Free Energy of Mixing of Acetone and Methanol: A Computer

Article pubs.acs.org/JPCB

Free Energy of Mixing of Acetone and Methanol: A Computer Simulation Investigation Abdenacer Idrissi,*,† Kamil Polok,‡ Mohammed Barj,† Bogdan Marekha,†,§ Mikhail Kiselev,∥ and Pál Jedlovszky*,⊥,#,▽ †

Laboratoire de Spectrochimie Infrarouge et Raman (UMR CNRS A8516), Université des Sciences et Technologies de Lille, Bâtiment C5, 59655 Villeneuve d’Ascq Cedex, France ‡ Laboratory of Physicochemistry of Dielectrics and Magnetics, Department of Chemistry, University of Warsaw, Zwirki i Wigury 101, 02-089 Warsaw, Poland § Department of Inorganic Chemistry, V. N. Karazin Kharkiv National University, 4 Svobody sq., 61022 Kharkiv, Ukraine ∥ Institute of Solution Chemistry of the Russian Academy of Sciences, Akademicheskaya 1, R-153045 Ivanovo, Russia ⊥ Laboratory of Interfaces and Nanosize Systems, Institute of Chemistry, Eötvös Loránd University, Pázmány P. Stny 1/A, H-1117 Budapest, Hungary # MTA-BME Research Group of Technical Analytical Chemistry, Szt. Gellért tér 4, H-1111 Budapest, Hungary ▽ EKF Department of Chemistry, Leányka utca 6, H-3300 Eger, Hungary ABSTRACT: The change of the Helmholtz free energy, internal energy, and entropy accompanying the mixing of acetone and methanol is calculated in the entire composition range by the method of thermodynamic integration using three different potential model combinations of the two compounds. In the first system, both molecules are described by the OPLS, and in the second system, both molecules are described by the original TraPPE force field, whereas in the third system a modified version of the TraPPE potential is used for acetone in combination with the original TraPPE model of methanol. The results reveal that, in contrast with the acetone−water system, all of these three model combinations are able to reproduce the full miscibility of acetone and methanol, although the thermodynamic driving force of this mixing is very small. It is also seen, in accordance with the finding of former structural analyses, that the mixing of the two components is driven by the entropy term corresponding to the ideal mixing, which is large enough to overcompensate the effect of the energy increase and entropy loss due to the interaction of the unlike components in the mixtures. Among the three model combinations, the use of the original TraPPE model of methanol and modified TraPPE model of acetone turns out to be clearly the best in this respect, as it is able to reproduce the experimental free energy, internal energy, and entropy of mixing values within 0.15 kJ/mol, 0.2 kJ/mol, and 1 J/(mol K), respectively, in the entire composition range. The success of this model combination originates from the fact that the use of the modified TraPPE model of acetone instead of the original one in these mixtures improves the reproduction of the entropy of mixing, while it retains the ability of the original model of excellently reproducing the internal energy of mixing.

1. INTRODUCTION Acetone and methanol are widely used as solvents and reagents in the pharmaceutical and fine industries.1 The mixture of these two compounds forms a maximum pressure azeotrope. Considerable scientific effort has thus been expended on separating the azeotropic mixture either by using a separating agent or by changing the thermodynamic conditions.2 From the theoretical point of view, this mixture offers the possibility to study the interplay between dipole−dipole, hydrogen bonding, and hydrophobic interactions due to the presence of, respectively, the strongly polar CO group of the acetone molecules, OH group of methanols (possessing with Hbonding donor and acceptor properties), and apolar CH3 © 2013 American Chemical Society

groups of both methanol and acetone. Because all of these groups also play important roles in a set of biologically relevant molecules, a local structural analysis of acetone−methanol mixtures may also give insight into the interactions in bioorganic and organic molecular systems. Computer simulation investigations contributed in a large part to shedding light on the microscopic structure of acetone− methanol mixtures. Thus, Venables et al.3 performed molecular dynamics (MD) simulations with the effective optimized Received: May 23, 2013 Revised: October 23, 2013 Published: October 28, 2013 16157

dx.doi.org/10.1021/jp405090j | J. Phys. Chem. B 2013, 117, 16157−16164

The Journal of Physical Chemistry B

Article

OPLS force field. Later, it has also been shown that the wellknown full miscibility of acetone and water cannot be reproduced by any combination of the TraPPE and AUA420 acetone and SPC,21 SPC/E, TIP4P, TIP4P/2005,22 TIP5P,23 and TIP5P-E24 water models, whereas the combination of the recently developed Pereyra−Asar−Carignano (PAC) acetone model15 with TIP5P-E water not only shows the desired full miscibility but also is able to rather accurately reproduce the energy, free energy, and entropy of mixing in the entire composition range.26 All of these findings outline the urgent need of testing combinations of various potential models working well for neat systems against experimental data obtained for the mixed systems. In the particular case of investigating the microscopic structure of acetone−methanol mixtures, such an assessment of various potential model combinations is clearly needed to proceed further in the structural and dynamical analysis of these systems on the molecular level in a meaningful way. A potential model pair that also works well for mixtures of these components could then be used to study issues such as the effect of acetone on the local structure on the hydrogen bonding network of methanol, cluster formation, inhomogeneity, orientational correlation between neighboring molecules, self-association, as well as vapor−liquid equilibrium or transport properties in these mixtures. We compare the performance of three combinations of the OPLS, TraPPE, and TraPPE-mod acetone and OPLS and TraPPE methanol models by comparing the change of their energy, free energy, and entropy upon mixing with experimental data27,28 in the entire composition range. The change of these quantities accompanying the mixing of the two neat components are calculated by the method of thermodynamic integration (TI),29,30 employing a thermodynamic cycle recently proposed by us31 for such calculations. The paper is organized as follows. The performed calculations, including Monte Carlo simulations, TI, and calculation of the change of various thermodynamic quantities accompanying the mixing of the two neat liquids are detailed in Section 2. The potential model combinations considered are also introduced here. The results obtained with different model combinations are then presented and compared to experimental data in Section 3.

potential for liquid simulations (OPLS) force-field parameters both for methanol4 and acetone5 to calculate dynamical properties (translational and rotational motions) related to the infrared spectra of such mixtures. They found that the ability of the acetone molecules of accepting hydrogen bonds is the primary factor that determines the molecular level details of both the structure and the dynamics of these mixtures. Their results also revealed that methanol molecules show a remarkably strong tendency to remain in chains, and these chains get shorter as the mole fraction of methanol in the mixture decreases.3 Later, Gupta and Chandra6 showed, using the three-site H1 methanol model of Haughney et al.,7,8 that the dynamical properties (diffusion coefficient, energy, and lifetime of hydrogen bonds between like and unlike molecules) essentially linearly depend on the composition of such mixtures. In other words, acetone−methanol mixtures behave ideally in this respect. Kamath et al. determined the phase diagram of acetone− methanol mixtures by Monte Carlo simulations 9 and reproduced the maximum pressure azeotropy behavior of this system by the transferable potentials for phase equilibria (TraPPE)10,11 potential model. In the course of these simulations they developed a modified version of the TraPPE force field for acetone (referred to here as TraPPE-mod), which gives a better reproduction of the experimental data for acetone−methanol mixtures when used in combination with the original TraPPE model of methanol.9 They also performed a structural analysis of the equimolar mixture and found that the limited association of the unlike molecules in this system has only a minor effect on the self-association behavior of the methanol molecules.9 Performing simulations with the OPLS force-field parameters and calculating the Kirkwood−Buff integrals from the long-distance behavior of the radial distribution functions, Perera et al. showed that this mixture is microheterogeneous.12 In a recent study,13 we confirmed, by performing a detailed Voronoi polyhedra analysis, the microheterogeneity of acetone−methanol mixtures and showed that acetone molecules show even stronger tendency for selfassociation than methanols. The strongest self-association of the acetone and methanol molecules was found to occur at the acetone mole fraction ranges of 0.2 to 0.5 and 0.4 to 0.5, respectively.13 The potential models used in these studies were originally developed to reproduce primarily the properties of the respective neat systems. Although potential models belonging to the same force field, such as OPLS or TraPPE, claimed to be transferable, that is, they can be equally used in different systems, recent evidence indicates that potential models working well for neat systems can reproduce the experimental properties of various liquid mixtures with a rather broad range of accuracy. Thus, for instance, we have recently demonstrated that the potential model proposed by Zhang and Duan for CO2,14 although working very well for the neat system in a broad range of thermodynamic states, cannot accurately reproduce the mixing properties of supercritical CO2 with either acetone or ethanol.15 Using the TraPPE force field for both components resulted in a rather accurate reproduction of the mixing properties of these systems.15 More strikingly, neither the OPLS nor the KBFF16 model of acetone turned out to be fully miscible with the SPC/E17 and TIP4P18 models of water,19 although KBFF was claimed to be developed by fitting mixing properties with SPC/E water,16 whereas the TIP4P water model is generally thought to be compatible with the

2. METHODS 2.1. Thermodynamic Integration. The method of TI29,30 can provide the Helmholtz free-energy difference between the state of interest (marked here as Y) and an appropriately chosen reference state, X. When applying the method for a given condensed phase, the reference state is typically chosen to be the ideal gas state of the same system. In this way, the excess Helmholtz free energy of the phase (relative to the ideal gas state) is obtained from the calculation. The free-energy difference between states X and Y is calculated as ΔA = AY − AX =

∫0

1 ⎛ ∂A(λ) ⎞

⎜ ⎟ dλ ⎝ ∂λ ⎠

(1)

where A stands for the Helmholtz free energy and λ is a coupling parameter that defines the continuous (and fictitious) transition path along which the system is brought from the state of interest Y (corresponding to λ = 1) to the reference state X (corresponding to λ = 0). Considering the fundamental relation between the Helmholtz free energy and the canonical partition function, Q, i.e., 16158



A(λ) = −RT ln Q (λ)

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continuous transition path connecting the system of interest and the reference system. In this study we evaluated the integrand of eq 7 at five different λ values corresponding to the Gaussian quadrature, i.e., at 0.046911, 0.230765, 0.5, 0.769235, and 0.953089, respectively, by performing computer simulations on the canonical (N,V,T) ensemble. In addition, a simulation at the real temperature of the system (i.e., at λ = 1) has also been performed to obtain the energy of the system as well. The integral of eq 7 has been calculated by fitting a fifth-order polynomial function to these data. Finally, having the internal energy and Helmholtz free energy, the entropy of the system (in excess to the ideal gas state) can simply be evaluated as

(2)

and writing Q as Q (λ ) =

∫ exp(−βU(λ)) dq N

(3)

the integrand of eq 1 can be written as ∂A(λ) ∂ ln Q (λ) = −RT ∂λ ∂λ

∫ ( ∂U∂(λλ) ) exp(−βU (λ)) dq N

=

∫ exp(−βU (λ)) dq N ⎛ ∂U (λ) ⎞ ⎜ ⎟ ⎝ ∂λ ⎠

=

S= λ

(4)

(5)

also taking into account the fact that in a 3D system where the repulsive interaction decays with r−12 the use of an exponent smaller than 4 would lead to a singularity of the integral in eq 1 at its λ = 0 end.29 Considering also that UX = 0 the ensemble average of eq 4 can be written as ∂U (λ) ∂λ

= 4λ 3⟨UY ⟩λ λ

uij =

∫0

1

4λ 3 < UY >λ dλ

where indices a and b run over the interaction sites of molecule i and j, respectively, qa and qb are the fractional charges carried by the respective sites, ∈0 is the vacuum permittivity, σab and εab are the Lennard-Jones distance and energy parameters, respectively, calculated from the values corresponding to the individual sites using the Lorentz−Berthelot rule,32 and ria,jb is the distance between site a of molecule i and site b of molecule j. The CH3 groups are treated in these models as united atoms, and thus methanol and acetone models consist of three and four interaction sites, respectively. The Lennard-Jones parameters and fractional charges corresponding to the interaction sites of the acetone and methanol models considered are summarized in Tables 1 and 2, respectively. All of the models considered in this study are rigid; the bond lengths and bond angles of the originally flexible TraPPE and TraPPE-mod models are fixed at their equilibrium values. Thus, the CH3−O and O−H bond lengths and CH3−O−H bond angle of methanol are 1.43 Å, 0.945 Å, and 108.5° in both models considered; the CH3−C−CH3 and CH3−CO bond angles of the planar acetone molecule are 117.2 and 121.4° in every case, whereas the CH3−C and CO bond lengths are 1.507 and 1.222 Å, respectively, in the case of the OPLS model, and 1.52 and 1.229 Å, respectively, in the TraPPE and TraPPEmod models of acetone. 2.3. Monte Carlo Simulations. Monte Carlo simulations of acetone−methanol mixtures of different compositions have

(6)

(7)

(8)

where kB is the Boltzmann constant and T * = T/λ 4

b

12 ⎡⎛ ⎛ σ ⎞6 ⎤ σab ⎞ 1 qaqb ⎢ ⎟⎟ − ⎜⎜ ab ⎟⎟ ⎥ + 4εab⎢⎜⎜ ⎥ 4π ∈0 ria , jb r ⎝ ria , jb ⎠ ⎦ ⎣⎝ ia , jb ⎠

(11)

The ensemble averaging of eq 7 should be made at a given λ value, i.e., with using the U(λ) potential function in the simulation instead of the full UY potential corresponding to the system of interest. However, using eq 5 and the relation UX = 0, the Boltzmann factor to be used in this ensemble averaging can be rewritten as exp(− U (λ)/kBT ) = exp(− λ 4UY /kBT ) = exp(− UY /kBT *)

∑∑ b

Substituting eq 6 to eq 4 and then to eq 1 one gets ΔA =

(10)

2.2. Potential Models. In this study we consider two potential models of methanol, namely, OPLS4 and TraPPE,10 and three models, i.e., OPLS,5 TraPPE,11 and TraPPE-mod,9 of acetone. The mixing properties of the two compounds are calculated describing both molecules by their OPLS model, both by their TraPPE model, and methanol by the TraPPE and acetone by the TraPPE-mod potential. These mixed systems will be referred to here by the corresponding acetone potential model, i.e., OPLS, TraPPE, and TraPPE-mod, respectively. All of these potential models are pairwise additive, and thus the total internal energy of the system can be calculated as the sum of the interaction energies of all molecule pairs. The uij interaction energy of the ith and jth molecules is the sum of the Lennard-Jones and charge−charge Coulomb contributions of their interaction sites:

In these equations R stands for the gas constant, T is the absolute temperature, β = 1/RT, U is the potential energy of the system, qN represents the full set of 3N position coordinates of the N particles, and the brackets ⟨....⟩λ denote ensemble averaging at a given λ value. To perform the derivation in eq 4, the continuous U(λ) function has to be defined in the entire 0 ≤ λ ≤ 1 range in such a way that U(1) corresponds to the potential energy of the system of interest and U(0) = 0, reflecting the fact that the potential energy of the ideal gas is zero. The U(λ) function is conventionally defined as a polynomial function, i.e., U (λ) = λ 4UY + (1 − λ)4 UX

U−A T

(9)

In other words, the simulation to be performed at the real temperature of the system, T, using the potential function U(λ) is technically equivalent to that performed at the virtual temperature T* = T/λ4 using the full potential function UY. It should be clearly pointed out that the simulations performed at the various virtual temperatures themselves do not correspond to any real system; instead, they represent fictitious systems corresponding to given λ values along the fictitious and 16159



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no more than 10°. More than 40% of the attempted moves have been successful in every case. All interactions have been truncated beyond the center−center cutoff distance of 12.5 Å. The long-range part of the electrostatic interaction has been taken into account by the reaction field correction method32,36,37 under conducting boundary conditions. Initial configuration of the λ = 1 run has been obtained by randomly placing the required number of molecules into the basic box. Simulations of a given system have been performed in the order of decreasing λ values, the starting configuration of the simulations done at the quadrature points has always been the final configuration obtained with the previous λ value considered. The systems have been equilibrated by performing 5 × 107 Monte Carlo moves; the ensemble average of eq 6 has then been evaluated over a trajectory of 108 equilibrium sample configurations. Error bars have been estimated by the method of block averages32 using 2.5 × 105 Monte Carlo steps long blocks. The integrand of eq 7 is shown in Figure 1 as calculated

Table 1. Interaction Parameters of the Acetone Potential Models Used potential model

interaction site

(ε/kB)/K

σ/Å

q/e

CH3 C O CH3 C O CH3 C O

80.5 52.8 105.7 98.0 40.0 79.0 98.0 27.0 79.0

3.91 3.75 2.96 3.75 3.82 3.05 3.75 3.82 3.05

0.062 0.300 −0.424 0.0 0.424 −0.424 −0.049 0.662 −0.564

OPLSa

TraPPEb

TraPPE-modc a

Ref 5. bRef 11. cRef 9.

Table 2. Interaction Parameters of the Methanol Potential Models Used potential model OPLSa

TraPPEb a

interaction site

(ε/kB)/K

σ/Å

q/e

CH3 O H CH3 O H

104.2 85.5 0 98.0 93.0 0

3.775 3.07 0 3.75 3.02 0

0.265 −0.700 0.435 0.265 −0.700 0.435

Ref 4. bRef 10.

been performed in the canonical (N,V,T) ensemble. The temperature of the real system, corresponding to the λ value of 1, has been 298 K; the virtual temperatures along the thermodynamic path connecting the real system to the corresponding ideal gas have been determined according to eq 9. Mixtures of nine different compositions, corresponding to the acetone mole fraction (xac) values of 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, and 0.9, have been considered. In addition, the two neat systems have also been simulated. The basic simulation box has consisted of 512 molecules; the number of acetone and methanol molecules (Nac and Nme, respectively) used in simulations of mixtures of different compositions is collected in Table 3. The edge length L of the cubic basic simulation box Table 3. Properties of the Systems Simulated xac

Nac

Nme

L/Å

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0 51 102 154 205 256 307 358 410 461 512

512 461 410 358 307 256 205 154 102 51 0

32.94 33.77 34.58 35.35 36.10 36.81 37.51 38.19 38.87 39.51 40.13

Figure 1. Integrand of the thermodynamic integration calculated at the six λ points considered in equimolar acetone−methanol mixtures (circles) as well as in the corresponding neat systems (methanol: squares; acetone: triangles), together with the fifth-order polynomial functions fitted to these data (solid curves) in the systems where both components are described by the OPLS force field (top panel), both are described by the TraPPE force field (middle panel), and methanol is described by the TraPPE while acetone is described with the TraPPE-mod potential model (bottom panel). Error bars are shown only when smaller than the symbols.

at the six λ points considered in the three equimolar mixtures and in the corresponding neat systems. The fifth-order polynomial functions fitted to these data are also shown. 2.4. Calculation of the Change of Thermodynamic Quantities upon Mixing. To characterize the thermodynamics of mixing acetone and methanol, we have calculated the change of the molar internal energy, Helmholtz free energy, and entropy of the system accompanying the mixing of the two components at various compositions. For this purpose, the mixing of the two neat components is assumed to occur along a fictitious thermodynamic path in the following three steps.31

has been determined in accordance with the experimental density of the simulated system;33,34 these values are also included in Table 3. Simulations have been performed using the program MMC.35 In a Monte Carlo step a randomly chosen molecule has been randomly translated by no more than 0.25 Å and randomly rotated around a randomly chosen space-fixed axis by 16160



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First, the neat components are brought to the ideal gas state. Because the potential energy of an ideal gas is zero, the energy change corresponding to this step is simply the internal energy of the neat liquids at the starting point. Furthermore, the change of the Helmholtz free energy and entropy in this step can be calculated by the method of TI, as described in subsection 2.1. (see eq 7) and by eq 10, respectively. In the second step, the two neat components are mixed in the ideal gas state. This step is not accompanied by any change of the energy, while the change in the Helmholtz free energy and entropy can simply be given as the corresponding changes of the ideal mixing. Finally, in the third step, the mixture is brought from the ideal gas to the liquid state; the change of the thermodynamic quantities accompanying this step can be calculated just as in the first step. Thus, the molar internal energy, Helmholtz free energy, and entropy of mixing the two components can be calculated as U mix = UAB − xAUA − x BUB

(12)

Amix = AAB − xAAA − x BAB + RT (xA ln xA + x B ln x B) (13)

and S mix = SAB − xASA − x BSB − R(xA ln xA + x B ln x B)

Figure 2. Helmholtz free energy (top panel), internal energy (middle panel), and entropy (bottom panel) of acetone−methanol mixtures (in excess to the ideal gas state) as calculated in the entire composition range using the OPLS force field for both components (blue triangles), the TraPPE force field for both components (green squares), and the TraPPE model for methanol and TraPPE-mod for acetone (red circles). The lines connecting the symbols are just guides to the eye. Error bars are shown only when smaller than the symbols.

(14)

In these equations the indices A, B, and AB refer to the two neat components and their mixture, respectively, xA and xB are the mole fractions of the two components in their mixture (being xA + xB = 1), and the last term of eqs 13 and 14 represents the change of the respective quantities upon ideal mixing.

3. RESULTS AND DISCUSSION The Helmholtz free energy, internal energy, and entropy of the simulated acetone−methanol mixtures (in excess to the ideal gas state) are shown in Figure 2 in the entire composition range, as obtained with the three potential model combinations considered. It is seen that the obtained curves deviate noticeably from the linear shape, indicating considerable nonideality of acetone-methanol mixtures. This thermodynamic indication of the nonideal mixing is in clear accordance with the results of previous structural analyses, indicating self-association behavior of the like components.12,13 Density data also show negative deviations from ideal mixing laws (with highest deviations observed at xac = 0.4 or 0.5 depending on the data source).33,34 It is also seen that the steepest change of all of the three thermodynamic functions with the composition is obtained with the original TraPPE potential model, whereas the less steep curve always corresponds to the OPLS force field. The use of the TraPPE-mod acetone model instead of the original TraPPE parameters resulted in less steep curves by somewhat lowering the free energy of systems containing acetone. This slight decrease in the free energy is clearly of energetic origin, as both the internal energy and the entropy of the TraPPE-mod system are lower than that of the TraPPE system in the entire concentration range. (See the two lower panels of Figure 2.) The observed differences in the various thermodynamic functions of the three systems are also reflected in the change of the corresponding quantities upon mixing the two neat components. The Helmholtz free energy, internal energy, and entropy of mixing values are shown, together with experimental data, in Figures 3−5, respectively, as obtained with the three

Figure 3. Free energy of mixing acetone and methanol, calculated in the entire composition range using the OPLS force field for both components (blue open triangles), the TraPPE force field for both components (green open squares), and the TraPPE model for methanol and TraPPE-mod for acetone (red open circles). Experimental data reported by Campbell and Kartzmark27 are shown by black full circles; data corresponding to the three-suffix Margules equation28 based on the 293 K data of Bekarek38 and on the 298 K data of Campbell and Anand39 are shown by solid and dashed lines, respectively. The lines connecting the symbols are just guides to the eye. Error bars are shown only when smaller than the symbols.

model combinations considered in the entire composition range. The simulation results are compared with the experimental data of Campbell and Kartzmark, obtained at slightly higher temperatures ranging from 329 to 338 K for mixtures of different compositions. 27 To describe the 16161



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two curves are also shown in Figure 3. It should be noted that in the original papers the experimental data are always reported without the ideal mixing term (see eqs 13 and 14), which can, however, be calculated analytically and be added to the reported experimental data for the present comparisons. Furthermore, experimental data correspond to the Gibbs rather than Helmholtz free energy; however, these quantities differ only in the pV term, which is negligibly small in condensed systems at atmospheric pressure (being on the order of a few J/mol). As is seen from Figure 3, all model pairs tested are fully miscible with each other, as the free energy of mixing is always resulted in negative values. It is also clear that the thermodynamic driving force of this mixing is very small; the magnitude of Amix (as well as of Umix) is always below the value of the average energy of the thermal motion along one degree of freedom of 0.5RT (its value being ∼1.25 kJ/mol at 298 K). Thus, the mixing of acetone and methanol is the result of an almost as subtle interplay of the energetic and entropic factors, as what was established between acetone and water.26 However, in contrast with the acetone−water system,19,26 the existing potential models are able to reproduce this delicate balance of the energetic and entropic terms and hence also the full miscibility in the case of the acetone−methanol system. The obtained results reveal that the thermodynamic driving force of mixing acetone and methanol is of entropic origin, as the mixing is accompanied by an increase in the internal energy of the system, which is compensated by the increase in the entropy. The fact that the mixing of these components is energetically unfavorable is in a clear accordance with the previously observed microheterogeneous structure, caused by the self-association of the like molecules of these systems.12,13 This self-association, also illustrated in the equilibrium snapshot of the equimolar TraPPE-mod system in Figure 6, can thus be

Figure 4. Energy of mixing acetone and methanol, calculated in the entire composition range using the OPLS force field for both components (blue open triangles), the TraPPE force field for both components (green open squares), and the TraPPE model for methanol and TraPPE-mod for acetone (red open circles). Experimental data reported by Campbell and Kartzmark27 are shown by black full circles. The lines connecting the symbols are just guides to the eye. Error bars are shown only when smaller than the symbols.

Figure 5. Entropy of mixing acetone and methanol, calculated in the entire composition range using the OPLS force field for both components (blue open triangles), the TraPPE force field for both components (green open squares), and the TraPPE model for methanol and TraPPE-mod for acetone (red open circles). Experimental data reported by Campbell and Kartzmark27 are shown by black full circles. The lines connecting the symbols are just guides to the eye. Error bars are shown only when smaller than the symbols.

composition dependence of the free energy of mixing, Wilsak et al. proposed the use of the three suffix Margules equation: Amix = RT[xac(1 − xac)(α(1 − xac) + βxac) + xac ln xac + (1 − xac) ln(1 − xac)]

(15)

where xac is the acetone mole fraction and α and β are the Margules parameters, the values of which were repeatedly determined using several different experimental data sets.28 Here we calculated the Amix(xac) curve according to this equation using the Margules parameters derived from the 293 K experimental data of Bekarek38 and those obtained using the 298 K data of Campbell and Anand.39 The corresponding

Figure 6. Equilibrium snapshot of the simulated equimolar mixture of acetone and methanol at 298 K in the system where methanol molecules are described by the original, whereas acetone molecules are described by the modified version of the TraPPE potential. Acetone and methanol molecules are denoted by green and red colors, respectively. 16162



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explained as having a like near-neighbor is an energetically more favorable situation than having an unlike one for both components. However, because this energy difference is small enough, larger scale self-association is prevented by the entropy gain of mixing the two components. Considering also the fact that this entropy gain is smaller than the entropy change associated with ideal mixing (see the last term of eq 14) up to rather high acetone mole fraction values in all three systems considered, we can conclude that the miscibility of acetone and methanol is governed by the entropy gain of ideal mixing, which is large enough to overcompensate the unfavorable changes occurring due to the interaction of the unlike molecules in their mixtures. It is also seen that the magnitude of the free energy of mixing is somewhat underestimated by all three model combinations, among which clearly the best results are given when the TraPPE methanol and TraPPE-mod acetone model is used. The deviation of the simulated curve corresponding to this model combination from the experimental data never exceeds 0.15 kJ/mol, which is comparable to the difference between the different experimental data sets. Interestingly, this excellent reproduction of the experimental free energy of mixing originates from the accurate reproduction of the delicate interplay of the energetic and entropic terms upon mixing the two components. Thus, the Umix(xac) curves obtained in the TraPPE and TraPPE-mod systems are almost identical to each other, giving an excellent reproduction of the experimental data up to high acetone mole fraction values, whereas the OPLS force field does a somewhat worse job in this respect. The entropy of mixing values obtained in the OPLS and TraPPEmod systems are very close to each other, giving a better reproduction of the experimental data than the original TraPPE force field. Thus, the use of the TraPPE-mod acetone potential instead of the original TraPPE one improved the reproduction of the entropy of mixing without worsening that of the internal energy of mixing when used in combination with the TraPPE model of methanol. Thus, the experimental Umix and Smix values are reproduced within 0.2 kJ/mol and 1 J/(mol K), respectively, by the TraPPE-mod system. It should also be noted that although among the three model combinations tested the use of the original TraPPE force field for both components clearly led to the less accurate reproduction of the experimental data, this reproduction can by no means be regarded as being poor because even this model combination reproduces the experimental free energy, internal energy, and entropy of mixing within 0.7 kJ/mol, 0.25 kJ/mol, and 2 J/(mol K), respectively, in the entire composition range. Finally, it should be emphasized that in this study we focused on one, although very important, property of the acetone− methanol mixtures. Although insufficient reproduction of the change of thermodynamic quantities, such as the free energy, upon mixing can prevent the meaningful use of a given model combination (as was previously seen for models of acetone and water19,26), there are a number of other properties of the mixtures to be modeled that also have to be reproduced by good model combinations. Investigation of such properties as well as that of the effect of various improvements of the force field (i.e., flexibility, polarizability) are clearly beyond the scope of this paper.

Article

AUTHOR INFORMATION

Corresponding Authors

*E-mail: [email protected] (A.I.). *E-mail: [email protected] (P.J.). Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work has been supported by the Hungarian OTKA Foundation under Project No. OTKA 104234, by the Hungarian-French Intergovernmental Science and Technology Program (BALATON) under project no. TéT_12_FR-1-20130013, by the Marie Curie program IRSES (International Research Staff Exchange Scheme, GAN°247500), by the Foundation for Polish Science MPD program cofinanced by the EU Regional Development fund. The Institut du Développement et des Ressources en Informatique Scientifique (IDRIS), the Centre de Ressources Informatiques (CRI) de l’Université de Lille, and the Centre de Ressource Informatique de Haute−Normandie (CRIHAN) are thankfully acknowledged for the CPU time allocation. P. J. is a Szentágothai János fellow of Hungary, supported by the European Union, cofinanced by the European Social Fund in the framework of TÁ MOP 4.2.4.A/2-11/1-2012-0001 “National Excellence Program” under grant number A2-SZJÖ -TOK-13-0030.

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