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From realistic to primitive models: a primitive model ofmethanolLUKÁ[Sbreve]C VLČCEK a & IVO NEZBEDA a b

a E. Hála Laboratory of Thermodynamics , Institute of Chemical Process Fundamentals,Academy of Sciences , 165 02, Prague, Czech Republicb Physics Department , J. E. Purkyně: University , 400 96 Ústí n. Lab., Czech RepublicPublished online: 18 Nov 2009.

To cite this article: LUKÁ[Sbreve]C VLČCEK & IVO NEZBEDA (2003) From realistic to primitive models: a primitive model ofmethanol, Molecular Physics: An International Journal at the Interface Between Chemistry and Physics, 101:19, 2987-2996,DOI: 10.1080/00268970310001605750

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MOLECULAR PHYSICS, 10 OCTOBER 2003, VOL. 101, No. 19, 2987-2996 + Taylor & Francis 0 TayloraFrandr G a p

From realistic to primitive models: a primitive model of methanol LUKAS VLCEK' and IVO NEZBEDAlS2*

' E. Hala Laboratory of Thermodynamics, Institute of Chemical Process Fundamentals, Academy of Sciences, 165 02 Prague, Czech Republic

2Physics Department, J. E. Purkyni: University, 400 96 Usti n. Lab., Czech Republic

(Received 26 April 2003; accepted 7 July 2003)

An attempt to develop a methodology to construct a primitive model which descends directly from a parent realistic short-range model and reproduces its structural properties has been made. The realistic three-site OPLS model of methanol has been chosen as a test case. The primitive model copies the geometry of the OPLS model and thus pictures the methanol molecule as a hard heteronuclear dumbbell (representing oxygen and carbon atoms) with one embedded hydrogen site. All sites interact as hard spheres with the exception of the oxygen- hydrogen pair which may form a hydrogen bond mimicked by a square-well attraction. To determine parameters of the model two routes have been followed: (i) theoretical, based on an effective sphericalized site-site potential obtained from the parent potential, and (ii) semi- theoretical which makes use of the knowledge of the structure of the dense parent fluid. Both sets of parameters provide similar results and reproduce the structure (sitesite correlation functions, distribution of H-bonds and H-bond geometry) of the parent OPLS fluid reasonably well.

1. Introduction Simple theoretical models of matter have contributed

a good deal to the advancement of our understanding of nature and to the development of elaborate theories, with the ultimate goal of quantitative understanding of the properties of real fluids.

In the statistical mechanics of fluids there are two basic routes leading to molecular models: (i) the traditional, and most common, route is based on intuitive specula- tions and on the knowledge of the properties of indi- vidual molecules and/or small clusters, e.g. shape and size of the molecules or their dipole moment; (ii) the other route arrives at simple models by a series of well-defined approximations to the originally complex formulation of the problem. These approximations are based on the known effect of various parts of the given total Hamiltonian on the properties of fluids, e.g. the well- known fact that the structure of non-polar fluids is determined primarily by the short-range repulsive part of the intermolecular interaction. Whereas the former method usually defies further refinement, the latter is part of a general rigorous scheme and is therefore open to further development. It is appropriate to remark that it is also possible that one and the same model may result from both types of considerations. A typical example is the model of hard spheres, originally intro- duced by van der Waals as a simple approximation for the (impenetrable) core of molecules. The same model

*Author for correspondence. e-mail: IvoNez@icpf.cas.cz

however arises from rigorous calculations as an approx- imation to the short-range soft-repulsive part of the intermolecular potential of non-polar fluids [ 1, 21.

As regards associating fluids, the phenomenon of hydrogen bonding (H-bonding) does not result from any common simple intermolecular interaction and early models of these fluids focused therefore on models of particular (macroscopic) properties [3]. Most of the effort targeted at water (which has actually become a synonym for associating fluids) and early attempts used then to relate its properties to strong dipoldipole interactions and first potential models reflected this view, see e.g. [4].

An intuitive simple molecular model of water (called the Mercedes-Benz model today by some researchers [5]) was apparently first proposed by Ben-Naim [6]. A similar model in three dimensions was then proposed by Bol [q. Both models are based on the Bjerrum concept [8] of distributed charges and model H-bonding by a strong short-range and strongly orientation-dependent attrac- tion between interaction sites of different kinds. A similar model for associating fluids, in general, was then proposed by Smith and Nezbeda [9]. As a further development along these lines, Kolafa and Nezbeda proposed similar models for water and methanol [lo], and later also for ammonia [ll]. All these models and their various modifications were then used as basic molecular models upon which a semi-empirical statistical association fluid theory (SAFT) was developed [12] (for a review on SAFT see [13]).

Molecular Physics ISSN 00268976 print/ISSN 1362-3028 online 0 2003 Taylor & Francis Ltd http://ww .tandf.co.uk/journals

DOI: 10.1080/00268970310001605750

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2988 L. VlEek and I. Nezbeda

The above molecular models capture the essence of H-bonding but because they are not linked to any realistic ihteraction model they yield a structure only in qualitative agreement with observations on real fluids. To bring the models closer to reality, Nezbeda and co-workers [1&16] introduced another class of primitive models, called extended primitive models (EPMs) which, in addition to the attraction between the unlike sites mimicking H-bonding, also incorporate the omnipresent repulsion between like sites. It turned out that the structure of EPM fluids was even in semi-quantitative agreement with that of real water. Moreover, recent findings on the effect of long-range forces on the properties of fluids [17-201 have made it legitimate to write various properties of associating fluids in a per- turbed form with the leading reference term given by a suitably chosen short-range reference (SRR) [21, 221. To implement this scheme, the properties of the SRR fluid must be available in an analytic form and this is the point where EPMs enter the scheme: it is convenient to estimate the properties of the SRR fluid by mapping them onto the properties of an appropriate EPM (cf. a similar scheme for simple fluids: Lennard-Jones fluid +. soft repulsive short-range reference fluid --+ hard-sphere fluid). To demonstrate the feasibility of this approach we mention the recently derived molecular-based equation of state for water [23].

To extend the above approach to other associating (and strong polar) fluids and to put it on a sound footing, extended primitive models descending directly from real- istic models (called parent models) must be developed first. The requirements imposed on the EPM are (i) to reproduce the structure of the parent model as faithfully as possible, and (ii) to satisfy certain conditions to make the application of the thermodynamic perturbation theory (TPT) of Wertheim [24] possible.

This is the first paper reporting results of a general project with the goal of developing a methodology to

construct primitive models and, consequently, to develop a molecular-based theory for a set of selected polar and associating substances. We begin this project with methanol which is another typical associating fluid and an important organic solvent. In comparison with water, its strength for H-bonding is weaker and it does not form such a well-developed H-bonding network, which suggests that it may be a suitable test case for developing a methodology of constructing primitive models. As a parent model we choose an SRR fluid descending from the three-site OPLS model of Jorgensen [25] which is known to reproduce the properties of real methanol quite well. The descending primitive model copies the geometry of this realistic model and it is shown that most of its parameters may also be obtained a priori from the parent model using theoretical arguments.

2. Basic definitions and computational details Although the full-atom concept (six interaction sites)

for methanol is feasible [26], it has been shown that it has no advantage over the united-atom concept [27] and the most common realistic effective pair-potential model of methanol is thus the three-site OPLS model of Jorgensen [25]. The mode1 has rigid geometry for nonlinear tri- atomics, see figure l(a), and the total interaction energy is given by site-site interactions as follows:

= U L J ( ~ , ~ ) + U C O U I ( ~ , ~ ) , (1)

where (1, 2) stands for the complete set of varia- bles defining the mutual position and orientation of molecules 1 and 2, subscripts i a n d j refer to the oxygen (0), methyl (Me) and hydrogen H sites, defined by the position vectors $) and rV = Ir,! ) - rI1)l. The Lennard- Jones (LJ) parameters, E and u, and the charges, q, for

1 ’

H t

(4 (b) Figure 1. Geometry of (a) the realistic 3-site OPLS model of methanol [25] and (b) the descending primitive model MeOH3.

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Table 1. Potential parameters of the realistic three-site OPLS potential model of methanol [25].

~

0 85.547 3.07 -0.700 Me 104.17 3.775 0.265 H 0.0 0.0 0.435

i = j are given in table 1; the geometric mean is used to define the cross-interactions:

The dipole moment of the model, pmod = 2.2D, is sig- nificantly greater than the experimental value p = 1.7 D in order to account for a rather large polarizibility but otherwise it reproduces both structural and thermody- namic properties of real methanol very well. The dimer global internal energy, U = 28.45kJmo1-’(U/kB =

3421.7K), is found for the linear hydrogen bond 0. aH-0 configuration at the 0-0 separation 2.73 A.

An SRR fluid results from the MeOH potential (1) by gradually switching off its long-range Coulombic part,

U S R R ( ~ , ~ ) = UMeOH(192) - S(roo; R’,R1’)uc,,i(1,2), (3)

where S(r; R‘, R”) is the switch function defined by

S(r; R’,”’) for r < R ’ ,

for r > R”. R‘)2(3R”- R’ -2r)/(R’’ - R’)3, for R‘ < r < R” ,

(4)

For the switching range (R’, R”) we use the values found in [20], i.e. (R, R”) = (5.5,7.5) A. The structure of both the above models, (1) and (3), is identical and their thermodynamic properties differ typically by several percent only [20].

The above short-range potential model of methanol, equation (3), serves as the parent model for an EPM to be constructed. The EPM should therefore respect its geometry and the basic features of the site-site interac- tions. Thus, because there is no LJ interaction associated with the H-site (and therefore no direct harsh short- range H-H repulsion), the methanol molecule will be represented by a heteronuclear dumbbell, formed by 0- and Me-spheres, with an embedded H-site within the O-sphere, see figure l (b) . For the Coulombic part of potential (1) we will then follow the strategy adopted for the derivation of successful primitive models of water [28]: the repulsive interaction between the like charges will be represented by a hard-sphere interaction, UHS, and

the attractive interaction between the unlike charges by a square-well interaction usw. Justification of this approx- imation stems from the findings that only the short-range part of the Coulombic interactions affects the properties of fluids [19]. Furthermore, to begin with as simple a model as possible, we neglect the charge on the carbon atom, i.e. a weak bond between the methyl group and oxygen. The three-site EPM of methanol, denoted as MeOH3 and shown schematically along with its interac- tions in figure 2, thus assumes the following functional form:

The first two terms define the repulsive hard core interaction (first term) and the repulsive interaction between the H-sites (second term),

and the last term represents H-bonding between the 0- and H-sites in the form of a square-well attraction,

To find the optimized values of the potential parameters, dv, 3, and E H B , is the goal of the paper and is dealt with in the next section.

Properties readily available and directly related to the potential model are those at low density. The low density limit for the site-site correlation functions is given by the average Boltzmann factor (ABF) defined with respect to the pair of sites of interest [29],

(eij) =/ exp[-u(l,2)/kBTld(l)d(2), (8) r y s o n s t

where kB is the Boltzmann constant and T is the temperature. The main contribution to the low-density thermodynamics is given by the second virial coefficient, which may also be obtained from the ABF through the relation

B(T) = -2n [(eq(rg)) - 116 drv , / (9)

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2990 L. VlEek and I. Nezbeda

uox

0

-&Ht

n

square-well attraction

h I

hard sphere repulsion

rox I

0

Figure 2. Schematic representation of the site-site interactions in the primitive MeOH3 model considered.

valid for any i-j pair. For evaluation of the multi- dimensional integrals in (8) and (9) we use the Conroy integration method with the root points developed by Nezbeda et al. [30].

Both the structural and thermodynamic properties at higher densities were obtained from computer simula- tions. We used the standard Metropolis Monte Carlo (MC) simulations in an NVT ensemble with N = 216 particles in the simulation box. The only exception was the independent study of cluster formation for which we used 512 particles. No difference in the structural properties of systems with 216 and 512 particles was observed. The simulations were arranged in cycles, with one cycle consisting of N trial steps. There were 4 x lo5 cycles generated for each system considered and the configuration at the end of each cycle was recorded for an analysis and evaluation of the desired quantities. The convergence of the simulations was controlled by the histogram of the internal energy, and statistical errors were assessed by the block method [31]. Although some data for the realistic SRR model are available from literature [20], for the purpose of this paper we carried out our own simulations to get all properties of interest at the state points considered. For these simulations we used the same set-up as above with N = 2 16 particles in the simulation cell and literature data were used to check correctness of the developed code. In addition

to the internal energy, U, pressure, P, usual site-site correlation functions, go@&, and coordination numbers Nii (i.e. the number of particles in the first coordination shell), we also measured two specific angles charac- terizing the geometry of the hydrogen bond: (i) angle 8 = LO - H . . 0, which is defined by the 3 vectors pointing from the hydrogen to the oxygens, one on the same molecule and the other on the hydrogen bonded molecule, and (ii) angle 4 = LH. - - 0 - H, defined by OH vectors pointing similarly from the oxygen to the hydrogens.

+

3. Results and discussion 3.1. Parametrization of the model

The conditions imposed on EPMs are, in general, that they reproduce as faithfully as possible the structure of the parent model and, at the same time, that they are treatable by means of the TPT. For this purpose we use as a benchmark the results for the site-site correlation functions, angles 8 and 4, and the probability distribu- tion of H-bonds at T = 298 K and density 0.7619 g ~ m - ~ (henceforth referred to as the ambient conditions (P = 1 atm)); the corresponding dimensionless number density is p = (iV/V)c& = 0.4143. It is also desirable to develop an EPM using theoretical arguments and to reduce the use of the known experimental data and ad hoc adjustment to a minimum.

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Primitive model of methanol 299 1

If the EPM to be developed is to correspond to reality, there should be (at least an approximate) direct connec- tion between its parameters and those of the parent model. We therefore first try to deduce the MeOH3 parameters directly from the realistic parent potential U S R R . Thus, geometry of the primitive model copies that of its realistic parent model: site-site distances are 10 - Me1 = 1.430A and (0 - HI = 0.945A, and the angle formed by the 0-H and &Me vectors is 108.5'.

To get the parameters of hard-sphere repulsions one might formally use directly the repulsive parts of the respective LJ site-site interactions (see table 2) but due to the presence of strong and long-range Coulombic interactions this choice would be unphysical: the short- range part of the Coulombic interactions is known to be indispensable for associating fluids and no site-site interaction can therefore be treated separately without any regard for the other sites [19].

A theoretical possibility to derive the diameters d of the 0- and Me-spheres, and of the H-H repulsive range from the parent model is to take into account the composite influence of all sites using ideas of the Reference Average Mayer-function (RAM) perturbation theory 1321. This theory defines a sphericalized effective simple reference potential obtained from the ABF of the original molec- ular potential:

upM = -kgT 1n(ev), (10)

which is known to provide quite accurate site-site correlation functions. Here we are concerned with effective short-range repulsions of the individual sites defined by these potentials. To this end, we use the combined Weeks-Chandler-Andersen and Barker- Henderson methods well known from theories of simple fluids [2]. Thus, we first separate the sphericalized potentials at their minimum, e, to obtain their repulsive and attractive parts, then shift the repulsive part to make it zero at (see figure 3), and then apply the Barker-Henderson recipe to finally obtain the theoretical effective hard cores of the sites:

The hard cores resulting from this procedure (and referred to as theoretical parameters) are listed in table 2. As one could expect, because they account indirectly for collective effects of all site-site interactions, they are considerably different from those given by the LJ parameters of the parent model. From examination of figure 4, where we show the ABFs and the effective potentials, we may gather that for the 0-0 and H-H pairs the procedure used may be reasonable because the

Table 2. Hard core diameters of MeOH3 models, dii (obta- ined directly from the LJ part of the sitesite interactions of the parent model, and by means of the RAM theory using the average Boltzmann factors (th) and the exper- imental sitesite correlation functions (g)), and the range of the SW attraction, 1.

Diameters dii/A

LJ th g

0-0 3.070 2.64 2.594 Me-Me 3.775 3.92 3.634 &Me 3.404 3.29 3.257 H-H 0.000 2.12 -

x-x - 2.11 2.075

1 1.636 1.686

PUij

0

rrnin r.. 'I

Figure 3. Construction of the effective site-site repulsive interaction and an effective hard core. The solid line is a sphericalized site-site potential, the dotted line is its shifted repulsive part and the dashed line is the hard- sphere potential obtained from equation (1 1).

repulsive range is really short-range and thus well defined. However, this need not be the case for the Me-Me pair. The first peak of (eMeMe) is very broad and the entire concept thus becomes only very approximate (cf. a similar problem to estimate the properties of soft- repulsive models (u/r)" by a perturbation theory [33]). Another remark concerns additivity of interactions. We have determined the dumbbell core by determining the diameters of the 0- and Me-spheres which also implies that the @Me interaction is pair-wise additive, doMe = (dm + dMeMe)/2. We may however also use the above method, equation (1 l), to directly de!ermine doMe. From this procedure we get &Me = 3.291 A which compares excellently with the arithmetic mean of do0 and dMeMe.

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2992 L. VlCek and I. Nezbeda

0-0

2.4 2.8 3.2 3.6

I

Me-Me

3.6 4.0 4.4 4.8

H-H

1.6 2.0 2.4 2.8 3.2

2.4 2.8 3.2 3.6 3.6 4.0 4.4 4.8

dA1 1.6 2.0 2.4 2.8 3.2

flA1 Figure 4. Upper row: normalized sitesite average Boltzmann factors, (eg)’ = (eg)/(eg)max (solid), and the correlation functions,

g’.. = gg/gu.max (dashed) for 0, Me-Me and H-H interactions. Lower row: corresponding repulsive parts of the effective site-site potentials. !J

With the hard core parameters now completely defined, it remains to estimate the parameters corre- sponding to the Coulombic interaction between the 0- and H-sites, i.e. the depth and range of the square-well attraction. Since this attraction is to approximate the slowly decaying Coulombic interaction it is natural to try to make its range as long as possible. On the other hand, there are certain general conditions which H-bonding must satisfy and which set upper l i t s on the SW range. One constraint results from a general requirement that two molecules cannot be double bonded, and from simple purely geometrical arguments we then find an upper limit for I :

Nonetheless, for R < A,,,,, the H-site can be simulta- neously engaged in two bonds and when this is the case, the TPT of Wertheim is not easily applicable. We will therefore impose on I a stronger condition that the H-site may not form more than one hydrogen bond. Using geometrical arguments again we then get that

To investigate the effect of the range I on the structural properties we carried out a series of test simulations for a range of I s at a number of temperatures and the observed general trends may be summarized as follows: (i) the bonding angle distribution rapidly deteriorates with increasing R and becomes unrealistic for La,, and (ii) the correlation functions are very narrow and strongly overestimate the height of the first peak; for I,,, they are then completely out of phase. These findings are understandable if we recall the geometry of the model. First, for multiple bonding of H-sites becomes significant and distorts the geometry of the H-bond network. Second, the H-site is located deeply inside the 0-sphere which makes the bonding angle over which H-bonds can be established very wide, which leads to loss of the required strong directionality of the H-bonding. This effect is also known from constructions of realistic models and we will therefore follow the pattern established by studies of realistic models and place the interaction site mimicking the interaction of hydrogen (and labelled as X-site) on the surface of the 0-site, i.e. we set 10 - XI = (1/2)d00. This X-site is considered as an auxiliary site only and therefore in the discussion of the properties related directly to hydrogen we will consider the ‘true’ H-site located at the proper separation 10-HI = 0.945A (cf. placement of

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Primitive model of methanol 2993

the negatively charged M-site away from the oxygen site in the TIP4P model of water [34] or an auxiliary site in the 3-site model of hydrogen fluoride [351). For the maximal range of attraction between the 0- and X-sites we get from (13) ;11HB = 1.636A and this value is used for the SW range henceforth. Since the X-site is not a part of the OPLS model, we cannot use recipe (11) to directly determine the optimal range of the repulsion between the X-sites. Therefore we performed test simulations and found that the influence of this param- eter on the structure is best reflected by the function gHH

and the probability distribution of bonding angles 4 and 8. In figure 5 we show the results for gHH and C O S ~ at a typical temperature with dependence on d n ; the average values of angles 4 and 8 are shown in table 3. On the basis of the observed dependence we set dn = 0.8 doo, which seems to provide the best fit to the properties obtained for the realistic model (and which is, accidentally, equal to the value found for the extended primitive models of water [28]). We will refer to the model with the above parameters as the theoretical model and denote it as ‘th’.

3.2. Summary of model properties To directly (and quantitatively) compare various

structural properties of the MeOH3 model with those of the uSRR fluid, we now need to relate the primitive model temperature scaled by EHB to real temperature, i.e. T(K) + kgT/€HB = 1/B. One possibility to determine EHB is to use the second virial coefficient, equation (9). For the ambient temperature the second virial coefficient of the SRR fluid is -4824.04A3 which, using the theoretical parameters given in table 2, gives /I%:. However, it is known that €HB/kB for water is about 2200-2400 K which gives B % 8. Since the H-bonding in water is stronger than in methanol, this value must be an upper limit of /I. The value obtained from the second virial coefficient is thus unacceptable pointing to the fact that the second virial coefficient reflects the properties of the dimer only. From the other side, Jorgensen [25] used for methanol energy 12.552kJmol-’ to define the H- bond and thus this value sets the lower limit for /I at about 5. From other single value properties we may mention the coordination numbers in the first shell. These numbers are given in table 4 for three different /Is. As seen, the best agreement is obtained for B = 8. It seems appropriate to recall now that in practical appli- cations the temperature is set so as to provide the best fit to the property at hand and any specification of EHB is therefore at this moment unnecessary. For the purpose of comparison, and as a reasonable compromise, we use the value /I = 7; nonetheless, we also show the results for /I = 8 to see the effect of temperature.

In figures 6 8 we compare various structural proper- ties of the ‘th’ model and the parent model at ambient

2.0 2.5 3.0 3.5 4.0 4.5 5.0

44 0.025

-1.0 -0.5 0.0 0.5 1 .o

cos cp

Figure 5 . The correlation function gHH (upper graph), and the probability distribution of cos q5 (lower graph) for the MeOH3 model ‘th’ at fl = 7 and p = 0.252 with depen- dence on d= (0.7 (solid), 0.8 (dotted) and 0.9 (short dashed)), and for the parent model at ambient conditions (filled circles).

Table 3. Average bonding angles 0 and q5 with dependence on d= for the ‘th’ model.

0.7 0.8 0.9

147 147 146

107 114 124

parent model 156 113

conditions. As can be seen, the overall structure of the ‘th’ model compares reasonably well with that of the USRR fluid. In discussion of the site-site functions we must differentiate between the bonding and non-bonding sites because they are affected by different factors. Concerning the latter sites, the worst agreement is obviously found for the Me-Me correlation function. This result should have been anticipated because of the form of the (eMeMe) mentioned earlier: larger

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2994 L. VlEek and I. Nezbeda

Table 4. Coordination numbers in the first coordination shell for the ‘th’ and ‘g’ models of MeOH3 at various inverse temperatures /3, and for the short-range parent model at ambient conditions.

~

‘th’ 6.0 1.65 0.69 1.77 7.0 1.80 0.79 1.93 8.0 1.97 0.90 2.15

‘g’ 6.0 1.83 0.79 1.97 7.0 2.00 0.89 2.17 8.0 2.11 0.96 2.32

parent model T = 298 K 2.00 0.98 2.19

goo

3 4 6 7

MeMe

2

1

0 3 4 5 6 7 8

r[AI Figure 6. The site-site correlation functions of the parent

short-range model at ambient conditions (filled circles), and those of the MeOH3 models ‘th’ at ,!? = 8 (dotted line) and B = 7 (solid line), and ‘g’ at ,!? = 7 (dashed line).

oscillations of gMeMe and g o M e functions suggest that the diameter of the Me-sphere is overestimated. As for the H-H and G H functions, they reflect the SW approx- imation: the first peaks of g o H and goo are too narrow and high.

Since the methanol molecules may establish an H-bond with different numbers of neighbouring molecules, the probability distribution of the number of H-bonds per molecule is another useful indicator of the quality of the primitive model to approximate the properties of OPLS methanol. In the bar chart in figure 8 we compare this distribution for the ‘th’ and USRR models at ambient conditions (an energetic criterion with EHB = 12.552 kJ mol-’ has been used for the OPLS model [25]). We see that both models give the maximum for two bonds which implies that in both models linear clusters are preferred over branched ones. However, this preference is slightly stronger in the parent model as one could deduce from the ratio of the occurrence of two bonds per molecule versus occurrence of either one or three bonds per molecule.

Since the ‘th’ model is based primarily on the dimer properties, the question is whether it could be further improved by incorporating the knowledge of the dense fluid properties. The average Boltzmann factors used to evaluate the dumbbell parameters are the low density limit of the corresponding site-site correlation functions. One straightforward possibility to improve the perfor- mance of the ‘th’ model is thus to account for the density dependence and to use recipe (1 1) in combination with effective spherical potentials given not by the ABFs but by experimental gVs, uFMpg = --kgTlngii, see figure 4. We have therefore also explored this route using the same rules to get the parameters as for the ‘th’ model and the resulting parameters (referred to as model ‘g’) are also given in table 2. As we could expect, the size of the 0- sphere and that of the H-H repulsion are very close to the theoretical ones since the repulsive part of the effective potentials for these pairs is very steep. The difference is, however, observed for the Me-site, whose effective repulsion is very soft and thus a considerable density dependence can be expected. Using gMeMe we obtain for the Me-Me repulsion ~ M ~ M ~ = 3.634A. Secondly, to improve the correlation functions involving the H-site, it is also tempting to increase 1 only slightly beyond the &B limit so that the condition of single bonding would remain practically satisfied; we use the value 1 = 0.65 &O which exceeds l l H B by a few per cent only (see table 2). The model obtained in this way is referred to as the ‘g’ model and its parameters are listed in table 2.

The properties of the ‘g’ model at /3 = 7 (obtained from the coordination numbers, see table 4) are also shown in figures 6-8. It is seen that the shape of

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Primitive model of methanol 2995

gHH

2 3 4 5 6

$41 5 r

4 1 ; .. 9 ’ .

2 3 4 5 6 7

rt4 Figure 7. The same as in figure 6.

80

60 C 0 .-

a Q 20

gMeMe and &Me functions have been improved consider- ably. Relatively, the worst results remain for the func- tions of the hydrogen bonded sites, namely for g o H

and goo, whose shape is negatively influenced by the short range of the square-well attraction, i.e. the fist minima of the distribution functions are located at shorter separations. The other sitesite correlation func- tions that are not directly linked to the square-well interaction are in quite good agreement with the realistic ones, especially beyond the contact region that is directly influenced by the step-wise character of the sitesite interactions. Furthermore, very good agreement for the gMeMe and g o M e curves in figure 6 (for larger separations almost quantitative) justifies the assumption to neglect the charge on the methyl group. Further structural information is supplemented by the number of H-bonds per molecule shown in figure 8; the result is similar to that for the ‘th’ model.

4. Conclusions We have attempted to develop a methodology for an

easy determination of reasonable parameters of primitive models of associating liquids descending directly from their parent realistic models and without resorting at large to the known experimental data. The method has been exemplified for the primitive model of methanol. Following the practice established by similar existing models for water, we have represented the Coulombic interactions at short intermolecular separations by a

0 0 1 2 3 4

number of H-bonds per molecule Figure 8. Comparison of the percentage distribution of H-bonds per molecule in the MeOH3 models ‘th’ (light grey) and ‘g’

(dark grey) at B = 7 with that of the parent short-range model for ambient conditions (black).

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L. VlEek and I. Nezbeda

hard sphere interaction (repulsion between the like charges) and by a square-well interaction (attraction between the unlike charges). The geometrical parameters of the model have been determined directly from those of the parent model, and the parameters defining the H-bond interaction have been optimized so as to give the best structural results when compared to the realistic parent model at ambient conditions and, at the same time, to satisfy the conditions required by applicability of the thermodynamic perturbation theory of Wertheim. An empirical improvement of the theoretical route has also been explored. We have found that both routes yield very similar results and that the model, despite its simplicity, is able to predict the structure in good semi- quantitative agreement with its realistic parent model, which supports the earlier findings that the structure of associating fluids may be well reproduced without considering any long-ranged forces.

This study represents a further step towards devel- oping a molecular-based theory of associating fluids using a perturbation expansion about a suitably chosen short-range reference fluid. In this approach primitive models are used to provide an analytic estimate of the properties of the reference. Further investigations will therefore focus on (i) development of similar models for further selected compounds of practical importance, and on (ii) developing a theory for such models to obtain their properties in a closed analytic form.

This work was supported by the Grant Agency of the Academy of Sciences of the Czech Republic (Grant Nos. IAA4072303 and IAA4072309) and by the NATO Collaborative Linkage Grant PST.CLG. 978 178/6343. Valuable discussions with and useful comments by Professor H. L. Voertler, University of Leipzig, Germany, are also greatly acknowledged.

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