Water–Methanol Mixtures: Simulations of Mixing Properties over the

Article pubs.acs.org/JPCB

Water−Methanol Mixtures: Simulations of Mixing Properties over the Entire Range of Mole Fractions Jean-Christophe Soetens* and Philippe A. Bopp Institut des Sciences Moléculaires, UMR CNRS no. 5255, and Department of Chemistry, Université de Bordeaux, 351 Cours de la Libération, 33405 Talence, France ABSTRACT: Numerous experimental and theoretical investigations have been devoted to the hydrogen bond in pure liquids and mixtures. Among the different theoretical approaches, molecular dynamics (MD) simulations are predominant in obtaining detailed information, on the molecular level, simultaneously on the structure and the dynamics. Water and methanol are the two most prominent hydrogen-bonded liquids, and they and their mixtures have consequently been the subject of many studies; we revisit here the problem of the mixtures. An important first step is to check whether a classical potential model, the components of which are deemed to be satisfactory for the pure liquids, is able to reproduce the known thermodynamic excess properties of the mixtures sufficiently well. We have used the available BJH (water) and PHH (methanol) flexible models because they are by construction mutually compatible and also well suited to study, in a second step, some dynamic property characteristic of hydrogen-bonded liquids. In this article we show that these models, after a slight reparametrization for use in NpT simulations, reproduce the essential features of the excess mixing and molar properties of water−methanol mixtures. Furthermore, in the pure liquids, the agreement of the radial distribution functions with experiment remains as satisfactory as before. Similarly, the translation self-diffusion coefficients D are modified by less than 10%. In the mixtures, they evolve nonmonotonously as a function of mole fraction.

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INTRODUCTION Water (H2O) and methanol (CH3OH, MeOH) are completely miscible liquids at room temperature and ambient pressure, as is to be expected from the polar character of these two molecules. The molecular dipole moments (in the gas phase μH2O = 1.86 D, μMeOH = 1.69 D) are similar; however, the static dielectric constant (relative permittivities) of the two neat liquids (εH2O ≈ 80, εMeOH ≈ 33) are markedly different because of the different structures. The presence of a hydrophobic group CH3 nevertheless contributes to peculiar properties such as the appearance of a minimum in the partial molar volume of methanol. Thus, alcohols are sometimes seen as soluble hydrocarbons, and models to explain the properties of their mixtures with water are still discussed.1−4 Molecular simulations are certainly the methods of choice to elucidate the properties of these systems on a microscopic level. Most theoretical efforts have used molecular dynamics (MD) simulations to study the structure, the dynamic properties, and the properties of mixing, separately or together.5−12 A number of experimental investigations have also been performed over the years, and most properties such as the density,13−15 the diffusion,16 the properties of mixing,17−20 and the structure1,21,22 are well known. Even though there is a wealth of studies on mixtures, not many combined efforts to obtain thermodynamic, structural, and dynamic properties in a consistent fashion have been reported so far. This has, however, been accomplished for some © XXXX American Chemical Society

neat liquids, for which detailed analyses of the hydrogen bond network have been carried out by combining MD simulations and vibrational spectroscopy experiments23−25 Applied to mixtures such as water−methanol, we expect that this strategy will lead to an improved qualitative and quantitative microscopic understanding of the peculiar properties of these systems. To achieve this goal, an important first step is to check whether a classical (flexible) potential model is able to reproduce the wellknown excess properties of these mixtures sufficiently well. Interaction models for mixtures are usually constructed by combining existing models developed and tested for the pure liquids, in most cases by keeping the partial charges of the initial models and applying empirical combination rules (Lorentz− Berthelot,26 Kong,27 and others) for the Lennard-Jones 12−6 terms. In the present study we have started from the available BJH28 and PHH29 flexible models for water and methanol, respectively; they have already been used extensively. (See ref 30 and the references therein.) Having been devised to be mutually compatible, they are also suited to study spectroscopic properties because they describe with good accuracy, in the approximation of classical mechanics, the vibrational modes of water and methanol and in particular the shifts in the OH stretching region.5 Received: April 7, 2015 Revised: May 29, 2015

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Table 1. Refitted Parameters for the BJH28 and PHH29 Models (Equations 1−4)

Most of the previous MD studies of water−methanol mixtures have, however, been performed at constant volume (NVE or NVT ensembles), mostly at the experimental density at room temperature and normal pressure. Because both models were not devised to reproduce the experimental densities, their use in NpT simulations is not satisfactory. In some cases (depending, for example, on mixture compositions) this may prove to be impossible as they then predict unrealistic pressures. For these reasons we did not investigate the energy properties in any depth. The mixing effects are small, and comparing NVE and NpT simulations, at different average pressures and temperatures and with different box sizes and truncation schemes, would require extensive work. Suffice it to say that we obtain the correct qualitative behavior of the potential energies as a function of the mole fractions. In this article we first introduce a new parametrization for the BJH and PHH models to allow their use in NpT simulations of both the pure liquids and the mixtures over the entire miscibility range, i.e. χ = 0 to 1. We have attempted to keep the modifications as small as possible. We show that we can in this way reproduce the essential features of the density, excess volume, and partial molar volumes of water and methanol over the whole range of molar fraction reasonably well. These properties of the binary systems are found to be quite sensitive to details of the intermolecular interactions, so the results presented below can be said to be a quite stringent test of the models. Of course, we also want to preserve the features that made the original models successful. We therefore proceeded to reevaluate some structural and dynamics properties of the pure liquids and mixtures in a consistent fashion. In the next sections we present first the modified model used in this study and the details of the MD simulations. Then we look at the properties that we considered to be important and that were consequently used to parametrize the model. After comparing the radial distributions functions (RDFs) for the modified and unmodified models in the pure liquids, we proceed to discuss the radial pair distribution functions and translational self-diffusion constants in the mixtures.

old parameters28,29

new parameters

1.045 418 3.47 3.86

1.463 10.45 3.61 3.52

p1 p2 p3 p4

kJ/mol kJ/mol Å, water−methanol interactions only Å

oxygen−oxygen (VOO), oxygen−hydrogen (VOH), and hydrogen− hydrogen (VHH) contributions are identical for the water− water, water−methanol, and water−methanol interactions, and any change in the parametrization should retain this homogeneity. VOO =

111 889 − p1 [exp(−4(r − 3.5)2 ) + exp(−1.5(r − 4.5)2 )] r 8.86

(1) VOH

26.07 41.79 16.74 = 9.2 − − 1 + exp(40(r − 1.05)) 1 + exp(5.439(r − 2.2)) r

(2) VHH =

p2 1 + exp(29.9(r − 1.968))

(3)

⎡⎛ p ⎞12 ⎛ p ⎞6⎤ VMeO = 4 × 0.3713⎢⎜ 3 ⎟ − ⎜ 3 ⎟ ⎥ ⎝ r ⎠ ⎥⎦ ⎢⎣⎝ r ⎠ ⎡⎛ p ⎞12 ⎛ p ⎞6⎤ VMeMe = 4 × 0.1883⎢⎜ 4 ⎟ − ⎜ 4 ⎟ ⎥ ⎝ r ⎠ ⎥⎦ ⎢⎣⎝ r ⎠

(4)

We note that the main modification concerns parameter p2, which essentially controls the H−H repulsion (cf. the original CF models32,33). The original large value of this parameter, which had essentially been carried over in later developments, was one of the main causes of the unsatisfactory pressures obtained originally. Earlier work34 aiming essentially at bringing the H−H radial distribution into better agreement with experimental data35 involved adjusting the distance (1.968 Å) where the repulsion sets in. It succeeded in improving the structure but failed otherwise, among other things, because the self-diffusion coefficient increased by almost 40% in the case where the best agreement of the gOO values was achieved. Simulation Runs. After a large number of trial runs to determine which parameters in the original model to keep and which ones to readjust (eqs 1−4) and determine the new values (reported in Table 1), we have first carried out MD simulations of the pure liquids in the NEV ensemble at ⟨T⟩ = 298 K and experimental density.17 The electrostatic interactions were treated here, as everywhere else, with the reaction (RF) method36,37 (here for charge−charge interactions); the dielectric constant of the surrounding continuum was set to εRF = ∞. A molecular cutoff equal to half of the edge of the box was applied. The other empirical potential terms were truncated so that numerical problems (e.g., under/overflows in the exponential functions) were avoided. These simulations were carried out with N = 1000 molecules. The equations of motions were integrated with the velocity Verlet algorithm38 with a time step equal to 0.2 fs. The simulations were run with the MDpol39 program package, and each simulation consisted of two independent calculations of 500 ps (2.5 M steps) each after appropriate equilibration periods. During the simulations, a small velocity rescaling has nevertheless been applied every 0.1 ps in order to reach the desired average temperature with a deviation of less than 1 K. The NpT simulations (at 298.15 K

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MODELS AND SIMULATION DETAILS Interaction Model. All simulations presented in this article make use of the BJH28 and PHH29 models for water and methanol, respectively. In both models, the molecules are represented by three interacting sites located on the oxygen and hydrogen atoms of the hydroxyl group and on the methyl group (which is thus a united atom, a drastic simplification). The total potential energy is the sum of intra- and intermolecular contributions. The intramolecular parts, the anharmonicity of which has been adjusted31 in the water case to yield, roughly, the observed vibrational frequency shifts between the gas and liquid phases,31 were not modified. The modifications of the intermolecular potentials were kept, as already stated, minimal and are not expected to affect the vibrational frequencies,31 which just should not require any modification of the intramolecular part. More details can be found in ref 5. The model for the intermolecular potential energy is derived from the central force model.32,33 The electrostatic part consists of Coulomb terms between partial charges (qO = −0.66e and qH = 0.33e for water; qO = −0.60e, qH = 0.35e, and qMe = 0.25e for methanol). We recall below the empirical parts of the intermolecular pair potential (in kJ mol−1 and Å); p1−p4 are the parameters that we selected to be refitted in this work, and they are summarized in Table 1. It is important to note that the B

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The Journal of Physical Chemistry B and 1 atm) were carried out in a similar manner; i.e., all of the technical details were the same excepted for the thermodynamic ensemble controlled using the Nosé−Andersen algorithm40−42 (with coupling constants of 5 and 1 ps−1 with a heat bath and piston, respectively). We studied nine mixtures with mole fractions varying from 0.1 to 0.9 as well as the two pure liquids in NpT-MD simulations (details in Table 3). In all cases, as above, two runs starting from different and uncorrelated initial conditions were first equilibrated before starting the 500 ps productions runs. The results reported below stem from these simulations. Thermodynamics. Mixtures of liquids such as the water− methanol binary system can be characterized by their excess mixing functions. For the volume, for instance, one can express the corresponding excess property VE as

V E = Vmix −

∑ nivi(0)

Figure 1. Density of water−methanol mixtures at T = 298 K, p = 1 atm as a function of the mole fraction of methanol: results from NpT-MD simulations (dashed line and circles) and experimental results17 (solid line). The ideal densities, shown for comparison (dashed−dotted line), are obtained from a linear combination of the mole volumes of the pure compounds.

(5)

i

where Vmix is the volume of the mixture, v(0) is the molar i volumes of the pure compounds, and ni is the number of moles of each component. In the present binary mixtures, the partial molar volume of each component, vi, can be obtained from the following relations Table 2. Comparison of Some Properties of Liquid Water and Liquid Methanol under Ambient Conditions Obtained from NEV-MD Simulations Using the Original Models (BJH28 and PHH,29Respectively) and NpT-MD Simulations with the New Parametrization for These Models, as Described in the Text

Water ρ (g cm−3) ⟨Epot⟩ (kJ mol−1) D (10−9 m2 s−1) ⟨p⟩ (atm) Methanol ρ (g cm−3) ⟨Epot⟩ (kJ mol−1) D (10−9 m2 s−1) ⟨p⟩ (atm)

NEV

NpT

⟨T⟩ =298 K

T = 298.15 K, p = 1 atm

BJH Model 0.99717 −41.9 1.5 4015 PHH Model 0.78717 −26.6 4.0 1000

Modified Model 0.997 −45.8 1.5 1 Modified Model 0.788 −28.2 4.4 1

Figure 2. Excess molar volumes obtained from NpT-MD simulations at T = 298 K, p = 1 atm (dashed line and circles), experimental results17 (solid line), and results from previous simulation work9 (dashed−dotted line).

Table 4. Parameters Obtained from a Fit Using Equation 7 on Experimental Data17 and Previous Work from Diego González-Salgado et al.9 and the Present Worka

a

Ak

exp

previous work

this work

A0 A1 A2 A3

−4.033 −0.294 0.478 0.851

−3.209 −0.291

−3.997 1.043 0.224 1.033

The corresponding curves are presented in Figure 2.

Table 3. Details of the NpT-MD Simulations of Water−Methanol Mixtures, Mole Fractions, Number of Molecules in the Simulation Boxes, Experimental17 and Simulated Densities, and Excess Volumes of the Systemsa density (g cm−3)

a

χMeOH

NMeOH

NH2O

exp

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

0 100 200 300 400 500 600 700 800 900 1000

1000 900 800 700 600 500 400 300 200 100 0

0.99710 0.97015 0.94719 0.92523 0.90339 0.88175 0.86073 0.84075 0.82195 0.80417 0.78690

VE (cm3 mol−1)

simulation

exp

± ± ± ± ± ± ± ± ± ± ±

0.0000 −0.3174 −0.6166 −0.8441 −0.9758 −1.0083 −0.9509 −0.8176 −0.6189 −0.3535 0.0000

0.9970 0.9656 0.9428 0.9204 0.9014 0.8820 0.8638 0.8437 0.8256 0.8077 0.7876

0.0001 0.0014 0.0005 0.0000 0.0007 0.0003 0.0002 0.0004 0.0007 0.0010 0.0003

simulation 0.000 −0.221 −0.510 −0.708 −0.904 −1.001 −1.040 −0.910 −0.749 −0.490 0.000

± ± ± ± ± ± ± ± ± ± ±

0.000 0.026 0.017 0.007 0.014 0.020 0.016 0.005 0.019 0.060 0.000

The results and uncertainties were obtained from two sets of uncorrelated calculations, as discussed in the text. C

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Figure 3. Partial molar volumes of water and methanol in their mixtures obtained from NpT-MD simulations at T = 298 K, p = 1 atm (dashed line and circles), experimental results17 (solid line), and results from previous simulation work9 (dashed−dotted line). The lines are given by eq 7, and the parameters are listed in Table 4.

Figure 5. Radial distribution functions of liquid methanol. Comparison of the results obtained from the original PHH water model29 under NEV conditions (dashed lines) with the modified model under NpT (T = 298 K) conditions (solid lines).

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RESULTS AND DISCUSSION Excess Properties. The simulated properties of the pure liquids in the NVE and NpT ensembles are given in Table 2. The goal of having the correct experimental pressure is achieved in both cases while the average potential energies of both systems are lowered. The self-diffusion constant of water remains the same, i.e., a bit too low, while that of methanol, already too high compared to experiment,16 is increased by about 10%. We present in Table 3 the densities and the excess volumes of the water−methanol mixtures as a function of the methanol mole fraction χMeOH. The variation of these properties is also shown in Figures 1 and 2 and compared with experimental results. The densities and their associated error bars have been obtained from the two independent 500 ps productions runs (as described in a previous section). The statistical errors appear to be negligible, and the agreement with experiment is fairly good as the maximum deviation is about 0.5%. The calculated excess volume VE also agrees well with experiment, and the maximum deviation with respect to ideal mixtures is about −1 cm3 mol−1. The extremum is slightly shifted toward larger mole fractions. Figure 2 also shows the curve resulting from a fit of simulated data using eq 7, and the corresponding optimized parameters are given in Table 4.

Figure 4. Radial distribution functions of liquid water. Comparison of the results obtained from the original BJH water model28 under NEV conditions (dashed lines) with the modified model under NpT (T = 298 K) conditions (solid lines). ⎛ ⎞ ⎛ ⎞ ∂V ∂Δv ⎟ vi = ⎜⎜ mix ⎟⎟ = vi(0) + Δv − χj ⎜⎜ ⎟ ⎝ ∂ni ⎠T , P , n ≠ n ⎝ ∂χj ⎠T , P j

i

(6)

where χi = ni/(n1 + n2) is the mole fraction of component i and Δv = ((VE)/(n1 + n2)). The following equation17 can be used to fit the simulated volumes: 3

V E = χ1 χ2 ∑ Ak (χ2 − χ1 )k k=0

(7) D

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Figure 6. Selected radial distribution functions (left ordinate) and corresponding integrals (right ordinate) for methanol−water mixtures as a function of the mole fraction of methanol. (Left) Top: gOwOw(r). Middle: gOmOm(r). Bottom: gOwOm(r). (Right) Top: gOwHw(r). Middle: gOmHm(r). Bottom: gOwHm(r). Subscripts w and m denote water and methanol, respectively. For clarity, only the results for the systems with the following mole fractions of methanol have been plotted: 0.1, 0.3, 0.5, 0.7, and 0.9. Because the sequence of the curves is monotonous, only the lowest mole fraction of each set is indicated.

selected self and cross g functions in mixtures of various compositions. We define as usual

Using these parameters one can calculate the partial molar volumes of water and methanol; the results are presented in Figure 3. As the densities of the pure liquids are very well reproduced the calculated partial molar volumes of water (vH(0)2O = 3 −1 18.07 cm3 mol−1) and methanol (v(0) MeOH = 40.66 cm mol ) virtually coincide with experimental results. For both species the overall agreement is also good for intermediate mole fractions. To the best of our knowledge the minimum in the partial volume of methanol at low concentration has been reproduced here for the first time. Because this feature depends on the slope of the excess volume curve, the present model seems to be able to describe the subtle changes occurring in the structure of the mixture when only a few alcohol molecules are mixed with water. Structure. The structures of the pure liquids and of the mixtures have been analyzed by computing radial distributions functions (RDFs) together with the corresponding running coordination numbers. In Figure 4 we present the three water− water partial RDFs in pure water, namely, gOwOw(r), gOwHw(r), and gHwHw(r), where subscript w denotes water (and later on subscript m denotes methanol). Figure 5 shows the corresponding three functions for pure methanol. Finally, Figure 6 shows

nαβ (r ) = 4π

Nβ V

∫0

r

r′2 gαβ (r′) dr′

(8)

where Nβ is the number of β particles in the system and ⟨V⟩ is the average volume (in NpT simulations, otherwise the fixed volume) of the simulation box. nαβ(r) is thus the total number of β-type neighbors around a particle of type α. The results presented in Figures 4 and 5 allow a comparison with the original model, BJH and PHH, respectively, under NEV conditions, with the present modified versions under NpT conditions. In both systems gOO(r) and gOH(r) remain constant and only very slight differences are observed in the gHwHw(r) and gMeMe(r) functions. The new parametrization obviously does not alter the interactions responsible for the hydrogen bonding yet modifies other parts of the potential enough to maintain a reasonable pressure in the systems. The cross g functions in mixtures, presented in Figure 6, show an increase in the height of the peaks as the mole fraction of methanol increases. The positions of the first and second peaks seem not to be affected by the change in composition in E

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Figure 7. Running coordination numbers (text and eq 8), nOwOw (solid line and triangles), and nOwOm (dashed line and circles) as a function of the mole fraction of methanol.

the system. The values of the integrals of the gOwOw(r) and gOwOm(r) RDFs at the first minimum are shown in Figure 7. The first peak in the gOwOw(r) functions becomes higher as χMeOH increases, but the coordination number associated with its integration decreases, which indicates the presence of methanol molecules around water molecules even at very low mole fractions. The “inversion” in the number of neighbors occurs around χMeOH = 0.65, i.e., the number of methanol molecules in the first solvation shell of water becomes larger than the number of water molecules. These results show that the correct evolution of the density of the system seen here results from a progressive mixing of the two species and not from a segregation process. Dynamics. We have characterized the translational mobility of the molecules by their self-diffusion constant D. This property was obtained from the mean square displacements using the Einstein relation.26 For the pure liquids under ambient conditions, the experimental self-diffusion constants of water and methanol are similar, namely, 2.3 × 10−9 and 2.4 × 10−9 m2 s−1, respectively.16 In addition to other properties it is well known that these values are difficult to reproduce by most potentials. (For water, see ref 43.) This remark applies in particular to the PHH model because it makes use of only three interacting sites. It has already been observed that the reduced steric hindrance resulting from this choice, compared to an all-atom six-site model, leads to an increase in the diffusion processes.44 The results are presented in Figure 8. In the mixtures, D goes through a minimum as a function of composition of both the water and methanol molecules. This feature is moderately well reproduced in our simulations for water; i.e., we see a minimum but not at the mole fraction where it is found experimentally. For methanol, the simulated self-diffusion increases monotonously with increasing mole fraction of MeOH and rises to a value that is too large for the pure system. The very slight minimum seen experimentally around χMeOH ≈ 0.3 does not appear.

Figure 8. Self-diffusion coefficient of water (top) and methanol (bottom) in water−methanol mixtures obtained from NpT simulations (dashed lines and circles) as a function of the mole fraction of methanol. Experimental data are from ref 16 (solid lines and triangles).

Because these models have essentially been constructed to study spectroscopic properties, they are particularly well suited to study in detail the hydrogen bond networks and its evolution: the shifts of the vibrational OH-streching frequencies induced by the presence of hydrogen bonds (or other perturbations) can be easily detected and correlated with structural features. We expect that the combination of such studies with vibrational spectroscopy experiments will lead to an improved qualitative and quantitative understanding of the peculiar properties of these systems.

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AUTHOR INFORMATION

Corresponding Author

*Phone: +33 (0)5 4000 2242. E-mail: jean-christophe. [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS Computer time for this study was provided by MCIA (Mésocentre de Calcul Intensif Aquitain), the public research HPC center in Aquitany, France.

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CONCLUDING REMARKS The modified versions of the BJH and PHH models presented in this article demonstrate that it is possible within the framework of pair interactions to describe the pure liquids and to reproduce the excess mixing and molar properties of mixtures. To the best of our knowledge, one of these properties, the minimum in the partial volume of methanol at low concentration in water, has been reproduced here for the first time. Only minor changes in the model parameters were required to use these models in NpT simulations, thus retaining their intrinsic qualities such as the interactions responsible for the hydrogen bonding and the adjusted intramolecular parts.

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DOI: 10.1021/acs.jpcb.5b03344 J. Phys. Chem. B XXXX, XXX, XXX−XXX