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Thermodynamics of simple models of associatingfluids: primitive models of ammonia, methanol,ethanol and waterLuká[sbreve] Vl[cbreve]ek a & Ivo Nezbeda a ba E. Hála Laboratory of Thermodynamics, ICPF, Academy of Sciences, 165 02 Prague, CzechRepublicb Department of Physics, J. E. Purkyne 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 Aug 2006.

To cite this article: Luká[sbreve] Vl[cbreve]ek & Ivo Nezbeda (2004) Thermodynamics of simple models of associating fluids:primitive models of ammonia, methanol, ethanol and water, Molecular Physics: An International Journal at the InterfaceBetween Chemistry and Physics, 102:8, 771-781, DOI: 10.1080/00268970410001705343

To link to this article: http://dx.doi.org/10.1080/00268970410001705343

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Thermodynamics of simple models of associating fluids:primitive models of ammonia, methanol, ethanol and water

LUKAS VLCEK1 and IVO NEZBEDA1,2,*1E. Hala Laboratory of Thermodynamics, ICPF, Academy of Sciences,

165 02 Prague, Czech Republic2Department of Physics, J. E. Purkyne University,

400 96 Ustı n. Lab., Czech Republic

(Received 26 January 2004; revised version accepted 1 April 2004)

Thermodynamic P-V-T properties of primitive models that descend directly from realisticHamiltonians and reproduce the structure of real fluids have been studied both by means oftheory and computer simulations. Analytic expressions for the Helmholtz free energy of fourtypical associating fluids, ammonia, methanol, ethanol and water, have been derived using thethermodynamic perturbation theory. Whereas for the models which allow only single bondingof each site the first-order theory is sufficient, for models in which some sites may formsimultaneously up to two bonds the theory has to be extended to the second order.Comparison with simulation data shows that the theory is very accurate and has therefore alsobeen used to determine vapour–liquid equilibria. We have found fundamental differences inthe behaviour of different models; these differences are linked to the properties of thehydrogen-bond network that are discussed in detail.

1. Introduction

There is no doubt that the most useful and universalmethod to compute the properties of fluids from a givenintermolecular potential model is a perturbation expan-sion. The expansion starts from a decomposition of thetotal potential function, u, into reference and perturba-tion parts. The decomposition is crucial in the sense thatit governs the rate of convergence and hence also successor failure of the method. Moreover, if the decomposi-tion is dictated by physical rather than mathematicalconsiderations, the reference fluid may provide a con-siderably simplified, but physically footed, model ofreality enabling one to elucidate the basic molecularmechanisms.Whereas for non-polar fluids it was well established

more than two decades ago that their properties aregoverned by strong short-range repulsive interactions,for polar fluids with strong long-range Coulombicinteractions the problem has remained open. Onlyrecent extensive computer simulations [1–4] haveshown that the properties of polar, both associatingand non-associating, fluids are determined also primar-ily by short-range forces which, however, may be bothrepulsive and attractive. This finding has made itlegitimate to write various properties of associating

fluids in a perturbed form with the leading referenceterm given by a suitably chosen short-range reference(SRR) [5, 6].

To implement the perturbation expansion, the proper-ties of the SRR fluid must be well known and available,preferably, in an analytic form. Since the SRR may beyet too complex for theory to handle, another simplerfluid may conveniently be employed onto which theproperties of the SRR fluid are mapped. This has beenthe case even for theory of simple fluids which makes useof the known properties of the fluid of hard spheres; see,for instance, the scheme: Lennard-Jones fluid ! softrepulsive short-range reference fluid ! hard-spherefluid [7, 8]. For associating fluids the role of such simplemodels (a counterpart of hard spheres) may be playedby the so-called primitive models (PM). Such simplemolecular models of association were introduced along time ago [9, 10]. However, these original models(and their various later modifications [11–13]) arerather speculative with the intention to capture onlythe essence of association without any link to reality.Consequently, they may be used only in (semi-)empiricalmethods such as, for example, the SAFT approach[14], but may hardly be used within any rigorousperturbation theory.

Having in mind a general perturbation theory ofassociating fluids, we have recently developed a method*Author for correspondence. e-mail: IvoNez@icpf.cas.cz

MOLECULAR PHYSICS, 20 APRIL 2004, VOL. 102, NO. 8, 771–781

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

DOI: 10.1080/00268970410001705343

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to construct PMs directly from the realistic parentmodels and developed the PMs for a number ofassociating fluids [15–17]. These models do reproducethe structure of the parent models which is the necessarycondition for the perturbation expansion to converge.Another ingredient for the theory to be accomplishedis its ability to estimate the properties of the primitivemodel fluids in an analytic form. This can be accom-plished by using the thermodynamic perturbation theory(TPT) of Wertheim [18], the theory particularly tailoredfor such systems. With only a few exceptions dealingwith polymers (see, e.g. [19]), the TPT has been usedso far, to our best knowledge, only in the first orderwhich was, however, shown [20] to be only fairlyaccurate. We have therefore developed recently amodified version of the second-order theory (TPT2)[21] which accounts for a possible double-bonding ofsome sites and improves the performance of the theoryconsiderably.In this paper we consider the above-mentioned

modified second-order theory and study the thermo-dynamic behaviour of the recently developed primitivemodels of ammonia, methanol, ethanol and water. Thetheory employs as a reference system the fluid ofpseudo-hard bodies (the repulsive part of the interactionpotential) whose properties are determined both byapproximate theories and computer simulations. Com-parison of the results for the homogeneous phase withcomputer simulation data confirms the accuracy of thetheory. Application of the theory to phase equilibriathen shows that only the models of water exhibit aliquid–vapour phase transition whereas other modelsremain only in the fluid phase; this phase behaviour ofthe models is then discussed in detail.

2. Theoretical background

2.1. The modelsPrimitive models descending directly from realistic

intermolecular potential functions result from the fol-lowing approximations applied to the parent models[15, 16].

(1) The non-electrostatic short-range repulsive inter-actions are represented by a hard-sphere (HS)interaction which means that the molecule ismade up of HSs of diameters dii forming a fusedhard-sphere (FHS) body.

(2) The Coulombic interactions are represented asfollows:(i) The repulsive interaction between the like

charges is represented by an HS interaction.(ii) The attractive interaction between the unlike

charges is represented by a square-well (SW)interaction.

Denoting the sites which bear charges as (þ ) (positivecharge) and (�) (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

uHS

�jrð1Þi � r

ð2Þj j; dij

�þ

Xi, j2fþg

i, j2f�g

uHS

�jrð1Þi � r

ð2Þj j; dij

�

þX

i, j2fþ,�g

i 6¼j

uSW�jrð1Þi � r

ð2Þj j; �

�, ð1Þ

where

uHSðr12; �Þ ¼ þ1 , for r12<�

¼ 0 , for r12>� , ð2Þ

and

uSWðr12; �Þ ¼ ��HB, for r12<�,

¼ 0, for r12 >� : ð3Þ

The first term in equation (1) defines the FHS body(hard core), and the first two terms in (1) together definea pseudo-hard body (PHB) [23] used as the referencesystem in the TPT. Throughout the paper we will useunits such that energies will be scaled by �HB anddistances by dref , where dref is the size of the appropriate‘reference’ site, i.e. dref � dNN for ammonia anddref � dOO for all other models.

It is appropriate to note that the location of sites (seatof interaction) may, but need not necessarily, coincidewith the location of atoms in the real molecule. In[15, 16] we showed that in order to better localize thehydrogen bond it is convenient to use auxiliary siteswhich take over the interactions of the real Coulombicsites. In this paper we consider the PMs of ammonia,methanol, ethanol and water. The parent models forammonia [23], methanol and ethanol [24] are the OPLSmodels; for water we use the three most commonmodels: SPC/E [25], TIP4P [26] and TIP5P (in theoriginal version developed by Mahoney and Jorgensen[27]). Geometry of the PMs is shown in figure 1 and theirparameters are given in table 1. In accordance with theprevious papers, auxiliary sites representing hydrogeninteractions are denoted as X and those representingoxygen or nitrogen interactions are denoted as Y.

2.2. Thermodynamic perturbation theoryThermodynamic properties of the associating fluids

with the interaction potential defined by (1) are suitably

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described by the TPT. The theory starts from adecomposition of the full potential, uð1, 2Þ, into therepulsive reference part and highly directional attractiveperturbation part,

uð1, 2Þ ¼ uref ð1, 2Þ þ upertð1, 2Þ : ð4Þ

In the case of the PMs given by equation (1), thereference fluid is made up of PHBs and the perturbationterm is given by the SW attraction.The residual Helmholtz free energy per particle for the

fluids made up of molecules containing m (hydrogen-like) (þ )-sites which satisfy the condition of singlebonding and n complementary (�)-sites (oxygen, nitro-gen or some auxiliary sites), which may form up to twobonds, assumes the form [21]

��A=N"HB ¼ mð1� �Þ þ ln x0, ð5Þ

where � ¼ "HB=kBT , T is the absolute temperature, kB isthe Boltzmann constant, x0 is given by

x0 ¼�m

ð1þ m��Iþ� þ ðm��Þ2Iþ�þÞn , ð6Þ

� is the number density, � ¼ N=V , and parameter �,measuring ‘non-saturation’, satisfies the following cubicequation:

m2�2Iþ�þ�3 þ ½ð2mn� m2Þ�2Iþ�þ þ m�Iþ�Þ��

2

þ½ðn� mÞ�Iþ� þ 1��� 1 ¼ 0 : ð7Þ

Here the quantities I are the fundamental integrals ofthe theory,

Iþ� ¼

Zfþ�ð12Þgref ð12Þdð2Þ ¼ ½exp ½�� � 1�Jþ�, ð8Þ

Figure 1. Schematic representation of the primitive models considered in this paper.

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and

Iþ�þ ¼1

2

Zfþ�ð12Þf�þð23Þgref ð123Þdð2Þdð3Þ

¼ ½exp ½�� � 1�2Jþ�þ, ð9Þ

where J is the temperature-independent part of I, grefis the correlation function of the reference system andfþ�ð12Þ is the Mayer function of the H-bond betweenthe (þ )-site of molecule 1 and the (�)-site of molecule 2,

fþ�ð12Þ ¼ exp ½�� � 1, if H-bond is established,¼ 0, otherwise : ð10Þ

In (9), fþ� and f�þ refer to the same (�)-site onmolecule 2.In the case when only single bonding is allowed, i.e.

Iþ�þ ¼ 0, equation (7) reduces to a quadratic equationof the first-order TPT. This is the case of PM/TIP5P andPM/NH3 models. For the remaining models equation(7) must be used in full.Once the residual Helmholtz free energy is known,

the thermodynamic properties of the models areobtained by the standard thermodynamic relationships.Differentiating (5) with respect to � we get theconfigurational energy per particle (recall that Uref ¼ 0),

U=N ¼mð1� �Þ

exp½��� � 1, ð11Þ

and by differentiation with respect to � we get the con-tribution to the compressibility factor due to H-bonding,

�P�

� �pert

� zpert ¼ �mð1� �Þ 1þI0þ� þ m��I0þ�þ

Iþ� þ 2m��Iþ�þ

� �,

ð12Þ

where

I0 ¼ �@I@�

: ð13Þ

For the models satisfying the single bonding conditionequation (12) reduces to

zpert ¼ �mð1� �Þ 1þ@ ln Jþ�

@ ln �

� �: ð14Þ

Interpretation of the properties of the individualH-bonds formed by the double-bonded oxygen sitesmay be facilitated if one views the double-bonded site ascomposed of two independent sites, each satisfying thesingle bonding condition. Then the quantity correspond-ing to the fundamental integral is defined as

Ii ¼ yiIþ�, ð15Þ

where

y1 ¼1

21þ ð1� 4Jþ�þ=J2þ�Þ

1=2� �

; y2 ¼ 1� y1 : ð16Þ

After this decomposition the primitive models can beformally treated by the first-order TPT expressions,whose complexity corresponds to mixtures.

2.3. Pseudo-hard body fluidConcerning the reference PHB fluid, to determine its

properties is not a simple task, particularly because ofthe non-additive nature of the interactions representingthe Coulombic forces. As a consequence of this featureone cannot easily define the excluded volume; the volumeof the hard core, i.e. of the FHS body, is thereforeused for the ‘volume’ VPHB of the reference pseudo-hardbody.

Two common approximate routes to the thermo-dynamic properties are (1) the improved scaled particletheory (ISPT) and (2) the perturbed virial expansion(PVE) [28]. The first route uses a parameter � tocharacterize non-sphericity of the body and evaluatespressure from

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

ð1� �Þ3, ð17Þ

Table 1. Potential parameters of the considered primitivemodels.

Model Parameter [A] [d]

PM/NH3 dNN 3.050 1.0

dXX 1.918 0.63

�XY 0.915 0.30

PM/MeOH dOO 2.594 1.0

dMeMe 3.634 1.40

dXX 2.075 0.80

�XO 1.686 0.65

PM/EtOH dOO 2.623 1.0

dC1C13.929 1.50

dC2C23.904 1.49

dXX 2.065 0.79

�XO 1.705 0.65

PM/SPC-E dOO 2.642 1.0

dXX 2.087 0.79

�XO 1.638 0.62

PM/TIP4P dOO 2.652 1.0

dXX 2.122 0.80

�XM 1.724 0.65

PM/TIP5P dOO 2.651 1.0

dXX ¼ dYY 1.326 0.50

�XY 0.663 0.25

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where � is the packing fraction, � ¼ �VPHB. Followingthe convex body fluid results, the parameter � isobtained from the second virial coefficient, B2, usingthe relation [41]

B2=VPHB ¼ 1þ 3�: ð18Þ

The other route evaluates the compressibility factorfrom the expansion

zPVEð �Þ ¼ zref ð �ref Þ þXMi¼2

ðBi�i�1 � Bref, i�

i�1ref Þ, ð19Þ

where Bi is the ith virial coefficient and subscript ‘ref’refers to a suitable reference system, typically thesystem of HS at the same packing fraction as the fluidof PHB’s:

zHS ¼1þ �þ �2 � �3

ð1� �Þ3: ð20Þ

3. Computational details

To obtain the pseudoexperimental data of theconsidered PMs we carried out two types of thestandard Metropolis Monte Carlo simulations in anNVT ensemble [30]. In the first type, we simulated thereference system of pseudo-hard bodies to determine thevalue of the density-dependent integrals J. In the secondtype of simulation we measured the configurationalenergy, U, and the compressibility factor, z, of the fullprimitive models. Both types of simulations withN ¼ 216 were arranged in cycles, with one cycleconsisting of 2�N trial steps. There were 2� 106 cyclesgenerated for the computation of integrals J and 8� 105

cycles for the determination of the thermodynamicproperties at each state point. The convergence of thesimulations was controlled by the histogram of theinternal energy and statistical errors were assessed bythe block method [31].

The compressibility factor was obtained from thevirial theorem for step-wise potentials [32]:

z ¼ 1þ 4�PMX

fHS repg

d2ij ½hRmijigij�dijþ

(

þX

fSWbondg

�2ðexp½��� � 1Þ ½hRmijigij���

), ð21Þ

where the first summation runs over all hard-sphererepulsions and the second one runs over the square-wellattraction. The symbol �PM stands here for the packingfraction of the PM fluid, �PM ¼ ðp=6Þ��3

ref , where�ref ¼ �NN for ammonia and �OO otherwise, R is theseparation between the reference sites and �ij is the unitvector along the connecting line between sites i and j.

To evaluate the virial coefficients required inequations (17) and (19) we used the MC integration[33] and generated 108 configurations of three moleculesgiving a non-zero contribution to B4.

4. Results and discussion

4.1. The reference PHB systemTo describe the properties of the reference PHB

system by means of equations (17) and (19), wecomputed the lowest three virial coefficients and thevolume VPHB. The results for all the considered models,along with the coefficients � defined by equation (18),are given in table 2. From previous applications it maybe expected that for small values of � the results maybe reasonably accurate but that they may rapidly dete-riorate with increasing �. To assess accuracy of theapproach, we compare in figure 2 the theoretical resultswith computer simulations for the PM/TIP4P model.In addition to the largest value of �, this particularmodel (more accurately, its repulsive part) has beenchosen because its VPHB is approximately as large as thevolume excluded by the HS repulsions of the X sites,and the non-additive character of the potential is thusvery significant and largest errors can be expected. Asseen in figure 2, both routes to zPHB give very similarresults and both rapidly deviate from simulation data

Table 2. Virial coefficients, Bi, volumes, VPHB, and the parameter of non-sphericity, �, for the pseudo-hard bodies underlying thegiven primitive models.

Model B2/VPHB B3/V2PHB B4/V

3PHB VPHB/d

3ref �

PM/NH3 6.329 23.501 57.737 0.524 1.776

PM/MeOH 4.339 11.496 21.696 1.658 1.113

PM/EtOH 4.345 11.488 21.568 2.924 1.115

PM/SPC-E 6.570 25.300 63.833 0.524 1.857

PM/TIP4P 6.635 25.876 66.926 0.524 1.878

PM/TIP5P 4.855 13.879 26.486 0.524 1.285

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with increasing density. Therefore, to avoid any addi-tional inaccuracies of the final theoretical resultsbrought about by an inaccurate estimate of the referencefluid properties, we have fitted the simulation data by afunction of the form typical for hard-body fluids:

z ¼1þ �1�þ �2�

2 � �3�3

ð1� �Þ3: ð22Þ

This equation with coefficients �i listed in table 3provides an excellent fit to the simulation data for thepacking fractions up to � ¼ 0:5.To use the thermodynamic perturbation theory, it

is also necessary to know the values of the integrals Jþ�

and Jþ�þ defined by equations (8) and (9), respectively,which depend on the correlation functions of the ref-erence system. We have evaluated these quantities bycomputer simulations and fitted the results by exponen-tial functions, because they were shown in [21] to givean excellent fit of experimental data. For Jþ�, we use

Jþ� ¼ a0 þ a1 exp ½a2��, ð23Þ

and two exponential functions are used for Jþ�þ,

Jþ�þ ¼ a0 exp ½a1�� þ a2 exp ½a3�� : ð24Þ

The coefficients ai are given in table 4.

4.2. Thermodynamics of the primitive modelsTo assess accuracy of the TPT, we carried out MC

simulations to obtain pseudoexperimental data for theinternal energy and compressibility factor, i.e. thequantities that are easily accessible from simulations.We have considered such a range of densities andtemperatures that the saturation of the H-bonds rangesfrom zero to almost one (we recall that in PMs satu-ration is directly related to internal energy per particle,i.e. H-bonds per particle; for the models of alcohols andammonia the full saturation corresponds to U=N ¼ 1,

and for water to U=N ¼ 2). The results of the theoryand simulations for all models are compared in figures3–8. It is seen that, in general, the theory predicts thethermodynamic properties in quantitative agreementwith experiment, with only a minor inaccuracy inthe case of the PM/TIP5P model for higher saturations.This inaccuracy is related to the formation of a regular

Table 3. Coefficients of the equation of state, equation (22),of the reference PHB fluids.

Model �1 �2 �3

PM/NH3 2.5235 17.5892 23.5947

PM/MeOH 1.2974 1.8011 2.3446

PM/EtOH 1.2861 2.0887 2.6728

PM/SPC-E 3.0355 15.2827 14.5921

PM/TIP4P 3.1736 15.0409 12.4316

PM/TIP5P 1.7759 3.3930 6.1641

Table 4. Coefficients of the numerical fit of integrals J defined by equation (23).

Model Integral a0 a1 a2 a3

PM/NH3 Jþ� 2.1533e-3 1.1776e-2 2.3565

PM/MeOH Jþ� 6.7711e-3 5.6161e-3 8.8949

Jþ�þ 1.2495e-5 1.6018e-8 3.0604e-6 18.3012

PM/EtOH Jþ� 4.9816e-3 2.3928e-3 16.8917

Jþ�þ 3.9805e-6 1.8396e-7 1.1218e-6 30.8985

PM/SPC-E Jþ� 4.7291e-3 4.4541e-3 4.4157

Jþ�þ 5.4535e-6 1.9678 1.2586e-6 9.6350

PM/TIP4P Jþ� 3.6637e-3 2.9941e-3 4.7036

Jþ�þ 3.1283e-6 3.2569 7.8745e-7 10.0691

PM/TIP5P Jþ� 2.4525e-3 4.1413e-3 2.3786

Figure 2. Compressibility factor of the reference PHB fluidunderlying the PM/TIP4P model from simulations(symbols) and from theory (ISPT (17)—solid line; PVE(19)—dotted line).

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tetrahedral H-bond network which forces the non-bonded interaction sites to direct each otherand thus enhances the probability of their bonding.This effect, not accounted for in the first-orderTPT, was already observed and discussed in the pre-vious study of the older five-site primitive model ofwater [20].For a deeper understanding of the behaviour of the

three models of water it is useful to compare the strengthof H-bonds in the individual models. For this purposewe may use the decomposition of the double-bondedoxygen site in the PM/SPC-E and PM/TIP4P modelsaccording to (16) because the strength of the H-bondcan be related to the value of the fundamental integral,I, which depends on temperature and density (seetable 4). We find that the bonds in the PM/TIP5P aremuch stronger than in the other models of water and,

as a result, the five-site model can form a more stable H-bond network. Moreover, the bonds in the latter modelsare of unequal strength (different values of I1 and I2defined by (16)). These qualitative differences, which arereflected in the properties of the H-bond network, havediverse consequences, one of which is discussed below.

Good performance of the theory justifies its use forthe prediction of the properties that are not directlymeasured in simple computer simulations, particularlyvapour–liquid equilibria. Only concerning the criticalregion, one has to be aware of the limited applicability,in general, of the mean-field theories to which the TPTbelongs. Nevertheless, such predictions have at leastqualitative validity and are suitable for comparison ofthe primitive models.

First, we have found substantial differences betweenthe models of water on one side and the models of

Figure 3. Density dependence of the internal energy perparticle (upper graph) and the compressibility factor(lower graph) of the PM/NH3 model for a number ofinverse temperatures � from theory (lines) and simulations(symbols): � ¼ 0 (solid; filled circles); � ¼ 2 (dotted; opencircles); � ¼ 4 (short-dashed; filled triangles); � ¼ 6(dashed-double dotted; open triangles); � ¼ 8 (long-dashed; filled boxes).

Figure 4. The same as figure 3 for the PM/MeOHmodel: � ¼ 0 (solid; filled circles); � ¼ 1 (dotted;open circles); � ¼ 3 (short-dashed; filled triangles);� ¼ 5 (dashed-double dotted; open triangles); � ¼ 7(long-dashed; filled boxes); � ¼ 9 (dashed-dotted; filledboxes).

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alcohols and ammonia on the other side as regardsvapour–liquid equilibria. Whereas all three primitivemodels of water exhibit the vapour–liquid phasetransition, the models of the other associating fluidsexhibit only the fluid phase, even at very low tempera-tures where almost all H-bonds are saturated. Thisfinding is in full accordance with the results of Kolafaand Nezbeda [12] who explained the difference in thebehaviour of the original empirical models of waterand methanol in terms of percolation and stability ofthe H-bond network. According to their results, formolecules with two positive and two negative sites thepercolation threshold (i.e. the onset of formation of aninfinitely large cluster which is supposed to exist in theliquid phase) appears for the probability of forming anindividual bond equal to 1/3 or slightly larger. On theother hand, for molecules with only one site of eithersign, this probability must be equal to 1, i.e. all bondsmust be saturated but that state cannot be reached atnon-zero temperatures. It follows that although theH-bonds are an important structure-forming factor,

they need not be sufficient for the formation of theliquid phase. (We remind in passing that for primitivemodels to be used as a reference system it is sufficientthat they reproduce the structure of the considered fluidand not necessarily the thermodynamic properties, cf.the fluid of hard spheres.) These results also demon-strate a qualitative difference between the H-bondnetwork formed by water and by other associatingfluids, a finding that is overshadowed in realistic modelsby complex interactions.

Second, there are also certain differences betweenthe models of water themselves. Temperature versusorthobaric densities and saturated pressures as a func-tion of the inverse temperature are shown in figure 9,and the corresponding critical conditions are listed intable 5. Using the Clausius–Clapeyron equation, wehave also estimated the enthalpy of vapourization fromthe temperature dependence of the vapour pressure.Practically no temperature dependence of �Hvap hasbeen observed and very similar values have beenobtained for all three models suggesting that similar

Figure 5. The same as figure 3 for the PM/EtOH model. Figure 6. The same as figure 3 for the PM/SPC-E model.

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processes occur at the phase transition and that thesame number of hydrogen bonds is disrupted whenmolecules pass from the liquid to gaseous phase.Nonetheless, there are two noticeable differences.First, in the values of the critical compressibilityfactors and, second, in the temperature range of theexistence of the liquid phase. As regards the criticalcompressibility factor, it is well known that the water-like behaviour is characterized, among others, by avery low value of zc (for real water zc � 0:21, the valuewhich is very difficult to reach by perturbed, mean-fieldequations of state; typically zc � 0:3). The PM/TIP5Pmodel reproduces this low value of zc and the othertwo models give even smaller values which is evenmore favourable from the point of applications:addition of any other interaction pushes fast zc tolarger values turning the model to normal-fluid like.As regards the range of existence, we see that the PM/TIP5P model remains in the liquid phase up to highertemperatures, which reflects larger stability of theH-bond network. As for the other two models (i.e. those

with one double-bonding negative site), it is interestingto note that the PM/TIP4P, which forms weaker H-bonds (both fundamental integrals, I, are for a smallergiven density) and stronger repulsions (larger B2 of itsPHB), remains in the liquid phase to higher tempera-tures when compared with the PM/SPC-E model. Thisfinding can be understood if we decompose the double-bonding site into two single-bonding ones accordingto (16). As already mentioned, formation of the liquidphase depends on the possibility of water molecules toform two H-bonds per molecule, i.e. both negativesites must be able to form a bond. Since the strength ofthe H-bond chain depends on its weakest link, it isthe strength of the weaker negative interaction site,measured by I2, that matters. Comparing I2s forboth models we have found that for � � 0:2 theinteraction of the weaker site is stronger for the PM/TIP4P and thus, despite weaker average interactions,it forms a stronger H-bond network. This finding canalso explain the shift of the critical density of the PM/TIP4P model to larger values.

Figure 7. The same as figure 3 for the PM/TIP4P model.Figure 8. The same as figure 3 for the PM/TIP5P model.

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5. Conclusions

We have explored the possibility of accurate theore-tical description of the thermodynamic properties of therecently developed primitive models of ammonia,methanol, ethanol and water. This is part of a generalproject aiming at developing a molecular-based theoryof associating fluids, in which the primitive models are

supposed to play the role of a reference system and theirproperties are therefore required in an analytic form.

The used thermodynamic perturbation theory wasconsidered in the recently developed modified secondorder and comparison with computer simulation datahas shown that it is quantitatively correct. The theore-tical equation of state has therefore been applied todetermine vapour–liquid equilibria and the fundamentaldifference in the behaviour of the models of water andother models has been found; similarly, a differencebetween the models of water themselves has also beenfound. These differences have been traced back to andelucidated by the properties of the hydrogen-bondnetwork. The differences in the H-bond networks for-med by the primitive models of water are likely tomanifest themselves also in phenomena, such as hydro-phobicity, which largely depend on the stability andflexibility of H-bonds. Studies based on the theoreticalanalysis of H-bonding in primitive models thus providevaluable information that can also shed light on thebehaviour of real water for which such data are notaccessible.

This work was supported by the Grant Agency ofThe Academy of Sciences of the Czech Republic (GrantNo. IAA4072303).

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Table 5. The critical conditions for the models of waterobtained from the theory and the enthalpy of vapouri-zation, �Hvap, estimated from the Clausius-Clapeyronequation.

Model �c �c zc �Hvap

PM/SPC-E 7.9705 0.1295 0.146 2.1087

PM/TIP4P 7.8843 0.1685 0.178 2.1105

PM/TIP5P 6.7309 0.1396 0.222 2.0047

Figure 9. Vapour–liquid coexistence curve (upper graph)and orthobaric pressures (lower graph) for the PM/SPC-E(solid), PM/TIP4P (dotted) and PM/TIP5P (dashed)models of water. The crosses denote the location of thecritical points.

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