Understanding the Thermodynamics of Hydrogen Bonding in Alcohol

Article pubs.acs.org/JPCB

Understanding the Thermodynamics of Hydrogen Bonding in Alcohol-Containing Mixtures: Cross-Association Wael A. Fouad, Le Wang, Amin Haghmoradi, D. Asthagiri, and Walter G. Chapman* Department of Chemical and Biomolecular Engineering, Rice University, Houston, Texas 77005, United States S Supporting Information *

ABSTRACT: The thermodynamics of hydrogen bonding in 1-alcohol + water binary mixtures is studied using molecular dynamic (MD) simulation and the polar and perturbed chain form of the statistical associating fluid theory (polar PC-SAFT). The fraction of free monomers in pure saturated liquid water is computed using both TIP4P/2005 and iAMOEBA simulation water models. Results are compared to spectroscopic data available in the literature as well as to polar PC-SAFT. Polar PC-SAFT models hydrogen bonds using single bondable association sites representing electron donors and electron acceptors. The distribution of hydrogen bonds in pure alcohols is computed using the OPLS-AA force field. Results are compared to Monte Carlo (MC) simulations available in the literature as well as to polar PC-SAFT. The analysis shows that hydrogen bonding in pure alcohols is best predicted using a two-site model within the SAFT framework. On the other hand, molecular simulations show that increasing the concentration of water in the mixture increases the average number of hydrogen bonds formed by an alcohol molecule. As a result, a transition in association scheme occurs at high water concentrations where hydrogen bonding is better captured within the SAFT framework using a three-site alcohol model. The knowledge gained in understanding hydrogen bonding is applied to model vapor−liquid equilibrium (VLE) and liquid−liquid equilibrium (LLE) of mixtures using polar PC-SAFT. Predictions are in good agreement with experimental data, establishing the predictive power of the equation of state.

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SAFT (SAFT-VR),58−60 SAFT-γ,61 and cubic plus association (CPA).9,10,18,20,62−69 Within SAFT, depending on the system under consideration, alcohols are usually modeled with two 2(1,1) or three 3(2,1) association sites. However, a clear basis for choosing a certain association scheme for alcohols has not yet been defined, thereby limiting the scope and generality of the equation of state framework. Spectroscopic data on the fraction of bonded chains can guide the selection of an association scheme, but such data is limited. Data from molecular simulations is likewise limited. Thus, perhaps not surprisingly, an in-depth examination of hydrogen bonding in alcohol + water binary mixtures has never been carried out before within the SAFT framework. Here we revisit the problem of predicting the VLE and LLE in 1-alcohol + water systems using the polar and perturbed chain forms of the statistical associating fluid theory (polar PC-SAFT).70,71 We perform molecular dynamic (MD) simulations to evaluate the distribution of hydrogen bonds and compare these results to predictions using polar PC-SAFT with different association schemes. The work exhibits the importance of extending the SAFT equation of state to allow multiple bonding per association site to capture accurately the distribution of hydrogen bonds in 1-alcohol + water binary systems.

INTRODUCTION Understanding the thermodynamics of 1-alcohol + water mixtures is crucial for many applications across the biochemical, chemical, and petroleum industries. For example, predicting accurate water content of the hydrocarbon phase is critical for natural gas dehydration and hydrate inhibition. In cases where hydrates are likely to form, aqueous solutions of methanol or ethylene glycol are injected into the pipelines to suppress hydrate formation. The latter represents one of many industrial applications for aqueous alcohol solutions. Consequently, predicting the phase behavior of 1-alcohol + water binary systems is essential for better process design, simulation, and optimization. Given the centrality of hydrogen bonding in the structure and thermodynamics of pure water and pure alcohols, selfassociation and cross-association among water and alcohol molecules is expected to play an important role in determining the thermophysical properties of the binary alcohol−water system. Incorporating the physics of such association in the equation of state modeling of the mutual solubility of water− alkane and water−acid gas systems has thus received extensive attention both in our group1−4 as well as in other research groups.5−44 Typically, the equation of state development is anchored using different versions of the statistical associating fluid theory (SAFT) equation of state45−47 including early versions,15,48−50 the perturbed chain version of SAFT (PCSAFT),20,51,52 the polar versions of SAFT,53 the simplified version of PC-SAFT,25,39,54−57 the variable range version of © 2016 American Chemical Society

Received: December 17, 2015 Revised: March 12, 2016 Published: March 15, 2016 3388

DOI: 10.1021/acs.jpcb.5b12375 J. Phys. Chem. B 2016, 120, 3388−3402

Article

The Journal of Physical Chemistry B

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MOLECULAR DYNAMIC SIMULATIONS Molecular dynamics simulations of pure 1-alcohol, pure water, and 1-alcohol + water binary mixtures were carried out primarily using the GROMACS 4.6.5 software package.72 To parametrize the 1-alcohols and water, we use the OPLS-AA force field73 for alcohols and TIP4P/200574 for water. To assess the importance of flexibility and polarizability, we also separately studied water modeled using the iAMOEBA75 force field. For simulations with GROMACS, the bond lengths were constrained using the LINCS algorithm76 for alcohols and SETTLE for water. Newton’s equations of motion were integrated using the leapfrog algorithm77 with a time step of 2 fs. The modified Berendsen thermostat78 and the Parrinello− Rahman barostat79 were used to maintain the temperature and pressure of the system. As usual, periodic boundary conditions were applied. Lennard-Jones interactions were cut off at 1.2 nm, and tail corrections were included. Long-range electrostatic interactions were described using the particle mesh Ewald (PME) summation method.80 The real-space cutoff for electrostatic interaction was 1.2 nm. All systems were initially energy-minimized, and after a short NVT simulation, the system was equilibrated in the NpT ensemble for over 5 ns. The NpT production run was also for over 5 ns. To compute the number of hydrogen bonds formed in the system, a bond angle and distance criteria was defined. From the oxygen−oxygen radial distribution function for pure 1-alcohols, the distance criterion was set to be 3.7 Å. From the oxygen−oxygen radial distribution function for pure water, the distance criterion was set to be 3.5 Å for TIP4P/2005 and iAMOEBA. A hydrogen bond was counted as formed only if the angle formed between the O−H bond and the vector between oxygens of different molecules is less than 30° and the distance between oxygens in two alcohol or water molecules is less than the specified distance criteria. For the hydrogen bonds formed between 1-alcohol and water molecules, the distance criterion was calculated using the arithmetic mean mixing rule. The fraction of molecules involved in self-association (at least once with the same kind of molecules), the distribution of hydrogen bond number per molecule of alcohol, and the average number of hydrogen bonds formed in the mixture were all calculated on the basis of the hydrogen bonding criteria mentioned previously.

chains comprising m segments, given by a hc = ma hs −

i

(2)

and the average chain length m̅ is given by m̅ =

∑ ximi

(3)

i

where xi is the mole fraction of molecules of component i, mi is the number of segments in a molecule of component i, ghs ii (dii) is the radial pair distribution function at contact for segments of component i in the hard sphere system, dii is the hard sphere diameter of component i, and the superscripts hc and hs indicate quantities of the hard-chain and hard-sphere systems, respectively. For mixtures of hard spheres,81,82 the residual Helmholtz free energy is given in terms of reduced densities by ξ2 3 1 ⎡⎢ 3ξ1ξ2 + ξ0 ⎢⎣ (1 − ξ3) ξ3(1 − ξ3)2

a hs =

⎤ ⎛ξ 3 ⎞ + ⎜ 22 − ξ0⎟ ln(1 − ξ3)⎥ ⎝ ξ3 ⎠ ⎦⎥

(4)

And the radial distribution function at contact is giihs(dii) =

⎛ didj ⎞ 3ξ 1 2 ⎟⎟ + ⎜⎜ (1 − ξ3) ⎝ di + dj ⎠ (1 − ξ3) ⎛ didj ⎞2 3ξ 2 2 ⎟⎟ + ⎜⎜ d d + (1 − ξ3)3 ⎝ i j⎠

(5)

where the reduced densities written in terms of ρ, the number density of molecules, are π ξn = ρ ∑ ximidin 6 i (6) Theory has shown that the soft core repulsion between molecules can be approximated using a temperature dependent hard sphere diameter. On the basis of Barker and Henderson’s perturbation theory,83 Chen and Kreglewski propose the temperature-dependent segment diameter, di(T), as

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⎡ ⎛ 3ε ⎞⎤ di(T ) = σi⎢1 − 0.12 exp⎜ − i ⎟⎥ ⎝ kT ⎠⎦ ⎣

POLAR PC-SAFT EQUATION OF STATE The SAFT equation of state models molecules as chains of spherical segments. These segments include short-range repulsion, long-range dispersion and polar interactions, and hydrogen bonding contributions. An advantage of the model is that the contributions to the free energy are derived from statistical mechanics based theory and validated versus molecular simulation results. Thus, the equation of state requires a minimal number of physical parameters that can be fit to pure fluid properties such as vapor pressure and saturated liquid density. The complete equation of state in polar PC-SAFT, in terms of the reduced Helmholtz free energy, is given as the sum of an ideal gas contribution, aid, a hard-chain contribution, ahc, a dispersion contribution, adisp, an association contribution, aassoc, and a polar contribution, apolar: a res ≡ a − a id = a hc + adisp + aassoc + apolar

∑ xi(mi − 1)ln giihs(dii)

(7)

where εi is the depth of square-well potential due to van der Waals attractions and k is the Boltzmann constant. The perturbation theory of Barker and Henderson is used to calculate the contribution to the free energy due to van der Waals (dispersion) interactions between molecules. The dispersion term of the Helmholtz free energy, adisp, is given as the sum of first- and second-order contributions

adisp =

A1 A2 + NkT NkT

(8) 84,85

where N is the number of molecules. Gross and Sadowski derived a set of equations appropriate for the square-well potential function

(1)

adisp = −2πρ[I1(η , m̅ )](m2εσ 3)m 2 2 3 − πρmC ̅ 1[I2(η , m̅ )](m ε σ )m

45,47

For the hard chain contribution, Chapman et al. developed an equation of state applicable for mixtures of hard-sphere 3389

(9)


Article

The Journal of Physical Chemistry B and κ

−1 ⎛ ∂Z hc ⎞ C1 = ⎜1 + Z hc + ρ ⎟ ∂ρ ⎠ ⎝

(10)

The parameters of a pair of unlike segments, σij and εij, are obtained by Berthelot−Lorentz mixing rules σij =

1 (σi + σj) 2

a2 = −

(13)

εiεj (1 − kij)

εij =

(14)

(m2εσ 3)m =

nc

⎛ εij ⎞ 3 ⎟σ ij kT ⎠

∑ ∑ xixjmimj⎜⎝ i

j

nc

nc

i

j

∑ ∑ xixjmimjx px p i

i

μi 2 μj 2 dij 2

j

j

(23)

∑ ∑ xixjxkmimjmkx px p x p i

i

j

j

I2, ij

k

μi 2 μj 2 μk 2 dij 2djk 2dik 2

I3, ijk (24)

In the aforementioned equations, I2,ij and I3,ijk are integrals over the angular pair and triplet correlation functions and μi is the dipole moment for component i. As shown by Jog et al.,70,71 they are related to the corresponding pure fluid integrals by

(15)

⎛ εij ⎞2 (m ε σ )m = ∑ ∑ xixjmimj⎜ ⎟ σij 3 ⎝ kT ⎠

2π ρ 9 (kT )2

5π 2 ρ2 a3 = 162 (kT )3

In eq 9, the mixing rules for (m2εσ3)m and (m2ε2σ3)m are defined by nc

(21)

(12)

∑ bj(m̅ )η j j=0

κ

where a2 and a3 are the second- and third-order terms in the perturbation expansion. Written for mixtures, and allowing for multiple dipolar segments, these terms have the following form:

6

I2(η , m̅ ) =

κ

(11)

6

∑ aj(m̅ )η j j=0

=

⎛ ⎞3 σiiσjj ⎜ 0.5(σ + σ ) ⎟⎟ ⎝ ii jj ⎠

A i Bi Aj Bj ⎜

The change in free energy due to polar interactions, as derived by Jog and Chapman,70,71 is accurately obtained by dissolving all the bonds in a chain and then applying the u-expansion to the resulting mixture of polar and nonpolar spherical segments. The polar contribution written in the Padé approximate has the following form a2 apolar = 1 − a3 / a 2 (22)

The value of C1 represents the compressibility of the hard chain fluid and I1(η , m̅ ) and I2(η , m̅ ) are given by power series in density where the coefficients are functions of the chain length and the packing fraction, η, which is equivalent to ξ3: I1(η , m̅ ) =

A i Bj

I2, ij = I2(η , m̅ )

(25)

I3, ijk = I3(η , m̅ )

(26)

2 2 3

(16)

The fugacity coefficient is given using the following expression

The water and alcohols will self- and cross-associate through the partially positively charged hydrogen on the hydroxyl group bonding with the partially negatively charged oxygen of another molecule. A strength of the SAFT approach is the ability to describe hydrogen bonding using specific regions (or sites) on a molecule that represent the hydrogen and oxygen association sites. Chapman et al.45,47 derived the change in Helmholtz free energy due to association, aassoc, for mixtures ⎡

⎡

∑ xi⎢⎢∑ ⎢ln X A

aassoc =

i

⎣

Ai

⎣

i

−

⎤ X Ai ⎤ 1 ⎥ ⎥ + Mi 2 ⎦ 2 ⎥⎦

ln φi =

j

where Δ

AiBj

(18)

is the “association strength” approximated as

⎤ ⎡ ⎛ ϵ A iBj ⎞ ΔA iBj = dij 3gij(dij)seg κ A iBj⎢exp⎜ ⎟ − 1⎥ ⎦ ⎣ ⎝ kT ⎠

45,47

(19) 49

The mixing rules suggested by Wolbach and Sandler are used for the association energy, εAiBi/k, and the effective association volume, κAiBi: ε A iBj =

1 A iBi (ε + ε AjBj) 2

(27)

PARAMETER OPTIMIZATION Following the notation of Yarrison et al.,86 alcohols were modeled in our previous publication87 using two association sites 2(1,1), one electron donor, and one electron acceptor. The chain length (m) for alcohols was fixed assuming that every hydroxyl group would occupy a spherical segment and every CH3/CH2 group would occupy half a spherical segment. The segment diameter (σ) and dispersion energy (ε/k) were fitted to activity coefficient data at infinite dilution for 1-alcohols in n-alkanes. This was done to minimize any contributions from self-association or polar interactions among alcohol molecules. The association parameters (εAiBi and κAiBi) and the fraction of polar segments in a chain (xp) were then fitted to pure component saturated liquid density and vapor pressure data. The dipole moment of alcohols was fixed to be 1.7 D. Table 1 illustrates the parameters fitted for the two-site alcohol model. The parameters were successfully used to predict the fraction of free monomers in pure alcohol and alcohol + n-alkane binary mixtures.87

i j

Bj

− ln Z

■

(17)

∑ ∑ ρi X B ΔA B ]−1 j

kT

where μres and Z are the residual chemical potential and i compressibility factor calculated through the derivatives of the Helmholtz free energy. Equations used in this work to calculate the distribution of hydrogen bonds in pure water, pure alcohols, and 1-alcohol + water binary mixtures can be found in the Supporting Information.

where the sum is over all components i and all association sites, Ai, on component i. Mi is the number of association sites per molecule, and the mole fraction of molecules that are not bonded at site A, XAi, can be determined as follows X A i = [1 +

μi res

(20) 3390


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The Journal of Physical Chemistry B Table 1. Pure Component SAFT Parameters Used in This Worka component

mi

σi (Å)

εi/k (K)

water

1.0

3.144

220

methanol ethanol 1-propanol 1-butanol

1.5 2.0 2.5 3.0

3.349 3.370 3.428 3.459

225 224 230 235

methanol ethanol 1-propanol 1-butanol 1-pentanol 1-hexanol

1.5 2.0 2.5 3.0 3.5 4.0

3.429 3.478 3.494 3.508 3.515 3.514

182.305 197.595 215.256 226.274 235.750 236.832

εAiBi/k (K)

κAiBi

μi (D)

Four-Site Water Model 4(2,2) 1631.600 0.0454 1.85 Two-Site Alcohol Model 2(1,1) 2439.295 0.0340 1.70 2333.936 0.0143 1.70 2515.569 0.0179 1.70 2507.637 0.0145 1.70 Three-Site Alcohol Model 3(2,1) 1906.309 0.0248 1.70 2058.916 0.0163 1.70 2024.482 0.0144 1.70 1999.958 0.0127 1.70 1801.102 0.0135 1.70 1796.613 0.0138 1.70

xpi

AAD%,b psat

AAD %,b ρsat

T range (K)

0.5

1.136

5.339

273−607

0.33 0.33 0.25 0.20

1.022 2.177 0.928 1.102

2.253 3.376 1.570 0.950

212−492 178−490 229−516 224−503

0.93 0.7 0.56 0.47 0.4 0.35

0.737 0.731 1.086 1.860 0.769 0.786

3.286 2.146 1.088 0.243 0.418 1.049

212−492 196−514 227−516 224−503 256−526 268−550

Parameters in bold are fixed and not fitted. The rest of the parameters were fitted to vapor pressure and saturated liquid density data. bAAD% is calculated using the following equation:

a

AAD % =

100 npts

npts

∑ i=1

|Xiexp − Xical| Xiexp

Figure 1. SAFT parameters for the two-site alcohol model as a function of carbon number.

for water could be fit independently of the polar and hydrogen bonding parameters to water content data in n-alkanes.1−4,88 Simulation efforts performed in our group88 to develop a

In molecular simulation, water is usually modeled as a LennardJones (LJ) sphere (m = 1) with a diameter that is close to 3 Å. In previous work, we showed that the LJ (dispersion) parameter 3391


Article


Figure 2. Distribution of the hydrogen bond number per molecule of pure methanol using MD and MC simulation.

Figure 4. Fraction of free monomers for saturated liquid water: (○) spectroscopic data measured by Luck;91 (blue ■) iAMOEBA simulation water model; (orange ■) TIP4P/2005 simulation water model; () polar PC-SAFT water model.

molecular based water model showed that an effective water Lennard-Jones energy (εW/k) of 220 K was required to match the experimental water solubility in TraPPE89 alkanes. As previously mentioned, Fouad et al.87 modeled the phase behavior and hydrogen bonding distribution in 1-alcohol + n-alkane binary systems using a two-site alcohol model. The difference in chemical structure between two consecutive compounds in the alcohol homologous series is just a CH2 group. Additionally, the difference in chemical structure between methanol and water is another CH2 group. Figure 1 demonstrates the SAFT parameters for the two-site alcohol models as a function of carbon number. At zero carbon number (CN = 0), it can be seen that the model was able to retrieve back parameters similar to those of what we would expect for water.

In addition to the two-site alcohol model, alcohols are also modeled here using three association sites 3(2,1), two electron donors and one electron acceptor. Water is modeled as a sphere (m = 1) with four association sites 4(2,2), two electron donors and two electron acceptors. On the basis of the work of Ballal et al.,88 the dispersion energy for water was fixed to 220 K. The dipole moment for water was fixed to be 1.85 D. The remaining pure component parameters were fitted to saturated liquid density and vapor pressure data, as provided by the Design Institute for Physical Properties (DIPPR).90 The water association energy was optimized to be as close as possible to that of the spectroscopic value91 of 1813 K. Table 1 presents the optimized parameters and the absolute average deviations (AADs) computed for each of the components.

Figure 3. Distribution of the hydrogen bond number per molecule of pure ethanol using polar PC-SAFT and MD simulation: (a) T = 298 K; (b) T = 373 K.

Table 2. Simulation Results for the Distribution of Hydrogen Bonds per Molecule of Pure Alcohol number of hydrogen bonds

methanol (T = 298 K)

ethanol (T = 298 K)

ethanol (T = 343 K)

ethanol (T = 373 K)

1-propanol (T = 298 K)

0 1 2 3

0.0345 0.2622 0.6269 0.0764

0.0203 0.2003 0.6884 0.0911

0.0541 0.2872 0.5832 0.0755

0.0937 0.3468 0.4993 0.0601

0.0348 0.2344 0.6667 0.0641

3392



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Table 3. Simulation and Equation of State Results for the Fraction of Free Monomers in Saturated Liquid Water temperature (K)

TIP4P/2005

iAMOEBA

polar PC-SAFT

293.5 327.38 354 389 426 456.39 486.3 528

3.57 × 10−5 0.000158 0.000388 0.000961 0.002314 0.004140 0.006766 0.013598

0.000527 0.001366 0.002702 0.004928 0.009043 0.015058 0.022527 0.039306

4.92 × 10−5 0.000170 0.000383 0.000955 0.002160 0.003878 0.006505 0.012435

ΔHvap (kcal/mol) at 298 K (corrected for polarization) μex (kcal/mol) at 293.5 Ka psat (bar) at 350 K psat (bar) at 350 K (corrected for polarization) ρ (g/cm3) at 298 K

experiment TIP4P/2005

iAMOEBA

polar PC-SAFT

10.52

10.89

10.94

10.39

−6.3

−6.1

−5.1

−6.3

0.42 0.42

0.13 0.53

0.43 0.43

0.41 0.41

0.997

0.993

0.997 (fitted)

0.936

RESULTS AND DISCUSSION

Hydrogen Bonding Distribution in Pure Alcohols. In this section, results from the polar PC-SAFT model are compared with molecular simulation results for the distribution of hydrogen bonds in pure alcohols. As a validation step, we compare MD simulation results, using the OPLS-AA force field,73 for the distribution of hydrogen bonds formed in pure methanol with that of Ferrando et al.92 using the AUA4 force field93 in Monte Carlo (MC) simulation. The AUA4 force field is known for its ability to predict accurately both thermodynamic saturated properties (density, vapor pressure, vaporization enthalpies) and liquid phase structure (radial distribution functions, hydrogen bond networks). Good agreement between both simulation approaches can be seen in Figure 2 for the average number of hydrogen bonds per molecule in pure methanol at atmospheric pressure and temperatures of 298 and 300 K. This supports the choice of OPLS-AA force field as a model for alcohols in this work. Figure 3 illustrates that, within the SAFT framework, a three-site alcohol model tends to overestimate the fraction of molecules forming three hydrogen bonds for pure alcohols. On the other hand, a two-site alcohol model overestimates the fraction of molecules forming two hydrogen bonds. Simulation data, as a function of temperature, for the distribution of hydrogen bonds in pure methanol, ethanol, and 1-propanol can be found in Table 2. Hydrogen Bonding Distribution in Pure Water. Luck’s91 spectroscopic data on the fraction of free OH groups in pure saturated liquid water have been widely used in the literature as a measure of evaluating the SAFT water models.

Table 4. Comparison between TIP4P/2005, iAMOEBA,and Polar PC-SAFT property

Article

a

The experimental estimate is obtained from the density of saturated liquid and saturated vapor. The TIP4P/2005 and iAMOEBA estimates are based on a quasichemical theory estimate.

Figure 5. Distribution of the hydrogen bond number per molecule of pure water using polar PC-SAFT and MD simulation: (a) T = 327.38 K; (b) T = 426 K.

Table 5. TIP4P/2005 Results for the Distribution of Hydrogen Bonds per Molecule of Pure Liquid Water at Saturation number of hydrogen bonds T (K)

0

293.5 327.38 354 389 426 456.39 486.3 528

3.57 × 1.58 × 3.88 × 9.61 × 0.0023 0.0041 0.0068 0.0136

10−5 10−4 10−4 10−4

1

2

3

4

5

0.0014 0.0047 0.0090 0.0178 0.0319 0.0477 0.0672 0.1028

0.0247 0.0532 0.0793 0.1187 0.1654 0.2041 0.2431 0.2903

0.1880 0.2655 0.3117 0.3550 0.3796 0.3878 0.3860 0.3670

0.7232 0.6048 0.5284 0.4430 0.3661 0.3105 0.2599 0.1999

0.0626 0.0716 0.0713 0.0646 0.0547 0.0457 0.0370 0.0264

3393


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The Journal of Physical Chemistry B However, how the data should be interpreted is still a matter of debate.94 Moreover, questions on the accuracy and reliability of these measurements have also been recently raised. Here, we performed MD simulations to understand better the hydrogen bonding distribution in pure water and compared it with predictions based on Luck’s data. In reference to Figure 4 and Table 3, iAMOEBA tends to agree better with Luck’s data while TIP4P/2005 agrees well with our polar PC-SAFT predictions. However, as seen in Table 4, both models reliably reproduce

water thermophysical properties, provided polarization corrections are explicitly included in the TIP4P/2005 calculations to improve the vapor pressure predictions. Figure 5 shows that water in TIP4P/2005 appears to be more structured with longer associated chains (higher fraction of molecules forming four hydrogen bonds) than that in iAMOEBA. Simulation data for the hydrogen bonding distribution across the water saturated curve is illustrated in Tables 5 and 6. It can also be seen that, based on the geometric hydrogen bonding criteria used here, structures with five hydrogen bonds are not excluded. This is in agreement with the analysis done by Prada-Garcia et al.95 using other classical water models such as SPC,96 SPC/E,97 TIP3P,98 TIP4P,99 and TIP4P-Ew100 and using six different hydrogen bonding definitions. The presence of a greater number of hydrogen bonds than would be suggested on the basis of quantum chemical considerations undoubtedly reflects a weakness in using a solely geometric based definition of hydrogen bonding. As a practical matter, this will influence the extents of association calculated using TIP4P/2005 and iAMOEBA water models relative to that inferred on the basis of spectroscopic data. In order to match Luck’s IR data, the association energy in the SAFT water model needs to be reduced down to a value close to 1300 K. This is far

Table 6. iAMOEBA Results for the Distribution of Hydrogen Bonds per Molecule of Pure Liquid Water at Saturation number of hydrogen bonds T (K)

0

1

2

3

4

5

293.5 327.38 354 389 426 456.39 486.3 528

0.0005 0.0014 0.0027 0.0049 0.0090 0.0151 0.0225 0.0393

0.0094 0.0193 0.0304 0.0485 0.0729 0.1009 0.1279 0.1787

0.0743 0.1151 0.1499 0.1940 0.2381 0.2756 0.3034 0.3368

0.2905 0.3360 0.3587 0.3714 0.3714 0.3595 0.3417 0.3014

0.5496 0.4574 0.3944 0.3278 0.2664 0.2167 0.1791 0.1277

0.0741 0.0691 0.0620 0.0515 0.0406 0.0311 0.0244 0.0155

Figure 6. Fraction of molecules involved in self-association for the binary mixture ethanol + water at 298 and 373 K. (A) Water and ethanol are modeled using a 2(1,1) association scheme. (B) Water is modeled using a 4(2,2) association scheme, while ethanol is modeled using a 2(1,1) association scheme. (C) Water is modeled using a 4(2,2) association scheme, while ethanol is modeled using a 3(2,1) association scheme. Black, ethanol; red, water; (■) MD simulations at 298 K; (▲) MD simulations at 373 K; () polar PC-SAFT at 298 K; (- - -) polar PC-SAFT at 373 K. 3394


Article

The Journal of Physical Chemistry B Table 7. Simulation Results for the Fraction of Molecules Involved in Self-Association (at Least Once) xalcohol

0.1

0.2

methanol water

0.1397 0.9999

0.2621 0.9985

ethanol water

0.1744 0.9997

0.3019 0.9974

ethanol water

0.1529 0.9990

0.2866 0.9957

ethanol water

0.1490 0.9979

0.2765 0.9931

1-propanol water

0.2092 0.9993

0.3209 0.9967

0.4

0.6

Methanol + Water at T = 298 K 0.4783 0.6759 0.9851 0.9135 Ethanol + Water at T = 298 K 0.5247 0.7028 0.9792 0.9069 Ethanol + Water at T = 343 K 0.4919 0.6741 0.9727 0.8916 Ethanol + Water at T = 373 K 0.4751 0.6410 0.9635 0.8748 1-Propanol + Water at T = 298 K 0.5315 0.6901 0.9771 0.9112

0.8

0.9

1

0.8613 0.6641

0.9308 0.4063

0.9656 0

0.8685 0.6549

0.9361 0.4070

0.1744 0

0.8291 0.6485

0.8951 0.3969

0.1529 0

0.7886 0.6328

0.8529 0.3830

0.1490 0

0.8440 0.6909

0.9652 0.4929

0.2092 0

Figure 7. Excess enthalpies for the binary mixtures 1-alcohol + water: (a) methanol at 298 K; (b) ethanol at 298 K; (c) 1-butanol at 303 K. (○) Experimental data;104−107 () polar PC-SAFT using a 3(2,1) association scheme for alcohol; (- - -) polar PC-SAFT using a 2(1,1) association scheme for alcohol.

away from the spectroscopic value of 1813 K.91 On the basis of the molecular dynamic simulation and modeling efforts done so far, it is difficult to make a definite conclusion on the accuracy of Luck’s data solely through comparing hydrogen bonding distributions. Hydrogen Bonding Distribution in 1-Alcohol + Water Binary Mixtures. It was shown in our previous publication87

that self-association in alcohols is best modeled using two association sites 2(1,1) for pure alcohol and alcohol + alkane binary systems. This can be attributed to steric hindrance caused by the alkyl chain attached to the hydroxyl group which prevents the oxygen atom from forming more than a single hydrogen bond. 3395


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Figure 8. Average number of hydrogen bonds formed by an ethanol molecule (E···X) and by the oxygen atom in an ethanol molecule (OE···HX−OX) as a function of composition in the binary mixture ethanol + water: (a) T = 298 K; (b) T = 373 K. Symbols: MD simulations; () polar PC-SAFT using a 3(2,1) association scheme for ethanol; (- - -) polar PC-SAFT using a 2(1,1) association scheme for ethanol.

Figure 9. Distribution of the hydrogen bond number per molecule of ethanol in a 10 mol % solution using polar PC-SAFT and MD simulation: (a) T = 298 K; (b) T = 373 K.

Table 8. Simulation Results for the Average Number of Hydrogen Bonds Formed in the Mixture xalcohol

0.1

0.2

OM···HX−OX M···X

1.4256 2.2945

1.3552 2.2299

OE···HX−OX E···X

1.4808 2.3583

1.3830 2.2601


1.3981 2.2012

1.2990 2.0977


1.3372 2.0940

1.2428 1.9944

OP···HX−OX P···X

1.3558 2.2014

1.2708 2.1157

0.4

0.6

Methanol + Water at T = 298 K 1.2460 1.1442 2.1311 2.0400 Ethanol + Water at T = 298 K 1.2351 1.1311 2.1174 2.0206 Ethanol + Water at T = 343 K 1.1634 1.0491 1.9622 1.8523 Ethanol + Water at T = 373 K 1.1086 0.9951 1.8585 1.7485 1-Propanol + Water at T = 298 K 1.1557 1.0736 2.0078 1.9350

To determine the optimum association scheme, the same strategy is followed in this work to study the thermodynamics of hydrogen bonding in alcohol + water binary systems.

0.8

0.9

1

1.0313 1.9363

0.9704 1.8777

0.9084 1.8168

1.0269 1.9270

0.9696 1.8755

0.9105 1.8211

0.9383 1.7480

0.8788 1.6919

0.8164 1.6328

0.8842 1.6422

0.8255 1.5864

0.7631 1.5263

0.9879 1.8604

0.9357 1.8110

0.8722 1.7445

Figure 6 depicts the fraction of ethanol and water molecules involved in self-association (fraction of ethanol/water molecules that are bonded at least once to another ethanol/water 3396


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concentrations, while a two-site association scheme 2(1,1) appears to be the most successful at high ethanol concentrations. Observations shown in Figure 6 were found to be independent of the number of carbons in the alcohol. Simulation results for methanol, ethanol, and 1-propanol systems are summarized in Table 7. On the basis of the results demonstrated in Figure 6, the best agreement was, in general, achieved using a two-site alcohol model 2(1,1). However, fractions of species involved in selfassociation can be misleading. Predicting accurate fractions using a SAFT two-site alcohol model does not necessarily mean that the correct thermodynamics of hydrogen bonding is captured. This can be shown through plotting the excess enthalpy of alcohol + water mixtures as a function of composition. In general, excess enthalpies are highly influenced by the fluid structure and the number of hydrogen bonds formed. In reference to Figure 7, it can be seen that a two-site alcohol model 2(1,1) predicts the opposite sign of the excess enthalpy (positive instead of negative). A better qualitative agreement was achieved using a three-site alcohol model 3(2,1). Consequently, we compared the average number of hydrogen bonds formed by both association schemes to MD simulation results. In reference to Figure 8, it can be seen that the average number of hydrogen bonds (nHB) formed by ethanol (E···X) in an ethanol + water binary mixture increases with increasing water concentration. Also, as expected, nHB decreases with

molecule) as a function of composition, temperature, and association scheme. Simulations at 298 K were performed at a pressure of 1 bar, while simulations at 373 K were done at a pressure of 3 bar. Fractions were found to be pressure independent. Simulations show that the fraction of water molecules involved in self-association stays almost constant at a value of unity up to a molar composition of 20% ethanol. A dramatic decrease can then be seen as ethanol molar composition increases. On the other hand, the fraction of ethanol molecules involved in self-association increases almost linearly with increasing molar composition. The fraction of water molecules involved in self-association was most accurately captured using four association sites 4(2,2) for water and two association sites 2(1,1) for ethanol in Figure 6b. In terms of ethanol concentration, it can be seen that an association scheme transition occurs at around 20 mol %. A three-site association scheme 3(2,1) appears to be the most successful at low ethanol Table 9. Simulation Results for the Distribution of Hydrogen Bonds per Molecule of Ethanol in a 10 mol % Solution number of hydrogen bonds formed with another ethanol

ethanol (T = 298 K)

ethanol (T = 373 K)

0 1 2 3

0.8192 0.1664 0.0140 0.0004

0.8492 0.1413 0.0092 0.0003

Figure 10. Isothermal VLE for the binary mixtures 1-alcohol + water: (a) methanol + water with kij = 0 (dashed line) and kij = 0.025 (solid line); (b) ethanol + water with kij = 0; (c) 1-propanol + water with kij = 0. (○) VLE experimental data;59,60,108 () polar PC-SAFT. 3397


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Figure 11. Isobaric VLE and LLE for the binary mixtures 1-alcohol + water at 101.33 kPa: (a) 1-butanol + water with kij = 0; (b) 1-pentanol + water with kij = 0; (c) 1-hexanol + water with kij = 0. (red ○) VLE experimental data;109−111 (black ○) LLE experimental data;112 () polar PC-SAFT.

increasing temperature. At 298 K, nHB increases almost linearly from 1.82 for pure ethanol to 2.36 for a 10 mol % ethanol solution. A two-site SAFT ethanol model 2(1,1) was able to well predict the nHB for pure ethanol. However, the composition dependence of the nHB could not be captured, since the maximum number of hydrogen bonds allowed by a 2(1,1) association scheme is two. On the other hand, a threesite SAFT ethanol model 3(2,1) was unable to well predict the nHB for pure ethanol. The composition dependence of nHB was qualitatively captured better using a 3(2,1) association scheme but quantitatively overpredicted. Figure 9 demonstrates that polar PC-SAFT was able to accurately predict the fraction of ethanol molecules bound in cluster with another ethanol molecule once, twice, and thrice for a 10 mol % solution. Consequently, we expect that overpredicting the nHB formed by ethanol in the mixture (as shown in Figure 8) is solely due to overpredicting the cross-association between ethanol and water. Simulation data, as a function of composition and temperature, for the nHB formed by methanol, ethanol, and 1-propanol containing systems can be found in Table 8. In addition, simulation data for the distribution of hydrogen bonds in the case of a 10 mol % ethanol solution can be found in Table 9. We also expect that quantitative agreement with simulation for the average number of hydrogen bonds formed can be achieved through extending the theory to allow multiple bonding at one site.101−103 In this case, alcohols would be modeled with an

electron acceptor site that can bond only once and an electron donor site that can bond twice. The maximum number of hydrogen bonds formed by an alcohol molecule would then be determined geometrically by the system and not specified as in the case of Wertheim’s first-order perturbation theory. Phase Behavior of 1-Alcohol + Water Binary Mixtures. On the basis of the previous comparisons against molecular simulation, we decided to model the phase equilibrium of 1-alcohol + water binary systems using a 3(2,1) association scheme for alcohols and a 4(2,2) association scheme for water. The isothermal and isobaric vapor−liquid equilibrium (VLE) and liquid−liquid equilibrium (LLE) of 1-alcohol + water binary systems are presented in Figures 10 and 11 respectively. Polar PC-SAFT was able to accurately predict the VLE without fitting binary interaction parameters. Results from the equation of state for VLE of binary aqueous solutions of methanol, ethanol, and propanol are in very good agreement with experimental data over a range of temperatures. In Figure 10b,c, the strong bubble point dependence on liquid composition at low alcohol concentration can only be matched with the 3(2,1) association scheme for alcohols. In terms of the VLE and LLE for the higher alcohols, polar PC-SAFT was able to accurately predict the VLE, the three-phase line, and the solubility of 1-butanol, 1-pentanol, and 1-hexanol in the aqueous phase at atmospheric pressure, but the model underestimates the solubility of water in the alcohol rich phase. 3398


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a Function of Water Content for Pressures up to 40 MPa. Ind. Eng. Chem. Res. 2015, 54, 9637−9644. (5) Yu, M.-L.; Chen, Y.-P. Correlation of Liquid-Liquid Phase Equilibria Using the SAFT Equation of State. Fluid Phase Equilib. 1994, 94, 149−165. (6) Galindo, A.; Whitehead, P. J.; Jackson, G. Predicting the HighPressure Phase Equilibria of Water+ n-Alkanes Using a Simplified SAFT Theory with Transferable Intermolecular Interaction Parameters. J. Phys. Chem. 1996, 100, 6781−6792. (7) Economou, I. G.; Tsonopoulos, C. Associating Models and Mixing Rules in Equations of State for Water/Hydrocarbon Mixtures. Chem. Eng. Sci. 1997, 52, 511−525. (8) Errington, J. R.; Boulougouris, G. C.; Economou, I. G.; Panagiotopoulos, A. Z.; Theodorou, D. N. Molecular Simulation of Phase Equilibria for Water-Methane and Water-Ethane Mixtures. J. Phys. Chem. B 1998, 102, 8865−8873. (9) Kontogeorgis, G. M.; Yakoumis, I. V.; Meijer, H.; Hendriks, E.; Moorwood, T. Multicomponent Phase Equilibrium Calculations for Water−Methanol−Alkane Mixtures. Fluid Phase Equilib. 1999, 158, 201−209. (10) Voutsas, E. C.; Yakoumis, I. V.; Tassios, D. P. Prediction of Phase Equilibria in Water/Alcohol/Alkane Systems. Fluid Phase Equilib. 1999, 158, 151−163. (11) Zhang, Z.-Y.; Yang, J.-C.; Li, Y.-G. Prediction of Phase Equilibria for CO2−C2H5OH−H2O System Using the SAFT Equation of State. Fluid Phase Equilib. 2000, 169, 1−18. (12) Voutsas, E. C.; Boulougouris, G. C.; Economou, I. G.; Tassios, D. P. Water/Hydrocarbon Phase Equilibria Using the Thermodynamic Perturbation Theory. Ind. Eng. Chem. Res. 2000, 39, 797−804. (13) McCabe, C.; Galindo, A.; Cummings, P. T. Anomalies in the Solubility of Alkanes in Near-Critical Water. J. Phys. Chem. B 2003, 107, 12307−12314. (14) Patel, B.; Paricaud, P.; Galindo, A.; Maitland, G. Prediction of the Salting-Out Effect of Strong Electrolytes on Water+Alkane Solutions. Ind. Eng. Chem. Res. 2003, 42, 3809−3823. (15) Li, X.-S.; Englezos, P. Vapor−Liquid Equilibrium of Systems Containing Alcohols, Water, Carbon Dioxide and Hydrocarbons Using SAFT. Fluid Phase Equilib. 2004, 224, 111−118. (16) Valtz, A.; Chapoy, A.; Coquelet, C.; Paricaud, P.; Richon, D. Vapour−Liquid Equilibria in the Carbon Dioxide−Water System, Measurement and Modelling from 278.2 to 318.2 K. Fluid Phase Equilib. 2004, 226, 333−344. (17) Ji, X.; Tan, S. P.; Adidharma, H.; Radosz, M. SAFT1-RPM Approximation Extended to Phase Equilibria and Densities of CO2H2O and CO2-H2O-NaCl Systems. Ind. Eng. Chem. Res. 2005, 44, 8419−8427. (18) Perakis, C.; Voutsas, E.; Magoulas, K.; Tassios, D. Thermodynamic Modeling of the Vapor−Liquid Equilibrium of the Water/Ethanol/CO2 System. Fluid Phase Equilib. 2006, 243, 142− 150. (19) Grenner, A.; Schmelzer, J.; von Solms, N.; Kontogeorgis, G. M. Comparison of Two Association Models (Elliott-Suresh-Donohue and Simplified PC-SAFT) for Complex Phase Equilibria of HydrocarbonWater and Amine-Containing Mixtures. Ind. Eng. Chem. Res. 2006, 45, 8170−8179. (20) Voutsas, E.; Perakis, C.; Pappa, G.; Tassios, D. An Evaluation of the Performance of the Cubic-Plus-Association Equation of State in Mixtures of Non-Polar, Polar and Associating Compounds: Towards a Single Model for Non-Polymeric Systems. Fluid Phase Equilib. 2007, 261, 343−350. (21) dos Ramos, M. C.; Blas, F. J.; Galindo, A. Modelling the Phase Equilibria and Excess Properties of the Water+Carbon Dioxide Binary Mixture. Fluid Phase Equilib. 2007, 261, 359−365. (22) Aparicio-Martínez, S.; Hall, K. R. Phase Equilibria in Water Containing Binary Systems from Molecular Based Equations of State. Fluid Phase Equilib. 2007, 254, 112−125. (23) Folas, G. K.; Froyna, E. W.; Lovland, J.; Kontogeorgis, G. M.; Solbraa, E. Data and Prediction of Water Content of High Pressure

CONCLUSION The polar and perturbed chain of the statistical associating fluid theory (polar PC-SAFT) and molecular dynamic (MD) simulations were used to study the thermodynamics of hydrogen bonding in 1-alcohol + water binary systems. Molecular simulation was used to compute the distribution of hydrogen bonds to evaluate the association term within SAFT. Results show that hydrogen bonding in pure alcohols is best captured using a two-site association scheme 2(1,1). However, increasing water concentration leads to an increase in the average number of hydrogen bonds formed by an alcohol molecule. As a result, a transition in association scheme occurs at higher water compositions and a three-site alcohol model 3(2,1) was required for better agreement with simulation. The latter association scheme was used to predict the vapor−liquid equilibrium (VLE) and liquid−liquid equilibrium (LLE) for a range of 1-alcohol + water binary systems. Polar PC-SAFT showed good agreement with experiment, demonstrating the predictive ability and strong fundamental basis of the equation of state. Future work will introduce an extension of the equation of state to allow for multiple bonds at an association site; thus, the theory will show the observed transition in association scheme.

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jpcb.5b12375. Derivation of the fraction of molecules involved at least once in self-association, derivation of the fraction of k-time bonded molecules due to self- and cross-association, and derivation of the fraction of k-time bonded molecules due to self-association only (PDF)

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Phone: 1-713-348-4900. Fax: 1-713-348-5478. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS W.A.F. gratefully acknowledges the Abu Dhabi National Oil Company (ADNOC) for financial support through a Ph.D. scholarship. L.W. gratefully acknowledges Rice University Consortium for Processes in Porous Media for financial support. A.H. gratefully acknowledges the financial support from the Robert A. Welch Foundation (Grant No. C-1241).

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REFERENCES

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