Extended primitive models of water revisited

MOLECULAR PHYSICS, 1998, VOL. 93, NO. 1, 25± 30

Extended primitive models of water revisited By MARTIN STRNAD² and IVO NEZBEDA² ³ ² E. HaÂ la Laboratory of Thermodynamics, Institute of Chemical Process Fundamentals, Acad. Sci., 165 02 Prague 6± Suchdol ³ Department of Theoretical Physics, Charles University, 180 00 Prague 8, Czech Republic ( Received 6 May 1997; revised version accepted 8 July 1997) A 4-site extended primitive model of water descending from the TIP4 potential has been reexamined over a range of the model parameters. It has been found, in contrast to recently reported results (Nezbeda, I., and SlovaÂ k, J., 1997, Molec. Phys., 90, 353), that the model clearly is superior to 5-site models descending from the ST2 potential and yields the liquid structure in very good agreement with that of real water.

1. Introduction Within the context of perturbation theories, simple models used in statistical mechanics of ¯ uids result from well de® ned simpli® cations of realistic potentials. To make the perturbation expansion converge it is required that the simple model ¯ uid reproduces the structure of the original ¯ uid of interest. It has been shown [1] that the structure of water is determined primarily by short range forces, which may be both repulsive and attractive, and the class of so-called primitive models aims to approximate, at a very simple level, the complex force ® eld generated by Coulombic site± site interactions for small particle± particle separations. The basic idea behind the original primitive models (PMs) of associated ¯ uids is to approximate the molecule by a hard core with embedded interaction sites of two kinds with a square-well attraction between the unlike sites (to mimic hydrogen bonding) [2± 4]. Recently we have introduced extended primitive models (EPMs) [5] in which, in addition to this attractive interaction, there is also a repulsive interaction between the like sites. The only acceptable primitive models of water so far have been 5-site models (PM5 and EPM5) descending from the ST2 potential [6]. These models can be used for developing both a molecular-based equation of state of real water [7, 8]and a theory of the ST2 water. However, the ST2 potential is inferior to newer potentials (as e.g. TIP models [9]) and therefore it would be desirable, both from theoretical and practical points of view, to have also a primitive model descending from these potentials. In [5]we used computer simulations to study both the structural and thermodynamic properties of 3- and 4site extended models descending from TIP potentials but within rather a limited range of their parameters; 0026± 8976/98 $12. 00

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speci® cally, only the smallest meaningful value of the hard sphere diameter s R for the repulsive site± site interaction has been considered. Results we obtained were rather disappointing in the sense that the models did not meet the requirements imposed on them, namely, that they reproduce the structure of the parent ¯ uid. Fortunately, later investigations showed that these results need not be correct in general and that the short range repulsive interactions (at least in these models) play as important a role as the attractive ones, and that the parameters s R and ¸ (range of the square-well interaction) must be considered simultaneously. This short communication completes our project on primitive models of water based on ideas put forward about ® fteen years ago [2, 10]by reinvestigating EPM3 and EPM4 ¯ uids. It is shown that by appreciating the substantial e ect of the short range repulsive interactions on the structure of PM ¯ uids it is possible to construct also an EPM4 which satis® es all the requirements imposed on the primitive models. Moreover, this EPM4 is simpler than the EPM5 model, better re¯ ects reality, and its hydrogen bonding range is e ectively shorter, which may be a considerable advantage for application of the theory. 2. Potential model The model considered here is a special case of EPM4 described in detail in [5] (and denoted as PM4 therein) and we therefore refer the reader to this paper for details. To summarize, the molecule is represented by a hard sphere of diameter s W with three embedded sites of two kinds (see ® gure 1 (a)). The complete pair potential consists of the following site± site interactions: (i) hard sphere repulsion (of diameter s W) between the centres of particles, (ii) hard sphere repulsion (of diameter 1998 Taylor & Francis Ltd.

26

M. Strnad and I. Nezbeda

Figure 1. (a) Geometry of the EPM4 rigid monomer comprised of the central sphere W, two ( + ) interaction sites, and one (- ) interaction site. Spheres R correspond to the hard sphere interaction between (+ )-sites. (b) A slice through the interaction ® eld of the EPM4 particle in the plane perpendicular to the + C+ plane and containing the C- vector. Circle RX denotes the `volume’ excluded by spheres R and the shaded area depicts the shape of the attractive potential well; R2 and R3 denote R spheres belonging to the second and third molecules in the `closest’ approach. R) between the ( + )-sites, and (iii) a coreless square-well attraction of range ¸ and depth e between ( + )- and ( - )sites. For the sake of simplicity we use henceforth units such that s W = e /kB = 1, where kB is the Boltzmann constant. The model is thus de® ned by three parameters, s R, ¸, and l- , from which only l- may be somewhat deduced from the parent model. In [5]it was shown for EPM5 that increase in s R may give rise to even qualitative changes in the structural properties of the EPM5 ¯ uid, and very convincing results were obtained for s R = 0.8. From the geometrical point of view, this value nearly coincides with 1 s R = 6 /2 /3 = 0. 816, the value corresponding to the tetrahedral arrangement of the ( + )-sites in which the repulsive spheres are in contact. It means that it is possible to exactly recover the tetrahedron-like structure of ice in which all the repulsive spheres are as close to one another as possible (see ® gure 1(b)). From this point of view the value s R = 0.8 seems thus very reasonable. Concerning l- , simple geometrical and physical considerations (it is required that the attractive well operates over the opposite side of the bisector + C+ , see ® gure 1 (b)) then dictate that l- must not exceed the value l- ,max = 31 /2 /6 = 0. 289. For the given s R and l- , the value of parameter ¸ must now be speci® ed. For both physical and mathematical reasons it has become common to impose two restrictions on these parameters, namely that (i) occurrence of a double bond between a pair of particles, and (ii) engagement of one (+ )-site in multiple bonding be prohibited. However, in some cases [5]these two physically independent conditions together may be too restrictive and the question arises as to what extent they must be strictly obeyed, especially when no such restrictions

s

occur for realistic potentials. Whereas condition (i) above, which for s R > ( 4 - ( 10) 1 /2 ) /3 1 /2 = 0.528 and the above speci® ed range of l- can be expressed as

[

¸ < ¸max =

[

s

2 R

2

+

]

1 - l 31 /2( 1 - s 4 -

[

2 R)

]

1 /2

- l- ] ,

( 1)

is the essential factor a ecting the structure and thus should be observed, the second condition may be lifted. We will not therefore adhere to it further, the consequences of which will be assessed a posteriori. Thus, according to equation (1) and for s R = 0. 8 and l- within the above range, the value ¸ = 0.7 has been chosen as a su ciently large range of hydrogen bonding without double bonds. Monte Carlo simulations on EPM4 models were performed in an NV T ensemble, details of which are given in [5]. The main quantities of interest were the average centre± centre (oxygen± oxygen) pair correlation function, gCC( r) = k gCC( r, X 1, X 2 ) l , where k ´l denotes angular averaging, and the average spatial arrangement of particles around an arbitrarily chosen particle. The structure is a ected also by modes of hydrogen bonding and because in the models considered the second condition of saturation is not strictly respected, we calculated the percentage of particles whose ( + )-sites were engaged in multiple bonding. Finally, we also calculated the structure factor [11] in order to keep the development of the simulated ¯ uids under control. Concerning the analysis of the spatial arrangement of molecules, we have adopted the same approach as in [5] with only a slight modi® cation of the coordinate system enabling us to prevent the contributions to the histogram in the vicinity of |cos µ| = 1 (for details see [5]). Instead of the bisector + C+ , we measure the angle µ, 0 < µ < p ,

Extended primitive models of water revisited

27

Figure 2. Average centre± centre correlation functions of EPM4 for h = 0. 35, b = 5, and di erent values of l- : (a) l- = 0. 0; (b) l- = 0. 05; (c) l- = 0. 1; and (d ) l- = 0. 2.

from the axis perpendicular to this bisector in the + C+ plane; the angle u , 0 < u < 2p , is then measured from the bisector + C+ itself. 3. Results and discussion Following the geometrical and physical arguments detailed in section 2, we have set s R to 0.8 and ¸ to 0.7, and considered l- within the range 0 £ l- < 0. 289. Two sets of computer simulations on EPM4 ¯ uids were then performed. The aim of the ® rst set was to obtain the dependence of the properties of the model on l- for a typical thermodynamic condition. On the basis of these results, one value of l- was chosen and the second set of simulations was performed for a number of thermodynamic conditions. In order to choose the thermodynamic condition for the ® rst set of simulations, we used the same arguments as those applied in our previous papers [2, 5]and set the packing fraction h (de® ned by the volume of the central sphere) to 0.35. Owing to the smaller number of pairs of repulsive spheres in comparison with EPM5, this density corresponds, roughly, to h = 0.3 for the latter model. As for the temperature T, it is known that the primitive models solidify not far from the Boyle temperature TB.

Since the inverse temperature b B = 1 / TB rapidly increases with increasing l- ( b B < 4 for l- = 0 whereas b B < 6.5 for l- = 0.15), the value b = 5 has been chosen as a reasonable representative. The results of the ® rst set of simulations are shown in ® gure 2. Unlike the case of the 5-site EPM5, the initial portion of gCC is always hard-sphere-like. However, outside the contact three qualitatively di erent types of behaviour, characterized by the location and shape of the ® rst minimum and the second maximum, are observed: for l- = 0.0, for l- = 0.05 and 0. 10, and for l- = 0.2. Although in the ® rst case the coordination number is water-like, gCC is too ¯ at, with the second peak at about 2. This behaviour is the consequence of the geometry of the model and re¯ ects the fact that a large number of particles (about 14.7%) violates the condition of the hydrogen bonding saturation. In the second case this number falls considerably (to 5. 7% and 1. 6% for l- = 0.05 and 0.1, respectively), and the observed behaviour becomes water-like. Further increase in l- causes the ® rst minimum close to contact to disappear, and we observe a very ¯ at gCC spread over a wide range of separations, with the second maximum past 2. It is clear from the geometry of the model that

28

M. Strnad and I. Nezbeda

Figure 3. Average centre± centre correlation functions of EPM4 for b = 5 (upper row) and b = 6 (lower row) and for packing fractions (a) h = 0. 3; (b) h = 0. 35; and (c) h = 0. 4.

with increasing l- the region in which a bond may be established is getting narrower and splits into two lobes resembling partly the geometry of the 5-site model (see ® gure 1 (b)). Overall hydrogen bonding thus becomes relatively weak, and temperature would have to be decreased to obtain the same structure as for smaller l- . To conclude, we see that, depending on the model parameters, EPM4 may give ¯ uids with qualitatively di erent behaviour. Over a narrow range of about l- < 0.10 EPM4 ful® ls (in a statistical sense) the conditions of saturation and reproduces the centre± centre correlation function. We have therefore set l- = 0. 15 and performed the second set of simulations at three di erent packing fractions and two inverse temperatures, b = 5 and b = 6. For all these conditions the hydrogen bond network is well established, as can be deduced from low values of the internal energy. Nonetheless, even for b = 6 the system undoubtedly remained in the homogeneous ¯ uid phase, as we could deduce from the structure factor. Whereas, in general, increase in b strengthens the e ect of association (resulting eventually in a glassy state), increase in density should enhance excluded volume e ects. These expectations have been con® rmed

and the results are shown in ® gure 3, from which it is seen that the interference of the excluded volume e ects and association at h = 0.4 distorts the shape of gCC, although the coordination number NC remains waterlike. Even though a typical water-like shape for gCC is obtained in the remaining cases, NC at h = 0.3 is quite low, which is mainly the consequence of the lower number density (due to the geometry of EPM4, the location of the ® rst minimum does not change). The location of other local extrema seems sensitive to both density (a shift of the second maximum and minimum towards smaller r is observed as h increases) and temperature (a slight shift of the above extrema towards larger r with decreasing temperature). It may be concluded that the present EPM4 yields both the water-like shape of gCC and the correct value of NC within thermodynamic conditions h < 0.35 and b ³ 5. Moreover, for the values considered of the geometrical parameters the problem of hydrogen bond saturation has been suppressed to only a statistically negligible level. Similarly as for EPM5, we compare in ® gure 4 the average centre± centre correlation function obtained from di raction experiments [12] with that of the EPM4 ¯ uid for b = 6 and h = 0.35. The agreement is


Figure 4. Comparison of the average centre± centre correlation function of real water obtained from di raction experiments (solid line) [12] with that of EPM4 for b = 6 and h = 0. 35 (dotted line).

very similar to that found for the pair correlation functions of the Lennard-Jones and hard sphere ¯ uids [13]. Outside the contact EPM4 reproduces gCC surprisingly well and even better than EPM5 [5], which seems to be a direct consequence of the spatial arrangement of molecules discussed in detail below.

29

Even if the function gCC( r) exhibits the typical waterlike shape, the true liquid water structure is not guaranteed due to the integral character of gCC which may hide possible defects. A deeper insight into the structure therefore may be gained by examining features of the full correlation function gCC( r, X 1 , X 2 ) . From the technical point of view, two simplifying methods can be found in the literature, both employing a partially angle-averaged correlation function gav º gCC( r, X 1 ) : (i) a three-dimensional map of the isosurface of gav [12, 14], and (ii) a slice through gav for a ® xed r [14]. An alternative used in our preceding paper [5]is to draw the spatial distribution of bonded particles around a central particle, although one could also consider all particles in the ® rst shell instead. In the case of the primitive models we have veri® ed that for the set of thermodynamic conditions under consideration these last three approaches yield almost the same pro® les, the typical representatives of which for EPM5 and the present EPM4 are shown in ® gure 5. While the EPM5 ¯ uid exhibits the strict tetrahedral arrangement re¯ ected by four equivalent peaks in ® gure 5 (a), this is not the case for EPM4. In contrast to EPM5, two `middle’ peaks of EPM4 are lower and, as a direct consequence of the shape of the potential well (see its slice at cos µ = 0 in ® gure 1 (b)), they are not so sharply separated as those of EPM5. This situation, re¯ ecting the interaction of the only ( - )-site of the reference molecule with ( + )-sites of neighbouring ones, thus resembles both the slice through gav at ® xed r mentioned in [14], and the three-dimensional maps of isosurface of

Figure 5. Average spatial arrangement of the bonded molecules about a central one for (a) EPM5 with ¸ = 0. 4, h = 0. 3 and b = 5, and (b) EPM4 with h = 0. 35 and b = 6.

30


gav in [12, 14]as well, in which the corresponding lobes of gav are joined together, too. It may be concluded then that the spatial arrangement of molecules in the EPM4 ¯ uid seems to reproduce the arrangement of real water molecules in a more satisfactory way than the EPM5. To summarize, it has been shown that a range of parameters may be found such that the EPM4 ¯ uid exhibits all properties the primitive models are required to exhibit. Moreover, because of the smaller number of sites employed in comparison with EPM5, the model enjoys relatively large ¯ exibility as regards the spatial arrangement of particles; as a result, the structure of the EPM4 ¯ uid is also more realistic and closer to that of real water even in details.

References [1] Nez bed a , I., 1997, J. molec. L iquids, in press. [2] Kolafa , J., and Nez beda , I., 1987, Molec. Phys., 61, 161.

[3] Kola fa , J., and Nez beda , I., 1991, Molec. Phys., 72, 777. [4] Walsh, G. M., Guedes , H. J. R., and Gubbins, K. E., 1992, J. phys. Chem., 96, 10 995. [5] Nez bed a , I., and Slovaík , J., 1997, Molec. Phys., 90, 353. [6] Stillinger , F. H., and R ahma n, A., 1974, J. chem. Phys., 60, 1545. [7] Chapma n, W. G., Gubbins, K. E., Jackson, G., and R adosz , M., 1989, Fluid Phase Equilibria, 52, 31. [8] PavliíCÏek , J., and Nez beda , I., 1996, Fluid Phase Equilibria, 116, 530. [9] Jorgensen , W. L., 1981, J. Amer. chem. Soc., 103, 335. [10] Bol , W., 1982, Molec. Phys., 45, 605. [11] Nez bed a , I., and Kolafa , J., 1995, Molec. Simulation, 14, 153. [12] Soper , A. K., 1994, J. chem. Phys., 101, 6888. [13] Hansen, J. P., and Mc Donald , I. R., 1976, Theory of Simple L iquids (London: Academic Press). [14] Svishc hev , I. M., and Kusalik , P. G., 1993, J. chem. Phys., 99, 3049.