siteinteraction can therefore be treated separately withoutany regard to other sites [6]. To account for the effectof the superposition of the interactions between all siteswe have suggested [24, 25] using the ideas of the RAM(reference average Mayer-function) perturbation theory[27] and shown that this is a reasonable route. Forany pair of sites, the RAM theory defines a simple effec-tive site–site reference potential ueff obtained by angularaveraging of the Boltzmann factor of the total inter-molecular potential:

ueff ðrijÞ � uRAMij ðrijÞ

¼ �kBT ln

Zrij¼const

exp½�uð1; 2Þ=kBT�dð1Þdð2Þ

* +;

ð10Þ

where the angular brackets denote an unweightedangular average, kB is the Boltzmann’s constant and T

is the absolute temperature. This method is known toprovide quite accurate site–site correlation functions[28]. Since we are concerned with effective short-rangerepulsions of the individual sites defined by thesepotentials, we may now follow the same route as fornormal fluids and use the effective site–site potentialueff ðriiÞ in place of the LJ (or any other non-electrostatic)potential. To obtain the diameters dii we then use thehybrid Barker–Henderson method [29]: (i) the site–sitepotential ueff ðriiÞ is decomposed at its minimum Rmin

ii

into the repulsive and attractive parts, with the repulsivepart, urep, shifted (see figure 2 for details), and (ii) thehard core diameter is then computed from

dii ¼Z Rmin

ii

0

f1� exp½�urepii ðrÞ=kBT�gdr: ð11Þ

It should be noted that for reasonable use of equation(11) the shape of the repulsive part of the effectivepotential, given by equation (10), must be sufficientlysteep. Certain intermolecular potentials need not pro-duce such effective potentials (the repulsive part is onlyslowly decaying) and the method may thus becomeinaccurate when applied to such models.

Once the FHS body determining the molecule isdefined, it remains 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 theparameters of the HS and SW interactions. However,simple physical considerations, supported by simulationresults in [24], show that this method does notnecessarily work sufficiently well. The problem is thatsome Coulombic sites (typically the hydrogen sites) maybe buried deeply within the LJ core and in this case thesubstituted SW attraction with an unlike site of the

ueff

rmin

0

εmin

urep

0rmin

ueff-εmin

Figure 2. Schematic representation of the procedure defining the repulsive part of interactions used to obtainthe hard-sphere diameters.


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other molecule will inevitably operate over a widebonding angle which leads immediately to the loss of thestrong directional effect typical for H-bonding (seefigure 3). This problem is actually similar to thatencountered already when constructing realistic modelswhere it has been bypassed by using (unphysical)auxiliary sites (cf., for instance, the three-site model ofhydrogen fluoride [30] or the TIP4P model of water). Weadopt the same approach for developing PMs and useauxiliary ‘Coulombic’ sites placed on the surface of theappropriate hard sphere (see figure 1).As for the Coulombic-like repulsions, we use for the

determination of the HS diameters the method describedabove for the hard core. It means we use equations (10)and (11) with the average Boltzmann factors computednow with respect to the Coulombic-like sites. It isappropriate to remark that this is a significant difference(and step forward) from our original method used formethanol and SPC water. In [24, 25] we used onlyempirical optimization to find suitable HS diameters.It turns out that the present theoretical route leads tonearly the same values of dii which lends strong supportfor this route. Concerning the parameters of SWattraction, there is actually only one parameter to bedetermined: the range of the SW attraction; the depth�HB scales the temperature and because the structuralproperties are only slightly temperature dependent overa wide range of temperatures, it need not be specifiedat this stage. Since the SW attraction is to approximatethe slowly decaying Coulombic interaction it is naturalto try to make its range as long as possible. On theother hand, for associating fluids there are certain generalconditions which H-bonding must satisfy and whichset upper limits on the SW range. For associating fluidsthe constraint results from the general requirement thattwo molecules cannot be double bonded. In order to beable to apply the thermodynamic perturbation theoryof Wertheim [31], we impose on � a stronger condition,namely that the H-site (or H-like auxiliary site) may notestablish more than one hydrogen bond, which is a

purely geometrical problem. It should, however, benoted that the range of the SW potential resulting fromthe conditions imposed by the TPT need not be theoptimal range. In fact, such a range may be very short,which may indicate possible problems when applied topotential models exhibiting weaker H-bonds, and alsocertain limitations for the use of the PMs at lowerdensities.

A remark on the notation now also seems appro-priate. The original primitive models introduced in the1980s and early 1990s [12, 13] were intuitive modelswithout any relation to realistic Hamiltonians. This firstfamily mimicked only H-bonding, meaning that therewas no repulsion between the like sites. The nextgeneration of PMs also incorporated the repulsionbetween the like sites and the models were calledextended PMs (EPMs). Because we dealt primarilywith water and what mattered was the number of sites,this number was explicitly given in the notation; e.g.EPM5 meant the five-site extended primitive model ofwater. The third generation dealt with in this paperdescends from Hamiltonians, and not only for water.The name of either the compound or the potentialmodel should therefore appear in the notation and wewill use the notation PM/model. Thus, for instance, PM/TIP4P will stand for the primitive model associated withthe TIP4P potential.

2.3. Computational detailsFor evaluation of the multidimensional integrals in

(10) we use the Conroy integration method with theroot points developed by Nezbeda et al. [32]. Thestructural properties were obtained from the standardMetropolis Monte Carlo (MC) simulations in an NVTensemble with N ¼ 216 particles in the simulation box.The simulations were arranged in cycles, with one cycleconsisting of N trial steps. There were 4� 105 cyclesgenerated for each system considered and the configura-tion at the end of each cycle was recorded for the analysisand evaluation of the desired quantities. The convergenceof the simulations was controlled by the histogram ofthe internal energy and statistical errors were assessedby the block method [33]. Although some data for therealistic SRR model have been available from literature[7], for the purpose of this paper we carried out ourown simulations to get all properties of interest at thestate points considered. For these simulations we usedthe same set-up as above with N¼ 216 particles in thesimulation cell and literature data were used to check thecorrectness of the developed code. For both the parentand primitive models we measured the usual site–sitecorrelation functions, gijðrijÞ, and, with the exceptionof ammonia, also two specific angles characteriz-ing geometry of the hydrogen bond: (i) angle

H

O

XX

H

Figure 3. Demonstration of the directional effect of thelocation of the square-well site within the hard sphere.The outer range is the same in both cases.


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� � ffO�H � � �O, which is defined by the HO�!

vectorspointing from the hydrogen to the oxygens, one on thesame molecule and the other on the hydrogen-bondedmolecule, and (ii) angle � � ffH � � �O�H, defined byOH

�!vectors pointing similarly from the oxygen to thehydrogens. To determine whether or not there is an H-bond between two molecules interacting via a realisticpotential, we used the energetic criterion according towhich the bond is formed when the pair energy is lessthan a certain energy limit. This limit (given in table 1) isindividual for each realistic model and was determined asthe energy of a local minimum or, when no local extremeis present, as the point of inflection on the pair energydistribution curve. In the case of ammonia, the pairenergy distribution has neither local minimum nor thepoint of inflection.

3. Results and discussion

Using the methods described in the previous section,we determined potential parameters of the primitivemodels of the considered associating compounds. Theresulting values of dij and lij are given in table 2 andcorrespond to the realistic models at ambient condi-tions. To assess the performance of the new primitivemodels, their structural properties, described in termsof the site–site pair correlation functions and the prob-ability distribution of angles characterizing the H-bond,are compared with those of their realistic counterpart.As already mentioned above, temperature does not

play the crucial role and we have therefore used for thePMs of ethanol and water the reduced inverse tempera-ture � ¼ 7, which was found in the previous study ofmethanol [24] to be the optimal temperature, and for thePM of ammonia we use � ¼ 4 which seems a reasonableoptimal value. Details and further refinement of therelation between the real and reduced temperatures wepostpone to another paper dealing with the thermo-dynamic properties of the models considered [34].Here we only mention in passing that for temperaturedependence considerations the parameters are scaledby a reference site (e.g. the oxygen sphere in thecase of water) which is made, eventually, temperaturedependent.

The primitive model of ammonia, PM/NH3, descendsfrom the realistic OPLS model of Rizo and Jorgensen[18]. Since the molecules of ammonia form weaker H-bonds, the attractions are less localized, which is alsoreflected in the rather wide and small first peak of thegNN correlation function (see figure 4). From the pointof construction of the PM it means that the range of theSW potential should be larger than in water or alcohols.To comply with this requirement while maintainingthe condition imposed by the applicability of TPT, wehave decided to introduce an auxiliary site Y mimickingthe Coulombic interaction of nitrogen and placed it onthe surface of the central sphere opposite to H sites sothat the bisector N–Y forms the normal to the H–H–Hplane (see figure 1). Without this choice the predictedstructural properties would be rather poor. Hydrogeninteractions are mediated by auxiliary X sites. Thecorrelation functions are shown in figure 4. As canbe seen, for the gNN and gNH functions the agreement isvery good. The gHH function of the PM exhibits adouble peak not observed for the parent model. Thereason for this discrepancy is the smooth repulsion inthe parent model which causes the valley between thetwo peaks to disappear.

The primitive model of ethanol, PM/EtOH, descendsfrom Jorgensen’s OPLS model. To keep the form ofthe primitive potential as simple as possible, we fixedthe dihedral angle at the most probable value of 180�.This choice has only a minimal influence on the averageBoltzmann factor and thus also on the hard-spherediameters derived from (11). As usual, the auxiliary Xsite is placed on the surface of the O sphere. Concerningthe SW parameters, we proceeded analogically to thecase of the previously considered primitive model ofmethanol [28], and slightly enlarged the range of attrac-tion to � ¼ 0:65dOO (as compared to � ¼ 0:62dOO

required by theory) to improve the performance of themodel. In practice, this minor change does not violatethe single bonding condition imposed on hydrogen.As can be seen from figures 5 (a) and (b), the site–site

Table 2. Parameters of the developed primitive models.

Model Parameter (A)

PM/NH3 dNN 3.050

dXX 1.918

lXY 0.915

PM/EtOH dOO 2.623

dC1C13.929

dC2C23.904

dXX 2.065

l 1.705

PM/SPC-E dOO 2.642

dXX 2.087

l 1.638

PM/TIP4P dOO 2.652

dXX 2.122

l 1.724

PM/TIP5P (th) dOO 2.651

dXX ¼ dYY 2.097

l 1.060

PM/TIP5P (emp) dOO 2.651

dXX ¼ dYY 1.326

l 0.663


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3 5 7 9

gNN

0

1

2

3

2 4 6 8

gNH

0.0

0.5

1.0

1.5

r[Å]2 4 6 8

gHH

0.0

0.5

1.0

Figure 4. The site–site correlation functions of the OPLS model of ammonia (filled circles) [18] at 240K and densityd¼ 0.697 g cm�3, and PM/NH3 at the same density and �¼ 4 (solid line).

3 4 5 6 7

gOO

0

2

4

6

1 2 3 4 5 6 7

gOH

0

2

4

6

8

10

r[Å]2 3 4 5 6 7

gHH

0

2

4

6

Figure 5. The site–site correlation functions of the OPLS model of ethanol [19] with the fixed dihedral angle (filled circles) at298K and density d¼ 0.748 g cm�3, and PM/EtOH at the same density and �¼ 7 (solid line).


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correlation functions involving oxygen and hydrogenare in very good agreement with those of the parentmodel. Somewhat larger discrepancies are found forthe Ci–Cj correlation functions but the agreement isstill satisfactory. The probability distributions of bond-ing angles, shown in figure 6, are in good qualitativeagreement. The differences observed for angle � may beattributed to the form of SW interaction whose flat wellis not able to localize interactions in the same way as theCoulombic potential.

Because of their undoubted preference in applicationsover other models of water, we have considered theSPC/E, TIP4P and TIP5P models and developedtheir associated primitive models. All three modelscontain two X sites mimicking the interaction ofhydrogen; the PM/TIP5P model also contains two Ysites, corresponding to the negative M sites of the parentmodel. All auxiliary sites are placed on the surface of theO sphere (see figure 1). The three-site model resemblesthe previously briefly considered model [29] descending

4 6 8 10

gC1C1

0.0

0.5

1.0

1.5

2.0

4 6 8 10

gC1C2

0.0

0.5

1.0

1.5

2.0

r[Å]4 6 8 10

gC2C2

0.0

0.5

1.0

1.5

2.0

r[Å]

3 4 5 6 7 8 9

gOC2

0.0

0.5

1.0

1.5

3 4 5 6 7 8 9

gOC1

0.0

0.5

1.0

1.5

2.0

r[Å]

Figure 5. Continued.


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from SPC water. In the present case, however, the rangeof the attraction satisfies exactly the condition of stericincompatibility, thus ensuring that the X sites may notform more than one H-bond. Moreover, we have foundthat there are only minor differences between the newPM/TIP4P and PM/TIP5P models and the old four- andfive-site speculative models (EPM4 and EPM5 models,respectively) whose parameters were set only intuitivelywithout any direct connection to realistic four- or five-

site models. The site–site correlation functions and thedistribution of bonding angles of the models consideredare shown in figures 7–12. As for the PM/SPC-E andPM/TIP4P models, the site–site correlation functionsare again in very good agreement with the corre-sponding functions of their realistic counterparts. As aconsequence of the step-wise character of the inter-actions, main discrepancies are found around the firstmaximum and minimum. Nonetheless, coordination

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Figure 7. The site–site correlation functions of the SPC/E model of water (filled circles) [20] at 298K and density d¼ 0.997 g cm�3,and the PM/SPC-E model at the same density and �¼ 7 (solid line).

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Figure 6. The same as figure 5 for the percentage distributions of the bonding angles � and �.


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numbers of the parent and primitive models are thesame. The distributions of the bonding angles agreequite well, particularly for the PM/TIP4P model.By examining the site–site correlation functions of

the five-site model, we have found that the structural

properties of the TIP5P model are better reproducedwith shorter X–X and Y–Y repulsions than with thoseobtained through our direct method based on equations(10) and (11). This finding may be explained by thefollowing intuitive arguments. First, in comparison with

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Figure 9. The site–site correlation functions of the TIP4P model of water (filled circles) [21] at 298K and density d¼ 0.999 g cm�3,and PM/TIP4P model at the same density and �¼ 7 (solid line).

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Figure 8. The same as figure 7 for the percentage distributions of the bonding angles � and � .


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the TIP4P or SPC/E models, the charges of hydrogenin TIP5P water are only about one half of the magni-tude and therefore their repulsion should be smaller.Secondly, the usual tetrahedral bonding configurationof water molecules is held by the four auxiliary sites

tetrahedrally positioned on the surface of the O sphereand, unlike the three- and four-site models, a large rangefor the X–X repulsions is therefore not needed. Thispoints to a certain weakness of our simple theoreticalmethod based on the average Boltzmann factors that

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Figure 11. The site–site correlation functions of the TIP5P model of water [22] (filled circles) at 298K and density d¼ 0.985 g cm�3,and two versions of the corresponding primitive models: empirical model PM/TIP5P(emp) (solid line) and theoretical modelPM/TIP5P(th) (dashed line) at �¼ 7 .

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incorporate the same interaction into each effectivesite–site potential. In addition to the theoretical model(denoted as ‘th’) we have therefore also developed amodel whose parameters have been adjusted empiri-cally. When there is no special reason, the latter modelbased on an empirical optimization of the X–Xrepulsion (and denoted as ‘emp’) and characterizedby dXX ¼ 0:5dOO and � ¼ 0:25dOO, should be preferredover the model obtained by the theoretical route.Potential parameters of both versions of PM/TIP5Psare given in table 2, and their site–site correlationfunctions and distribution of bonding angles areshown in figures 11 and 12. The performance of theempirical PM/TIP5P model is comparable to that ofother models; only the location of the peaks is somewhatout of phase.

4. Conclusions

As part of a long-term multistage project aimedat developing a molecular-based theory of associatingfluids, we have developed primitive models for commonassociating fluids, ammonia, ethanol and water, inaddition to a recently developed model of methanol.These models have been derived directly from realisticmodels using the method which relies on the potentialfunction of the parent model and theoretical arguments.This is a considerable improvement over earlier simplemodels which were constructed using only intuitionwithout any explicit link to intermolecular interactions.It has been shown that despite their simplicity theprimitive models reproduce the structure of the realisticmodels very well. Moreover, the models satisfy theconditions imposed by applicability of the thermo-dynamic perturbation theory and their thermodynamicproperties should therefore be obtainable in a closedform using this theory.

All obtained results make it clear that the developedPMs exhibit the same features as the fluid of hardspheres (when compared with models of simple fluids)and indicate that they may thus play for associatingfluids the same role which hard spheres have playedfor simple fluids. Specifically, (i) because of their directlink to realistic potential models they may be used as areference fluid in perturbation expansions, and (ii) usingeither theory or computer simulations, they may beused to study various complex phenomena involvingwater or other associating fluids, such as solubility orstructure of fluids at interfaces. Investigations alongthese lines are underway and results will be reported indue course.

This work was supported by the Grant Agency ofthe Academy of Sciences of the Czech Republic (GrantsNo. IAA4072303 and No. IAA4072309), and by theGrant Agency of the Czech Republic (Grant No. 203/02/0764).

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CUMMINGS, P. T., 2002, J. phys. Chem. B, 106, 7537.[8] NEZBEDA, I., and LISAL, M., 2001, Molec. Phys., 99, 291.

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from realistic to simple models of associating fluids. ii. primitive models of ammonia, ethanol and...

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