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THE JOURNAL OF CHEMICAL PHYSICS 127, 224501 共2007兲

Reference interaction site model and molecular dynamics study of structure and thermodynamics of methanol D. Costaa兲 and G. Munaó Dipartimento di Fisica, Università di Messina, Contrada Papardo, 98166 Messina, Italy and CNISM, Contrada Papardo, 98166 Messina, Italy

F. Saija CNR—Istituto per i Processi Chimico-Fisici, Sede di Messina, Salita Sperone, Contrada Papardo, 98158 Messina, Italy

C. Caccarno Dipartimento di Fisica, Università di Messina, Contrada Papardo, 98166 Messina, Italy and CNISM, Contrada Papardo, 98166 Messina, Italy

共Received 3 August 2007; accepted 4 October 2007; published online 11 December 2007兲 Thermodynamic and structural properties of various models of liquid methanol are investigated in the framework provided by the reference interaction site model 共RISM兲 theory of molecular fluids. The theoretical predictions are systematically compared with molecular dynamics simulations both at ambient conditions and along a few supercritical isotherms. RISM results for the liquid-vapor phase separation are also obtained and assessed against available Gibbs ensemble Monte Carlo data. At ambient conditions, the theoretical correlations weakly depend on the specific details of the molecular models and reproduce the simulation results with different degrees of accuracy, depending on the pair of interaction sites considered. The position and the strength of the hydrogen bond are quite satisfactorily predicted. RISM results for the internal energy are almost quantitative whereas the pressure is generally overestimated. As for the liquid-vapor phase coexistence, RISM predictions for the vapor branch and for the critical temperature are quite accurate; on the other side, the liquid branch densities, and consequently the critical density, are underestimated. We discuss our results in terms of intrinsic limitations, and suitable improvements, of the RISM approach in describing the physical properties of polar fluids, and in the perspective of a more general investigation of mixtures of methanol with nonpolar fluids of specific interest in the physics of associating fluids. © 2007 American Institute of Physics. 关DOI: 10.1063/1.2803059兴 I. INTRODUCTION

Methanol is the smallest organic compound able to exhibit a hydrogen bond and is largely used as organic solvent in experimental setups. It constitutes a typical example of associating fluid, which displays short-range and strongly directional attractive interactions. In comparison with water, the methanol does not give rise to such a completely welldeveloped network, due both to only one-donor configuration and to a weaker strength of the hydrogen bond.1 This simplified aggregation topology makes the methanol appealing for the development of statistical models, relating the macroscopic features of association with the details of molecular interaction, and in the construction of perturbation schemes 共see Refs. 2 and 3 and references therein兲. The influence of the hydrogen bond on the liquid structure is well characterized in the experimental literature, with NMR techniques4–6 and x-ray and neutron spectroscopy.7–10 Saturated and liquid density data have been also widely produced 共see, e.g., Refs. 11 and 12兲, and results for the liquid-vapor phase coexistence are reported in Ref. 13. The association properties have important consequences on the liquid struca兲

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ture, which is well characterized by winding chains of hydrogen-bonded monomers with a rather low fraction of branching points and an average number of six/ten molecules per chain.10,14 The radial distribution functions exhibit welldefined signatures of inter- and intrachain correlations, and typically display a sharp first peak followed by a rapid decay to the asymptotic values, without pronounced oscillations.15,16 This work is devoted to an extended theoretical investigation of structural and thermodynamic properties of several models of fluid methanol, from highly supercritical states down to normal conditions, including an overview of the liquid-vapor phase coexistence. We have adopted for this purpose the framework constituted by the reference interaction site model 共RISM兲 theory, formulated by Chandler and Andersen in the early seventies,17 and later extensively applied to investigate the structural properties of molecular fluids 共see, e.g., the reviews in Refs. 18–20 and the recent textbook of Ref. 21 that gives a good account of the RISM method兲. The RISM formalism constitutes a matrix generalization of the well-known Ornstein-Zernike equation for simple fluids,22 and is naturally suited when a molecular system is viewed as an assembly of interaction centers spatially disposed to realize a fixed geometry.

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© 2007 American Institute of Physics

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Current models of methanol are mostly based on a “united atom” representation of the molecular structure, where the CH3 group constitutes a single interaction site. In these three-site models, which do not show any significant differences with respect to the full-atom concept,23 the methyl group, the oxygen atom, and the hydroxyl proton bear a Lennard-Jones interaction plus a Coulomb partial charge. We have explicitly considered in this work the models J1 共Ref. 15兲 and his improvement J2,16 both introduced by Jorgensen in the context of the transferable intermolecular potential functions 共TIPS兲 and optimized potential for liquid alcohols 共OPLS兲 classes of molecular potentials, respectively, together with the H1 model by Haughney et al.14,24 Among other suggestions, not reviewed in this work, Nezbeda have introduced a family of “primitive” models, particularly useful to develop perturbation schemes, where the short-range repulsion is represented by an effective hard sphere, and the effects of hydrogen bond are accounted for by an attractive square-well interaction 共see the recent review in Ref. 2 for references兲. Van Leeuwen and Smit25 and Kvamme26 have proposed minor modifications to the J2 model to improve the overall appearance of the liquid-vapor coexistence curve and the correlation functions at ambient conditions, respectively. Classical simulation studies have characterized the fluid phase of different models of methanol 共see, e.g., Refs. 14–16, 23, and 26–30兲, including the liquid-vapor coexistence.25,31 As for theoretical investigations, Pettitt and Rossky32 carried out a RISM study of the J1 model addressing the effects of the hydrogen bond on the liquid structure. In a series of papers, Kvamme made a wide comparison between RISM theory and computer simulations for several mixtures of small polar molecules, including methanol 共see Ref. 30 and references therein兲. An extended assessment of RISM predictions coupled with a partially linearized version of the hypernetted chain 共HNC兲 closure 共named KH after Kovalenko and Hirata, see Refs. 33 and 34兲, concerning the fluid phase equilibria of several molecular fluids, including methanol, has been carried out in Ref. 35. The molecular Ornstein-Zernike formalism has been used by Richardi et al.36 to calculate the rotational invariant coefficients which define the liquid structure. Kiselev et al.37 have investigated the equation of state of methanol in the context of the selfassociating fluid theory. Recently, the dielectric properties of methanol in nonpolar solutes have been investigated through a mean-field approach by Pieruccini and Saija.3 Turning back to the subject of this work, we aim to discriminate the model which gives the best agreement between RISM theoretical predictions and simulation results. At the same time, we ascertain the most appropriate tools, among the various strategies offered by the RISM formalism 共different closures and various routes to thermodynamics兲, to characterize the structure and the thermodynamics of the model. Our predictions are systematically gauged against newly generated molecular dynamics data both at normal conditions and along two supercritical isotherms. The RISM liquid-vapor phase separation is compared with available Gibbs ensemble Monte Carlo data.25 We consider this study as a preliminary step in order to investigate from a theoretical point of view solutions of methanol with other nonpolar

solvent, over a wide range of concentrations, such as, in particular, mixtures with carbon tetrachloride, which have been the subject of recent experimental determinations in the group of one of the authors.38–41 In these studies it has been observed that, upon successive dilution of methanol, the dielectric and calorimetric results can be consistently described in terms of an increasing number of closed chains which form ring aggregates. Molecular dynamics simulations of mixtures of J2 methanol and CCl4 have also been carried out by Veldhuizen and de Leeuw.42 We are mostly interested in finding signatures of such properties in the framework provided by the RISM theoretical approach. The paper is organized as follows: in the next section the various models for methanol and the RISM theory are introduced; details of the molecular simulations are also given. In Sec. III we present and discuss RISM and simulation results for the structure and the thermodynamic properties, along with the calculation of the liquid-vapor phase coexistence. Last section is devoted to the conclusions and directions for future works. II. MODELS, THEORETICAL APPROACH, AND SIMULATION STRATEGIES A. Models for liquid methanol

The molecular geometry of all models envisaged in this work, namely, the J1,15 J2,16 and H1,14,24 is sketched in Fig. 1. All models are based on the vapor phase values of the intramolecular bond lengths and angles. The interaction potential between a site ␣ on a methanol molecule and a site ␤ on a different molecule at distance r is written as LJ coul 共r兲 + v␣␤ 共r兲 v␣␤共r兲 ⬅ v␣␤

= 4⑀␣␤

冋冉 冊 冉 冊 册 ␴␣␤ r

12

−

␴␣␤ r

6

+

q ␣q ␤ . r

共1兲

Sites ␣ and ␤ can be equally a methyl group 共CH3兲, an oxygen atom 共O兲, or a hydroxyl proton 共H兲. The first term accounts for the short-range interaction in the form of a Lennard-Jones 共LJ兲 potential 关where ⑀␣␤ is the attractive well LJ 共␴␣␤兲 depth and ␴␣␤ is the “collisional diameter,” i.e., v␣␤ = 0兴, while the second contribution arises from the presence of a partial charge q␣ on each site. Table I shows the various prescriptions for different models. J1 is the first potential developed for liquid methanol15 and shows some obvious

FIG. 1. Geometry of methanol models investigated in this work. The diameters of the methyl group and of the oxygen atom correspond to ␴CC and ␴OO of model J2 reported in Table I, respectively. The distances among the interaction sites are, irrespective of the model, LCH3O = 1.4246 Å, LOH = 0.9451 Å, and LCH3H = 1.9437 Å; ⬔CH3OH = 108.5°.

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weaknesses; in particular, it underestimates the strength of the hydrogen bond. In order to improve upon the model J1, the model H1 共Refs. 14 and 24兲 has different values of the partial charges. In the model J2 共Ref. 16兲 both the LJ parameters and the partial charges are slightly modified. The intramolecular bond lengths and angles are fixed for all models at issue. The LJ parameters for different types of sites are calculated according to the usual Lorentz-Berthelot rules,22 ⑀␣␤ = 冑⑀␣␣⑀␤␤ and ␴␣␤ = 共␴␣␣ + ␴␤␤兲 / 2. Although the O and H sites experience a net attractive interaction during the simulation runs, they cannot overlap due to the fact that the hydrogen site is well inside the repulsive core of the oxygen atom 共see Fig. 1兲. In the RISM calculations, instead, the r-space is sampled uniformly, and we need to introduce an “auxiliary site,”43,44 which adds an extra repulsion between the O and H sites. Following Pettitt and Rossky32 we have set for all models a purely repulsive LJ interaction with ⑀OH = 0.2 kcal/ mol and ␴OH = 1.85 Å. B. The RISM integral equation theory

In the RISM formalism17–19 the pair structure of a fluid of identical molecules, each carrying n distinct interaction sites, is characterized by a set of n共n + 1兲 / 2 site-site intermolecular pair correlation functions h␣␤共r兲 = g␣␤共r兲 − 1, where g␣␤共r兲 are the site-site radial distribution functions. The h␣␤共r兲 are related to a set of intermolecular direct correlation functions c␣␤共r兲 by a matrix generalization of the OrnsteinZernike equation for simple fluids,22 which reads in k-space as

c␣␤共r兲 =

再

H共k兲 = W共k兲C共k兲W共k兲 + ␳W共k兲C共k兲H共k兲.

In Eq. 共2兲, H ⬅ 关h␣␤共k兲兴, C ⬅ 关c␣␤共k兲兴, and W are n ⫻ n symmetric matrices and ␳ is the number density of the system. The elements of W ⬅ 关w␣␤共k兲兴 are the Fourier transforms of the intramolecular correlation functions. Provided that the molecules are rigid, we have explicitly w␣␤共k兲 =

otherwise.

The KH closure amounts using the full HNC approximation in the density depletion region, where g␣␤共r兲 ⬍ 1, and to replace it with a mean spherical closure when g␣␤共r兲 ⬎ 1, combining the advantages of both concepts. It has been employed to study the fluid phase equilibria of several polar molecular fluids, including water, methanol, and hydrogen fluoride 共see Ref. 34 for a recent account of the progress of this approach兲. The coupled set of Eq. 共2兲 and 共4兲, or 共5兲, has been solved numerically with a standard Picard iterative procedure. Calculations are performed on a grid of 4096 points, with a mesh size of 0.02 Å. The Coulomb interaction is splitted into a short-range part and a long-range contribution according to the Ng method 共see Ref. 19 for details兲. In the next section we shall present results for the sitesite radial distribution functions g␣␤共r兲 and for the site-site running coordination numbers, defined as

sin共kL␣␤兲 , kL␣␤

共3兲

where L␣␤ is the bond length between sites ␣ and ␤ on the same molecule. As appropriate for charged systems,20,45 the RISM equation has been complemented by a HNC closure for the direct correlations c␣␤共r兲, c␣␤共r兲 = exp关− ␤v␣␤共r兲 + ␥␣␤共r兲兴 − ␥␣␤共r兲 − 1,

共4兲

where v␣␤共r兲 is the pair potential introduced in Eq. 共1兲, ␤ is the inverse of the temperature T in units of the Boltzmann constant, kB, and the gamma functions are defined as ␥␣␤共r兲 = h␣␤共r兲 − c␣␤共r兲 共see, e.g., the review in Ref. 46兲. The RISM/HNC approach has been successfully applied to predict the structural and thermodynamics properties of several systems,20 but shows severe limitations when charged models are investigated in the dilute regime or near the critical region. For such reason we have also studied the partially linearized version of HNC named KH, which has been alluded to in the Introduction,33–35 and assumes for c␣␤共r兲,

exp关− ␤v␣␤共r兲 + ␥␣␤共r兲兴 − ␥␣␤共r兲 − 1, if g␣␤共r兲 ⬍ 1 − ␤v␣␤共r兲,

共2兲

冎

共5兲

N␣␤共r兲 = 4␲␳

冕

r

g␣␤共r⬘兲r⬘2dr⬘ ,

共6兲

0

which give the average number of ␤ 共or ␣兲 sites inside a sphere of radius r centered on the ␣ 共or ␤兲 site. The internal energy per particle is given by U ␳ = 兺 N 2 ␣,␤

冕

g␣␤共r兲v␣␤共r兲dr.

共7兲

As for the pressure, a closed formula can be deduced in the context of the HNC approximation,47

␤P ␳ =1− 兺 ␳ 2 ␣,␤

冕

2 dr关c␣␤共r兲 − 21 h␣␤ 共r兲兴 +

1 2共2␲兲3

冕

dk

⫻ 兵␳−1 ln det关I − ␳W共k兲C共k兲兴 + Tr关W共k兲C共k兲兴关I − ␳W共k兲C共k兲兴−1其,

共8兲

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TABLE I. Intermolecular potentials for liquid methanol. Parameters are taken from Ref. 15 for model J1, Ref. 16 for J2, and Refs. 14 and 24 for H1. Partial charges q are given in e units; ⑀ in kcal/mol; ␴ in angstroms. In all models, ⑀HH and ␴HH were originally set equal to zero; in the RISM calculations an extra repulsion is added between the O and H sites, see text. Model

qCH3

qO

qH

⑀CH3CH3

⑀OO

␴CH3CH3

␴OO

J1 J2 H1

−0.285 −0.265 −0.297

0.685 0.700 0.728

−0.400 −0.435 −0.431

0.181 13 0.207 01 0.181 13

0.174 75 0.170 01 0.174 75

3.860 93 3.775 00 3.860 93

3.082 78 3.071 00 3.082 78

with a similar expression for the KH closure.33 This expression has been compared along supercritical isotherms with the equation of state obtained through the compressibility route to thermodynamics,22

␤P =

冕

␳

d␳⬘关S共k = 0兲兴−1 ,

共9兲

has been fixed to ⌬t = 1 fs, and the electrostatic interaction is treated by the Ewald method 共see, e.g., Ref. 50兲. For each density and temperature, after an equilibration run of typically 500 ps, we have computed the relevant structural and thermodynamic averages over trajectories spanning a time interval of 500 ps– 1 ns.

0

where the k → 0 limit of the static structure factor, S共k = 0兲, is calculated according to the expression for interaction site fluids, S共k兲 =

1 兺 关␻␣␤共k兲 + ␳h␣␤共k兲兴. n 2 ␣,␤

共10兲

As for the Helmholtz free energy, a closed formula analogous to Eq. 共8兲 reads47 −

␤Aex ␳2 = 兺 2 ␣,␤ N −

冕

2 dr关c␣␤共r兲 − 21 h␣␤ 共r兲兴

1 2共2␲兲3

冕

dk兵Tr␳W共k兲C共k兲

+ ln det关I − ␳W共k兲C共k兲兴其.

共11兲

We have compared the expression 共11兲 with the free energy obtained from standard thermodynamic integrations along constant-temperature or constant-density paths,

␤A共␳兲 ␤A共␳0兲 = + N N

冕

␤A共T兲 ␤A共T0兲 = − N N

冕

␳

␤ P共␳⬘兲 d␳⬘ , ␳⬘ ␳⬘ ␳0 T

T0

U共T⬘兲 dT⬘ . NkBT⬘ T⬘

共12兲

共13兲

C. Molecular dynamics simulations

In order to test the accuracy of the RISM calculations, we have carried out extensive molecular dynamics 共MD兲 simulations. We have used for this purpose the program MOLDY 共Ref. 48兲 共see also the manual at the web page of Ref. 49 for a complete description of this package兲. Simulations have been performed in the canonical ensemble 共i.e., at constant temperature T, volume V, and number N of particles兲 at room conditions and along two supercritical isotherms, T = 550 and 950 K. We have used a standard sample composed of N = 512 methanol molecules; further runs with N = 256 and N = 2916 particles have been occasionally analyzed in order to detect any dependence of our results on the size of the system. The time step of the numerical integration

III. RESULTS AND DISCUSSION

For clarity sake we have divided the presentation and discussion of our results into two separate parts. In the first one we introduce and discuss the structural properties at normal conditions, i.e., T = 298 K and ␳ = 0.0148 Å−3, the molecular number density experimentally determined at pressure P = 1 atm. In the second part we report results for thermodynamics properties, both along the supercritical isotherms T = 950 K and T = 550 K, and at ambient conditions, and we discuss the overall appearance of the liquid-vapor phase coexistence. In what follows we shall use the notation C to indicate the methyl group CH3, O for the oxygen atom, and H for the hydroxyl proton. A. Liquid structure

In order to elucidate the differences among the various models, we show in Fig. 2 the MD results and RISM/HNC predictions for the site-site radial distribution functions 共RDF兲 g␣␤共r兲 at room conditions. It appears that the MD RDF involving the C site are almost indistinguishable from each other, independently of the model considered 共for such reason, we show at this stage the CO RDF only兲, whereas small differences emerge when the O and H correlations are considered. The high degree of short-range order, due to the presence of the hydrogen bond is especially visible in the high and sharp peak of the OH RDF. These features turn out to be slightly underestimated in the J1 model, characterized by a weaker electrostatic interaction between the O and H sites, in comparison with the J2 and H1 models 共see Table I兲. Although the J1 model has been practically superseded by the more refined J2 interaction potential, data are reported here essentially with reference to previous studies.32 The RISM correlations show the same trend of the MD results with even smoother differences among various models. Hinging upon this preliminary survey, and by comparing separately the RISM and MD results for all models at issue, we are led to the conclusion that, as far as the local structure is concerned, the best agreement between theory and simulations is obtained when the J2 model is considered. For this reason, we concentrate in Fig. 3 on a detailed comparison of all RDFs for the J2 model alone. It appears that the RISM

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J. Chem. Phys. 127, 224501 共2007兲

FIG. 2. 共Color兲 MD 共top兲 and RISM/ HNC 共bottom兲 site-site radial distribution functions g␣␤共r兲 at ambient conditions 共T = 298 K, ␳ = 0.0148 Å−3兲 for all methanol models envisaged in this work. In the left panels, OH RDF are grouped on the left and CO RDF on the right; in the right panels, HH RDF are grouped on the left and OO RDF on the right; All distances are given in angstroms. Note the different vertical scale in the bottom-right panel.

scheme reproduces the MD results with different levels of confidence, depending on the correlation function involved, with the notable accuracy of the CC RDF. The CC correlation is almost insensitive to the presence of the hydrogen bond and displays practically the same features with or in absence of the Coulomb interaction, as can be seen in the top-left panel of Fig. 3, where we show also the predictions obtained if the charges in the J2 potential are set equal to zero. Moreover, as again visible in the same panel, the CC correlation is well compatible with the RDF of a simple Lennard-Jones fluid 共as resolved through a Ornstein-Zernike/ HNC scheme兲, with the same ⑀ and ␴ parameters of the CC interaction. The position and the strength of the hydrogen bond, as indicated by the first peak in the OH RDF, shown in the top-right panel of Fig. 3, are also satisfactorily reproduced. On the other hand, the intermediate-distance secondary OH peak appears essentially misplaced in comparison with the simulation data. When looking for possible improvements of the OH interaction parameters, it emerges that this dephasing is not an artifact of the OH potential itself, and survives almost unaffected if we switch off the Coulomb interaction, and even when the full OH interaction completely reduces to zero 共see Fig. 3兲. A main feature in the CO and HH distribution functions is that the theoretical disposition of particles around a central site is satisfactory, with the positions of the first peaks and wells correctly predicted, although the height of the peaks and the depth of the correlation minima are smoother in comparison with the simulation data. This “on average” agreement is well reflected in the quantitative accurate RISM predictions of the integrated measure of correlations which

defines the running coordination numbers N␣␤共r兲 关see Eq. 共6兲 and the central panels of Fig. 3兴. The loss of resolution in the short-range structure, especially after the first correlation peak, reflects a general deficiency of the RISM approach, already documented in the past literature, as, for instance, when dealing with common water models.51,52 Predictions worsen for the CH and OO correlations. In the CH RDF the first peak at r ⬃ 2.8 Å, signaling a linear disposition of particles along the CO axis, and hence essentially related to intrachain correlations, is almost missed, barely replaced by a tiny shoulder at r ⬃ 3 Å; the second and higher peak, to be attributed to the CH correlations involving sites of molecules on different chains,53 is instead fairly well reproduced. The OO correlation retains a poor resemblance with MD results. All theoretical predictions have been obtained within the HNC closure. Our results are practically unchanged if we use the KH relation instead, but for the height of the first correlation peaks in all g␣␤共r兲, which appears invariably lower than the HNC counterpart. Since, as documented in Fig. 3, the RISM/HNC scheme already underestimates the importance of correlations at close contact, we conclude that at normal conditions the HNC closure performs slightly better than the KH one. On the other hand, as observed in Ref. 54, the KH scheme satisfactorily reproduces the features of the first solvation shell, and indeed, we have observed a complete agreement between KH and HNC results over the whole range of the coordination number function illustrated in Fig. 3. The general picture emerging from our study testifies that the RISM approach produces the most accurate predictions when the local structure is mainly determined by the short-range correlations, even in the presence of long-range

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J. Chem. Phys. 127, 224501 共2007兲

FIG. 3. 共Color兲 MD 共circles兲 and RISM/HNC 共full lines兲 g␣␤共r兲 for the J2 model at ambient conditions. All distances are given in angstroms. In the top panels, the dashed lines correspond to the neutral model, i.e., with q␣ = 0 in Eq. 共1兲. In the CC panel, the dotted line is the RDF of a LJ fluid. In the OH panel, the dotted line is obtained when the OH interaction is completely switched off. In the central panels, the running coordination numbers N␣␤共r兲, see Eq. 共6兲, are also shown 共squares: MD; dashed line, RISM兲, with the corresponding scale on the right.

electrostatic interactions, such as in the case of the CC RDF. In fact, the hypothesis that the molecular shape provides the dominant mechanism for the local order in the liquid underlies the original formulation of RISM 共where a molecule is essentially described as a collection of overlapping hard spheres17兲 and its fruitful applications. In this sense RISM can be thought as a natural extension to molecular fluids of the “Van der Waals picture” of simple fluids, successfully formalized in the same years 共see, e.g., the discussion in Ref. 55兲. Moreover, the comparison of the CC RDF with that of a simple Lennard-Jones fluid reveals that the best theoretical predictions are obtained when the site-site correlations weakly depends on the presence of surrounding sites. On the opposite side, the worsening of RISM predictions seems related to the influential presence of indirect correlations among the molecular sites: the OH secondary peak barely depends on the OH potential itself, the CH short-range structure is mainly mediated by the presence of the central oxygen and by the strong attraction between the O and H sites, and the correlation between the innermost O sites eventually depends in a complicated way from the global arrangements of all sites around a central molecule. The close resemblance of RISM predictions for all models envisaged in this study

makes our discussion coherent with the previous work of Pettitt and Rossky on the J1 model,32 where the focus was mainly on the influence of the hydrogen bond in determining the liquid structure. In that study, much emphasis is devoted to the evidence that, although a hydrogen-bonded structure exists for the model at issue, packing requirements associated with excluded-volume effects eventually dominate the whole appearance of the local order in the fluid. Several approaches have been proposed to improve upon some important deficiency of the original formulation of the RISM theory adopted in this work, notably the “diagrammatically proper” formulation of Chandler et al.56 and the use of more refined sets of closures52,57–61 with, in particular, the effective inclusion of approximate bridge functions.52,57,59 Hinging upon the body of our results, we suggest that the RISM approach could greatly benefit from the inclusion of the concept of “molecular closures,”62 to be considered as a remedy to the inherent impossibility of simple, atomiclike closures—which deal with a single pair of sites at once—to account for the observed indirect, sitemediated effects on the short-range structure of the fluid. The idea of molecular closures has been positively set forward by Schweizer and Yethiraj62 in the context of the reduced RISM

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Structure and thermodynamics of methanol

J. Chem. Phys. 127, 224501 共2007兲

FIG. 4. 共Color online兲 RISM and MD equations of state 共top兲 and free energy 共bottom兲 for the J2 model at T = 950 K 共left兲 and T = 550 K 共right兲. Symbols in the legends. In the top panels smooth interpolations of the simulation data are also shown. The MD free energies are obtained by thermodynamic integration through Eq. 共12兲 of the corresponding simulation pressures.

approach known as polymer-RISM,63,64 as a mean to include explicitly, also at the level of the closure relation, the “indirect” pathways by which two particular sites on different molecules can interact. In close connection, Raineri and Stell65 have preliminary introduced a family of “nonlocal” closures in the r-space, in an effort to correct the well-known trivial, ideal gas predictions of RISM concerning the dielectric constant of charged fluids 共see, e.g., Ref. 22兲. Work in this direction is currently under way. B. Thermodynamic properties and the liquid-vapor coexistence

The RISM equation of state and the free energy along the supercritical isotherms T = 950 and 550 K are compared with the corresponding MD results in Fig. 4. In order to calculate these properties as functions of the density within the KH and HNC schemes, we have either used the compressibility route of Eq. 共9兲 and the thermodynamic integration formula 共12兲 to obtain the pressure and the Helmholtz free energy in turn, or we have directly employed the closed formulae 共8兲 and 共11兲. In the case T = 550 K, we display the

HNC data obtained through the closed formula only, given the well-known difficulties of such a closure to converge in the low-density regime, and close to the critical region. The KH closure instead, explicitly devised to circumvent these problems,33 allows for a full integration all over the density interval investigated, ranging from ␳ → 0 up to the ambient density ␳ ⯝ 0.015 Å−3. It appears from Fig. 4 that, when favorable conditions for the HNC scheme to work properly are met, both closures give comparable results, and theoretical predictions tend to bracket the MD data with an overall better agreement for the free energy rather than for the pressure. At the lower temperature 共T = 550 K, right panels of Fig. 4兲, discrepancies among various theoretical schemes definitely emerge. In particular, the closed formula free energy intersects the corresponding MD results, whereas the thermodynamic integration procedure yields generally overestimated predictions. In comparison with the MD data, the theoretical pressure is generally too high, over a broad fluid density range. These results reflect well the high sensitivity of the pressure to the approximations introduced by the theory and confirm the general trend observed in Ref. 33 for the SPC

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TABLE II. RISM critical temperatures 共in kelvins兲 and densities 共in kg/ m3兲 for all models envisaged in this work, together with the corresponding Gibbs ensemble Monte Carlo results of Ref. 25.

FIG. 5. RISM/KH liquid-vapor coexistence points for the J1 共full squares兲 and J2 共full circles兲 models of methanol. Open symbols represent the corresponding Gibbs ensemble simulation results from Ref. 25. Interpolating curves are best fits of the theoretical predictions according to the scaling law for the densities and the law of rectilinear diameters, with the full diamonds indicating the critical points 共see the text for more details兲.

model of water: in that paper, the main source of errors was attributed to a general overestimate of the core repulsion contribution to the pressure 共and the chemical potential兲 when HNC-like closures are adopted. Moreover, as reported in Ref. 52 for the same model of water, the inclusion of “illegal”diagrams in the RISM formalism drives equally the pressure to higher values. A similar composition of effects is, hence, likely to manifest in our calculations. On the other hand, the analysis of g␣␤共r兲 carried out in the previous section shows that the strength of the hydrogen bond is generally well predicted, with possibly minor influence on the observed discrepancy among theoretical and simulation results. In the top-right panel of Fig. 4, we finally observe—perhaps due to a fortuitous cancellation of errors during the integration procedure—a nice agreement between the compressibility route pressure and the MD results in the density regime ␳ ⯝ 0.015 Å−3 which corresponds, at lower temperatures, to the liquid pocket of the system. Summing up all observations emerging at T = 550 K, it appears that the closed formulae eventually lead to the best predictions in the low-to-intermediate density regime where the liquid-vapor phase separation takes place.25,31 This conclusion has been corroborated by further trial thermodynamic integrations of Eq. 共13兲 along several constant-density paths in the liquid phase starting from the calculated values of the free energy at T = 550 K 共not reported here兲. As for the case T = 550 K, the application of this procedure generally results in a degradation of theoretical performances in comparison with the closed formula predictions. In Fig. 5 we report our estimate of the liquid-vapor coexistence in comparison with the Gibbs ensemble Monte Carlo 共GEMC兲 results of van Leeuwen and Smit.25 To avoid overcrowding, we have displayed in the figure results concerning models J1 and J2 only. All theoretical predictions are obtained within the KH scheme, which ensures the conver-

Model

Tcr 共RISM兲

Tcr 共GEMC兲

␳cr 共RISM兲

␳cr 共GEMC兲

J1 H1 J2

425 462 469

446 489 498

169 164 176

269 251 274

gence of the numerical procedure in the critical region and on the low-density side of the coexistence curve. When possible, we have verified that HNC predictions show no appreciable differences in comparison with their KH counterpart. Data for the critical point of all models are reported in Table II. In order to obtain the liquid-vapor coexistence curve, we have adopted a standard procedure: once the pressure and the free energy are calculated along several isotherms in the liquid and vapor sides of the binodal, the requirements of equal pressures and chemical potentials in both phases allow us to determine the coexisting densities at a given temperature. We have located the critical point by fitting the RISM results with the scaling law for the densities, ␳L − ␳V ⬀ 兩T − Tcr兩␤, where ␳L and ␳V are the densities of the coexisting liquid and vapor phases, respectively, Tcr is the critical temperature, and ␤ is the critical exponent for which we take the nonclassical value of 0.32; we have then used the law of rectilinear diameters to determine the critical density ␳cr. The main issue emerging from the analysis of Fig. 5 is that the RISM approach yields too low coexisting liquid densities, in comparison with the simulation results. The vapor branch is generally correctly estimated, but for the region closest to the critical point, especially for the J2 model. As a result, see Table II, the critical temperatures are reasonably well predicted, with a 4%–6% discrepancy with respect to the simulation counterpart, whereas the critical densities are definitely underestimated. RISM predictions generally preserve the trend of GEMC results in passing from model J1 to J2, with an increase of both the critical temperature and density. The theoretical approach barely discriminates among the small differences observed for the J2 and H1 binodals in the GEMC calculations.25 Our predictions agree well with the theoretical estimate of the methanol binodal given in Ref. 35, where a slightly higher critical temperature is reported; we have indeed verified that this small discrepancy depends 共i兲 mainly on the use of a mean-field value for the critical exponent ␤ in the scaling law for the densities, in place of the nonclassical choice adopted here, and 共ii兲 weakly on the different prescription for the auxiliary hydrogen site proposed in Ref. 35. More generally, the results of our investigation follow the trend reported in previous works33,35,52 for the liquid-vapor coexistence curve of the SPC model of water and other polar fluids, and clearly reflect the tendency of the RISM approach to overestimate the pressure, already commented on for the case T = 550 K. It is worth noting—see Ref. 52—that the theoretical predictions for the binodal do not definitely improve when the proper formulation of RISM 共Ref. 56兲 is adopted 共the liquid branch is slightly better, the vapor branch

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Structure and thermodynamics of methanol

FIG. 6. Internal energy per particle, calculated for ␳ = 0.0148 Å−3 and various temperatures. Dotted, solid, and dashed lines are RISM predictions for models J1, J2, and H1, respectively. Squares, circles, and triangles are the corresponding MD data.14 Stars are present simulation results for the three models at T = 298 K.

follows the opposite trend兲, even if a zeroth-order bridge function correction is enforced in the closure. We surmise that a valuable route to the thermodynamic properties could be provided by the use of different free energy functionals, which have been recently demonstrated to improve upon Eq. 共11兲 in the context of the interaction site formalism.66 A more extended comparison with simulation data than that reported in Ref. 66, especially involving the phase equilibria, would be at this stage highly desirable. We plan to develop such calculations for the methanol soon in the future. As far as the thermodynamic properties at ambient conditions are concerned, we compare in Fig. 6 the theoretical predictions for the internal energy of various models according to Eq. 共7兲, with present and past14 MD estimates. The best accuracy is observed for the J2 model, for which RISM predictions closely interpolate between simulation results, whereas the internal energy for the J1 and H1 models appears slightly overestimated. As for the pressure, simulation results tend to overestimate the experimental value, that ranges approximately around 1 bar. Our MD value, P = 35 MPa, with 10 MPa error bar—obtained with the largest sample employed in the simulations, composed of 2916 molecules—is consistent with the range P = 10/ 30± 10 MPa in the interval T = 268/ 338 K reported in previous calculations.14 As can be expected from our discussion about the binodal and the equation of state at T = 550 K, the closed formula predictions obtained with the use of Eq. 共8兲 severely overestimate the simulation datum by about a factor of 6. On the other hand, thermodynamic integration yields the much more reasonable value P ⬃ 70 MPa, slightly worsening at ambient temperatures the accuracy observed at T = 550 K in the top-right panel of Fig. 4. IV. CONCLUSIONS

We have presented extended theoretical RISM predictions of thermodynamic and structural properties of several models of liquid methanol. Our study encompasses the in-

vestigation of the liquid structure at ambient conditions, the analysis of the thermodynamic behavior at supercritical and normal temperatures and densities, and the calculation of the liquid-vapor phase coexistence. All results are assessed against newly generated molecular dynamics data and discussed with reference to previous theoretical calculations.32,33,35,52 We have compared our predictions for the liquid-vapor equilibrium with existing Gibbs ensemble Monte Carlo simulations.25 We have generally coupled the RISM equations with a HNC closure, but for the low-density regime and around the critical region, where the KH partially linearized version of HNC is more fruitful for the determination of phase equilibria.33 Most accurate predictions for the liquid structure are obtained when packing requirements constitute the dominant mechanism to organize the local order in the fluid, and apparently when the indirect correlations, mediated by the presence of surrounding sites around a given interaction center, play only a minor role. The internal energy at ambient conditions is well captured, whereas the pressure is generally overestimated. As a result, the liquid branch of the binodal 共and in turn the critical density兲 is misplaced to low densities, whereas predictions for the vapor branch and for the critical temperature are accurate within a few percents. In the discussion of the results we have indicated several putative routes to improve the RISM performances. With particular reference to the issues emerging for the present class of polar models, we suggest that these directions should include the use of more elaborated closures, possibly involving the concept of “molecular closure.”62 The adoption of more refined free energy functionals66 could also constitute a useful tool for the calculation of the thermodynamic properties and phase equilibria. Investigations along these lines are currently underway. As mentioned in the Introduction, this work paves the way to a more elaborate theoretical analysis of mixtures of methanol, and possibly other lower alcohols, in solution with nonpolar solvent such as carbon tetrachloride, for which experimental and theoretical studies3,38–42,67 have suggested the onset of different aggregation structures, in comparison with what is observed in the pure fluid. ACKNOWLEDGMENTS

This work makes use of results partly produced by the PI2S2 Project managed by the Consorzio COMETA, a project cofunded by the Italian Ministry of University and Research 共MIUR兲 within the Piano Operativo Nazionale “Ricerca Scientifica, Sviluppo Tecnologico, Alta Formazione” 共PON 2000-2006兲. More information is available at http://www.pi2s2.it and http://www.consorzio-cometa.it 1

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