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Insight into the Local Composition of the Wilson Equation at High Temperatures and Pressures through Molecular Simulations of Methanol−Water Mixtures Takumi Ono,* Kyouhei Horikawa, Masaki Ota, Yoshiyuki Sato, and Hiroshi Inomata Research Center of Supercritical Fluid Technology, Department of Chemical Engineering, Graduate School of Engineering, Tohoku University, 6-6-11-403, Aoba, Aramaki, Sendai 980-8579, Japan S Supporting Information *

ABSTRACT: Methanol−water mixtures are known to be microscopically inhomogeneous and show nonideal properties at ambient conditions and also at higher temperatures and pressures. To understand the nonideality of methanol−water mixtures, the behaviors of local compositions were studied through evaluating the local mole fractions by the well-known activity coefficient equation, Wilson equation, and by molecular dynamics (MD) simulation at three conditions: 25 °C−0.1 MPa, 300 °C−25 MPa, and 350 °C−25 MPa. The Wilson parameters were redetermined to generate the reference values for the comparison in this study. The deviation of local mole fraction from bulk mole fraction was selected as an indicator for local composition. The deviations by the Wilson equation were positive in all compositions, and its magnitude for water was much larger than that for methanol, which is principally independent of temperature and pressure and was completely supported by the results by MD simulation. The MD simulation provided the dependence of the local mole fraction deviation on the intermolecular distance and indicated that the immediate neighbor in the Wilson local mole fraction approximately corresponds to the distance range from first valley to second peak in the radial distribution function. In addition, the general trends of local mole fraction by MD simulation are similar at ambient and high temperature and pressure conditions, suggesting the applicability of the Wilson equation for methanol−water mixture to high temperature and pressure conditions in terms of representing the local mole fraction.

1. INTRODUCTION Alcohol−water mixtures are scientifically intriguing and important for many fields ranging from fundamental molecular science to widespread industrial applications. Simple alcohols such as methanol and water mix completely from a macroscopic viewpoint, but their solution is known to be microscopically inhomogeneous.1 This heterogeneity is induced by hydrogen bond interactions existing in solution and can be described in terms of negative excess enthalpy and negative excess entropy,2,3 which are attributable to ice-like or clathrate-like structures of water molecules surrounding methyl group called hydrophobic hydration. It has been experimentally reported that hydrophobic hydration regimes vary with methanol concentration.4 The molecular level nonuniformity of methanol−water mixtures induces nonrandom structure in molecular config© XXXX American Chemical Society

uration which is important for understanding nonideal behaviors of thermodynamic properties of liquids. The nonideality of liquids is expressed by excess function in thermodynamics, such as excess Gibbs energy or activity coefficient. There have been many proposed relationships between activity coefficients and mole fractions for representing solution nonideality. Three equations are widely used for many practical calculations: Wilson, 5 nonrandom two-liquid (NRTL),6 and universal quasichemical (UNIQUAC)7 equations. These models consider nonrandomness by introducing Special Issue: In Honor of Grant Wilson Received: August 16, 2013 Accepted: January 9, 2014

A

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related to the nonideality or inhomogeneity of a solution at high temperatures and pressures and induces a strong interest in how the inhomogeneous structure at high temperatures can be expressed in terms of local composition that is widely used in representing liquid phase nonideality. In this work, we determined the Wilson parameters from the reported VLE data for methanol−water mixtures and calculated local mole fraction from eqs 1.1 and 1.2. Local compositions based on potential model were directly computed by MD simulation of methanol−water mixtures at ambient temperatures and at high temperatures and pressures to discuss the intermolecular separation ranges that the Wilson parameters can represent and the temperature and pressure ranges that the Wilson equation can cover according to the solution structure in terms of local mole fraction. Methanol−water mixtures are used as an example since sufficient experimental data are available.

local composition that differ from the bulk composition in the liquid mixture. For example, Wilson proposed5 a relation between local mole fraction X11 of molecules 1 and local mole fraction X21 of molecules 2 which are in the immediate neighborhood of molecule 1: X 21 x exp( −λ 21/RT ) = 2 X11 x1 exp( −λ11/RT )

(1.1)

X12 x exp( −λ12 /RT ) = 1 X 22 x 2 exp( −λ 22 /RT )

(1.2)

where λ12 is the interaction energy between component 1 and 2. The well-known Wilson equation is conclusively given as follows for a binary mixture. ⎛ ⎞ Λ12 Λ 21 ln γ1 = −ln(x1 + Λ12x 2) + x 2⎜ − ⎟ Λ 21x1 + x 2 ⎠ ⎝ x1 + Λ12x 2

2. WILSON PARAMETERS There have been reported a lot of Wilson parameters for methanol−water mixtures,15−21 which were determined to represent the VLE data of the authors’ interest. The VLE data are both isothermal and isobaric at various temperature and pressure conditions, leading to the variation of Wilson parameter values. In calculating the local composition of methanol−water mixture by the Wilson equation, one has to determine the parameter values to generate the reference local composition data because the reported parameters depend on the data set used for parameter optimization. In this work, the Wilson parameters for reference data were determined by the following four steps: (1) Collection of VLE data for methanol−water systems reported. (2) Selection of data based on standard deviation, σ, of activity coefficients correlated by the Wilson equation with σ ≤ 0.05. (3) Redetermination of the Wilson parameters by regression using the selected data. (4) Confirmation of the parameters to represent the experimental values in terms of T ∼ x, y, and x ∼ y diagrams for each selected data set. Selected reference data sources are listed in Table 1, and the activity coefficients reported in each data source are shown in Figure 1. Solid lines in Figure 1 show the activity coefficients evaluated from the Wilson parameters determined by the above procedure. The Wilson parameters determined in this work were λmw−λww = 455.514 J mol−1 K−1, and λwm−λmm = 170.509 J mol−1 K−1. The validity of the determined Wilson parameters

(1.3)

⎞ ⎛ Λ 21 Λ12 ln γ2 = −ln(x 2 + Λ 21x1) + x1⎜ − ⎟ Λ12x 2 + x1 ⎠ ⎝ x 2 + Λ 21x1 (1.4)

Λ12 =

v2 ⎡ λ12 − λ11 ⎤ vX x exp⎢ − = 2 21 / 2 ⎥ v1 ⎣ RT ⎦ v1X11 x1

(1.5)

Λ 21 =

v1 ⎡ λ 21 − λ 22 ⎤ vX x exp⎢ − = 2 21 / 2 ⎥ v2 ⎣ RT ⎦ v1X11 x1

(1.6)

where v1 and v2 are the molar volumes of component 1 and 2, and γ1 and γ2 denote respectively the liquid-phase activity coefficients of the components 1 and 2. From eqs 1.3 to 1.6, the Wilson equation is regarded as a local composition activity coefficient model, and NRTL and UNIQUAC equations can be similarly considered to different local composition activity coefficient models. In this sense, it should be noted that the local composition consideration is one of the key reasons for high applicability of the three activity coefficient equations in practical fields. In other words, local composition is an important concept for representing the nonideality of liquids. The local composition must be related to liquid phase solution structure and molecular configuration, and it is governed by short-range forces that can be expressed by molecular interaction in the immediate neighborhood of a molecule. In these decades, there have been many reports on the fundamental properties of subcritical and supercritical water and industrial applications.8,9 Some experimental and theoretical approaches revealed the fact that hydrogen bonding still exists in sub- and supercritical water and methanol.10 Since the inhomogeneous structure attributed to hydrogen bonding is influenced by temperature and pressure, the solution structure of methanol−water mixtures is influenced by the effect of temperature and pressure, bringing about some unique features in physical properties of alcohol−water mixture at high temperature and pressure including sub- and supercritical conditions.11−13 Our group has measured the density of the methanol−water mixture and performed MD simulations for understanding of the relationship between the macroscopic and microscopic properties at (300 to 400) °C, and at high pressures,14 and showed that volumetric behavior and solution structure changed dramatically with only the change of mole fraction at high temperatures. This interesting behavior is also

Table 1. Wilson Parameters and Goodness of Fit for the Correlation of the Methanol−Water Mixture at 0.1 MPa ref Uchida15 Green and Vener16 Olevsky15 Swami15 Kojima et al.17 Verhoeye and De Schepper21 this work B

λmw−λww

λwm−λmm

no. data points

standard deviation

366.1964 550.7402 459.6143 485.3358 435.1416 554.0494

291.6273 20.5449 137.9394 139.1436 243.9483 19.2547

13 12 12 12 21 13

0.05 0.04 0.03 0.03 0.03 0.04

448.8620

179.5027

83

0.04

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(H), and methyl group (−CH3). The potentials of both flexible models were formulated as a sum of intra- and intermolecular potentials. The intramolecular potential was given by the angular form of the Toukan−Rahman (TR) potential. The intermolecular potential used was Lennard−Jones (LJ) 12−6 potential and a Coulombic potential. In our simulation, a cutoff radius of 0.997 Lbox (Lbox is the simulation box) was applied for all LJ interactions. The Reaction Field method24 was used to account for long-ranged electrostatic interactions. Unlike interactions were computed using the Lorentz−Berthelot combination rules.

Figure 1. Activity coefficients as a function of liquid compositions for the methanol−water mixture. ■, Kojima et al.;17 ●, Olevsky et al.;15 ▲, Swami et al.;15 ▼, Uchida et al.;15 ◆, Verhoeye et al.;21 ◀, Green et al.;16 the solid line is fitting results from the Wilson equation in this work. Open symbols indicate the activity coefficient of methanol, and filled symbols indicate these of water.

σij = (σii + σjj)/2

(3.1)

εij =

(3.2)

εiiεjj

The potential parameters for water and methanol were those given in the original studies.22,23 All simulations were performed with NVT-ensembles for several methanol mole fractions. Although the Wilson equation parameters do not depend on pressure, this work compares the local composition between the Wilson equation and MD simulation at ambient and higher temperatures where the properties tend to show pressure dependences. The temperature and pressure conditions in the simulation were three sets as ambient (25 °C−0.1 MPa), 300 °C−25 MPa, and 350 °C−25 MPa. The densities of the latter two conditions were evaluated from the measurement data in our group.14 Hence, the number densities (N/V) in the simulation were set to the experimentally determined densities to allow isobaric analysis at any given temperature.14,25−27 Table 2 shows the length of the box in this simulation. The

can be suggested as good reproducibility for the T ∼ x, y, and x ∼ y diagram of the methanol−water system in the Supporting Information. This work focuses on the deviation between local mole fraction and bulk mole fraction; Xii−xi, as an indicator of representing local composition. Figure 2 shows the deviation,

Table 2. Length of the Box in This Simulation 25 °C−0.1 MPa

300 °C−25 MPa

350 °C−25 MPa

xm

Lbox/Å

xm

Lbox/Å

xm

Lbox/Å

0.10 0.20 0.40 0.60 0.80

40.8 42.1 44.6 47.1 49.5

0.12 0.21 0.40 0.59 0.78

46.0 48.1 52.7 56.6 60.0

0.10 0.19 0.38 0.55 0.80

48.9 52.7 61.8 67.6 71.9

total number of molecules was 2048. The temperature was initially (∼60 ps) controlled by momentum scaling for rapid approach to equilibrium and afterward (∼60 ps) by the NosèHoover thermostat for computational stability. The simulations began from initial configurations with random distributions of molecular positions. The equation of motion was integrated using the velocity Verlet technique. Time steps were 1 fs for the intermolecular motion and 0.2 fs for the intramolecular motion. Statistical sampling in the simulation was done for 500 ps after 100 ps equilibration. Local mole fractions, Xji and Xii, were defined as the composition of component j and i locating the inner shell within the distance r from the center of mass of object molecule i.

Figure 2. Deviation between local mole fraction and bulk mole fraction around (a) methanol and (b) water molecules evaluated by the Wilson equation at ambient condition.

Xii−xi, evaluated by eqs 1.1 and 1.2 with the determined parameters. It can be clearly seen that the local mole fractions of methanol and water are larger than the bulk compositions over the entire composition range, indicating that molecules tend to aggregate with similar molecules as expected. It can be observed that the deviation of water is larger than that of methanol in Figure 2.

3. COMPUTATIONAL DETAILS The molecular models used in this study were the three-site flexible models proposed by Honma et al. for methanol22 and by Liew et al. for water.23 These molecular models can give quantitative representation of the PVT and saturated properties of methanol and water in their gas, liquid, and supercritical states. The three sites for methanol are oxygen (O), hydrogen

4. RESULTS AND DISCUSSION Except for high pressure conditions, in general, the Wilson parameters are independent of pressure because of the small compressibility of the liquid. The densities which considered in this study are not so different because of liquid conditions. For example, ρxm=0.2 is equal to 0.95 g·cm−3 at ambient, while ρxm=0.2 C

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is equal to 0.64 g·cm−3 at 300 °C−25 MPa and ρxm=0.2 is equal to 0.48 g·cm−3 at 350 °C−25 MPa.14,25,26 Therefore, we bring the following discussion assuming the Wilson parameters do not depend on pressure. First, the self-diffusion coefficients of methanol and water mixture were calculated from the time dependence of the mean square displacement using the Einstein relation28 and compared with experimental data to confirm the reliability of our simulation results. The self-diffusion coefficients of methanol and water in the mixture at ambient condition are shown in Figure 3. The self-diffusion coefficient of each molecule exhibits a minimum and were quantitatively in agreement with experimental data.29,30

Figure 4. RDF between the centers of mass for same type of molecule described by normalized distance rc (= r/σ) for the xm = 0.1 at each condition. (A) methanol; (B) water: (a) ambient condition; (b) 300 °C−25 MPa; (c) 350 °C−25 MPa. Figure 3. Self-diffusion coefficient of methanol (Dm) and water (Dw) in mixtures at ambient condition. ■, this work; ●, Hawlicka et al.30 (experimental values); ▲, Derlacki et al.29 (experimental values).

RDF behaviors shown in Figure 4. They are 1.2, 1.4, 1.6, 1.8, 2.0, 2.5, and 3.0. Smaller five points correspond to the distances from first and second peaks in the RDF, and rc = 2.5 and 3.0 correspond to the longer distances. Similarly to Figure 2, the evaluated deviations between local mole fraction and bulk mole fraction, (Xii−xi), at ambient conditions were plotted against bulk mole fraction in Figure 5. Figure 5 revealed that the deviations (Xii−xi) are positive for both cases and their

Radial distribution functions (RDF) are commonly used to study the molecular configuration in molecular simulations. It is possible to evaluate several types of RDF such as atom−atom pairs and molecular−molecular pairs from the MD simulation data due to three-site potential models. Since this work focuses on local mole fraction, the molecular pair RDF was considered to be an appropriate function for examining local composition effects. In deriving the RDF, this work normalized the distance r from the center of mass with hard-sphere diameter σ of each molecule (σmethanol = 3.63 Å, σwater = 2.64 Å)31 to take into account the difference in molecular size between methanol and water. Figure 4 shows the RDF between same type molecules described by normalized distance rc (= r/σ) for the xm = 0.1 at three conditions. It can be seen that the peaks are observed at 0.9σ and 1.3σ for the methanol molecule and at 1.1σ for the water molecule. These peaks may be attributable to the molecular configuration caused by hydrogen bond interaction. The heights of these peaks decrease with increasing temperature, especially the peaks for methanol molecule. This is because the intermolecular interaction between methanol molecules is not as strong as water. The local mole fractions Xji were calculated from the probabilities of molecule j around molecule i, which were evaluated by integrating each RDF for methanol−methanol, methanol−water, water−water, and water−methanol. The molecule probabilities can be evaluated principally at any distance rc; however we selected seven points of rc from the

Figure 5. Deviation between local mole fraction and bulk mole fraction at inner shell within the distance rc from the center of mass of (a) methanol and (b) water molecules at ambient (25 °C−0.1 MPa). ■, 1.2σ; ●, 1.4σ; ▲, 1.6σ; ▼, 1.8σ; ◆, 2.0σ; ◀, 2.5σ; ▶, 3.0σ. Solid lines denote the values by the Wilson equation at the same condition. D

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at ambient condition. However, values of the deviation for methanol (Xmm−xm) were fairly small compared with those for water (Xww−xw), and the values show strong distance dependence, which is the same trend at ambient conditions. A possible reason why the deviations for methanol (Xww−xw) change from positive at ambient temperature to negative at higher temperatures may be the strength of intermolecular interactions. As discussed previously in Figure 2, the methanol−methanol interaction may be weaker than that for water−water and is easily influenced by thermal energy, leading to weakening nonrandom structure, namely, local composition, and approaching bulk composition. For more detailed analyses of the distance rc dependence of the deviation (Xii−xi) behavior, values were plotted at xm = 0.4, 0.6 against rc for three conditions in Figure 7. The reason for

magnitude for water was much larger than those for methanol, which agrees with the behavior given by the Wilson equation. The composition of maximum deviation (Xii−xi) is xm = 0.4 for methanol and xm = 0.6 for water, which is also same as the Wilson equation. Focusing on the rc dependence, small rc tends to give larger deviations, and the deviation approaches zero at longer rc according to the criteria of the way of evaluating local composition. The exceptionally different behavior at rc = 1.2 for methanol may result from statistics of the small number of data at small rc. It should be noticed, however, that there is the region giving similar values for the deviation (Xmm−xm), which is rc = 1.6−2.0 and may correspond the distance from the first valley to the second peak in RDF. In comparing the values of the local mole fractions around methanol and water molecule calculated from MD simulation, those at 1.4σ show the closest relationship to those of the Wilson equation. Further discussion on this discrepancy between the MD simulation and the Wilson equation is not appropriate in this work because of different kind of assumptions being applied. Figure 6 shows the deviations (Xii−xi) around water and methanol molecules at 300 °C−25 MPa and 350 °C−25 MPa calculated by MD simulation. Comparing the behavior with those at ambient conditions (Figure 5), the behaviors for water (Xww−xw) are very similar at three conditions, while the deviations around methanol (Xmm−xm) at high temperatures and pressures are slightly negative in contrast to the behaviors

Figure 7. Deviation between local mole fraction and bulk mole fraction (a) around methanol at xm = 0.4 and (b) around water at xm = 0.6 against rc. ■, ambient; ● 300 °C−25 MPa; ▲, 350 °C−25 MPa. Dotted lines denote the maximum values of Xii−xi by the Wilson equation at ambient condition.

selecting xm = 0.4, 0.6 is that this composition gives the maximum by the Wilson equation. Although the behavior for three conditions are slightly different each other, the deviations generally change with increasing rc rapidly in shorter range and change very gradually from rc >1.6. The temperature dependence of the deviations at a certain distance rc were plotted against the bulk mole fraction in Figure 8. The selected rc values are rc = 1.2, 1.4, and 1.6. It should be emphasized that the general trends of local mole fractions are approximately similar at ambient conditions, 300 °C−25 MPa, and 350 °C−25 MPa although temperature and density of each condition differ from each other. The local composition in the short range may be governed by intrinsic molecular interactions and does not seem to depend on the state condition so much. This is one reason why the Wilson equation has been so successful in correlating vapor−liquid equilibrium data. The maximum values of Xww−xw slightly increased with increasing temperature. This may result from the difference of the molecular number density.

Figure 6. Deviation between local mole fraction and bulk mole fraction at the inner shell within the distance rc from the center of mass of (a) methanol and (b) water molecules at 300 °C−25 MPa and 350 °C−25 MPa. ■, 1.2σ; ●, 1.4σ; ▲, 1.6σ; ▼, 1.8σ; ◆, 2.0σ; ◀, 2.5σ; ▶, 3.0σ. Solid lines denote the values by the Wilson equation at ambient conditions. E

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corresponds to the distance range from first valley to second peak in RDF. The local mole fractions evaluated by MD simulation at 300 °C−25 MPa and 350 °C−25 MPa revealed that general trends of local mole fractions are approximately similar at ambient condition and 300 °C−25 MPa and 350 °C−25 MPa, although the temperature and density of each condition differ from each other. The results mentioned above implied that the local composition in the immediate neighbor may be governed by intrinsic molecular interaction and does not depend on the state condition so much. Hence, the results shown in this work support the applicability of the Wilson equation for methanol− water mixture at high temperature and pressure conditions in terms of representing local composition, leading to the importance in considering local composition in estimating various thermodynamic properties of high temperatures and pressures.

■

ASSOCIATED CONTENT

S Supporting Information *

T−x, y diagrams for the methanol−water mixture at 0.1 MPa. This material is available free of charge via the Internet at http://pubs.acs.org.

■

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Tel. and fax: +81-22-7957282. Figure 8. Deviation between local mole fraction and bulk mole fraction at inner shell within the distance rc from the center of mass of (a) methanol and (b) water molecules. ■, ambient; ●, 300 °C−25 MPa; ▲, 350 °C−25 MPa.

Funding

This work was supported by JSPS KAKENHI Grant No. 13319571. Notes

The authors declare no competing financial interest.

■

Consequently the results shown in this work support the applicability of the Wilson equation for methanol−water mixtures at high temperature and pressure conditions in terms of representing local composition, leading to the importance of considering local composition in estimating various thermodynamic properties of high temperatures and pressures. In addition, it is likely the Wilson parameter could be calculated roughly from only potential models and molecular dynamics simulation without vapor−liquid equilibrium data.

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5. CONCLUSIONS Wilson parameters of methanol−water mixture are redetermined from the selected VLE data of six references at atmosphere pressure in order to provide the local mole fractions which were used to compare with those from MD simulation. At ambient conditions, the deviations of local mole fraction around methanol and water evaluated by the Wilson equation were positive over the entire composition range and indicated that molecules tend to aggregate with similar molecules as expected. MD simulations enabled us to evaluate to the local mole fraction as a function of the distance from the center of mass of methanol and water. The distance dependencies of the local composition by MD simulation also indicated that local compositions evaluated the Wilson equation and MD simulation show a generally similar trend, and the immediate neighbor in the Wilson local mole fraction approximately F

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