Thermodynamic properties and diffusion of water + methane binary mixtures.

Thermodynamic properties and diffusion of water + methane binary mixtures I. Shvab and Richard J. Sadus Citation: The Journal of Chemical Physics 140, 104505 (2014); doi: 10.1063/1.4867282 View online: http://dx.doi.org/10.1063/1.4867282 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/140/10?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Intermolecular potentials and the accurate prediction of the thermodynamic properties of water J. Chem. Phys. 139, 194505 (2013); 10.1063/1.4832381 Thermodynamic properties of supercritical n-m Lennard-Jones fluids and isochoric and isobaric heat capacity maxima and minima J. Chem. Phys. 139, 154503 (2013); 10.1063/1.4824626 Thermodynamic properties of liquid water from a polarizable intermolecular potential J. Chem. Phys. 138, 044503 (2013); 10.1063/1.4779295 On the mutual diffusion properties of ethanol-water mixtures J. Chem. Phys. 125, 104502 (2006); 10.1063/1.2244547 The effect of force-field parameters on properties of liquids: Parametrization of a simple three-site model for methanol J. Chem. Phys. 112, 10450 (2000); 10.1063/1.481680

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THE JOURNAL OF CHEMICAL PHYSICS 140, 104505 (2014)

Thermodynamic properties and diffusion of water + methane binary mixtures I. Shvab and Richard J. Sadusa) Centre for Molecular Simulation, Swinburne University of Technology, PO Box 218 Hawthorn, Victoria 3122, Australia

(Received 29 November 2013; accepted 17 February 2014; published online 11 March 2014) Thermodynamic and diffusion properties of water + methane mixtures in a single liquid phase are studied using NVT molecular dynamics. An extensive comparison is reported for the thermal pressure coefficient, compressibilities, expansion coefficients, heat capacities, Joule-Thomson coefficient, zero frequency speed of sound, and diffusion coefficient at methane concentrations up to 15% in the temperature range of 298–650 K. The simulations reveal a complex concentration dependence of the thermodynamic properties of water + methane mixtures. The compressibilities, heat capacities, and diffusion coefficients decrease with increasing methane concentration, whereas values of the thermal expansion coefficients and speed of sound increase. Increasing methane concentration considerably retards the self-diffusion of both water and methane in the mixture. These effects are caused by changes in hydrogen bond network, solvation shell structure, and dynamics of water molecules induced by the solvation of methane at constant volume conditions. © 2014 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4867282] I. INTRODUCTION

The solvation of nonpolar solutes in water makes an important contribution to the thermodynamics of gas clathrate formation, membrane formation, and protein folding.1, 2 The low solubility of nonpolar solutes like methane or noble gases in water is known as a hydrophobic effect.3 However, despite more than a half-century of experimentation, theory, and simulation since the introduction of the “iceberg” concept, our understanding of hydrophobic solvation remains incomplete.3 Solvation is generally defined as the process of transferring a single molecule from a specific position in an ideal gas phase to a specific position in an aqueous phase.3–5 A number of simulation and theoretical investigations have been devoted to solvation changes in chemical potential, Gibbs energy, entropy, and structural changes around an inserted test particle.4, 6, 7 In contrast, thermodynamic response functions of aqueous nonpolar solute mixtures have been studied much less widely with some properties receiving more attention than others. For example, data for the Joule-Thomson coefficient, speed of sound, and adiabatic compressibility are almost completely absent in the literature.8, 9 In contrast to extensive experimental investigations10 of pure water, aqueous solutions of hydrocarbons and salts at critical conditions, studies of nonpolar solutes like methane and noble gases are very scarce and usually limited to either ambient conditions or very dilute mixtures.5, 11–13 The absence of data means the peculiar thermodynamic behavior accompanying the dissolution of both polar and nonpolar solutes is not well understood.3 For example, the

a) Author to whom correspondence should be addressed. Electronic mail:

[email protected]

0021-9606/2014/140(10)/104505/10/$30.00

hydration of a single nonpolar particle is accompanied by an unusually large decrease in entropy (S < 0), a decrease in enthalpy (H < 0), and an increase in constant pressure heat capacity (Cp > 0). In contrast, hydration of most polar and ionic species results in a negative change in constant pressure heat capacity (Cp < 0) and a decrease of entropy (S < 0).3, 5 Most studies of hydrophobic solvation CP changes have been limited to systems containing only one nonpolar particle at ambient conditions.5 The behavior of Cp upon the solvation of a high concentration (> 1%) of nonpolar particles from ambient to supercritical temperatures has not been investigated extensively. A thorough knowledge of the transport properties of water and its mixtures is particularly significant for geochemistry and engineering applications.11 The transfer of methane or noble gases into water is strongly dependent on the diffusion coefficient of these gases dissolved in water. Experimental and simulation data are available only for dilute mixtures at ambient pressures and temperatures lower than the normal boiling point.14 Expanding our knowledge of aqueous mixtures involving methane is of increasing importance to the energy industry.15 The main aim of this work is to comprehensively investigate the thermodynamic and diffusion properties of water + methane mixtures in the temperature range of 298–650 K, at a constant density of 0.998 g/cm3 , and methane concentrations up to 15%. To the best of our knowledge, there are no previous experimental or simulation data for water + methane binary mixtures for such a range of states. For the first time, this work provides Joule-Thomson coefficient and speed of sound data for aqueous solutions of methane. We also expect the results obtained for the water + methane binary mixture to be useful in elucidating the solvation of nonpolar solutes in general.

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II. MOLECULAR SIMULATION

B. Intermolecular potentials

A. Brief overview of the method

The study of methane + water mixtures requires suitable intermolecular potentials19 to describe methane-methane, water-water, and water-methane interactions.

In this work, we make use of the method of statistical averages developed for the microcanonical and canonical ensembles by Lustig.8, 9 The method and its application to water has been discussed in considerable detail elsewhere16, 17 and only a brief outline of the salient features is given here. The logarithm of the canonical (NVT) partition function (β, V , N ) is directly proportional to the Helmholtz function (A) − βA (β, V , N ) = ln (β, V , N ) ,

(1)

which depends on the number of particles (N), temperature (T), and volume (V). In Eq. (1), β = 1/kT and k is Boltzmann’s constant. In the canonical NVT ensemble, any thermodynamic property can be obtained from some combination of partial derivatives of the function βA (β, V , N ), or equivalently of the right-hand side of Eq. (1). The derivation of this procedure was described in detail by Lustig8 and is therefore not described here. The partial derivative of the partition function (β, V , N ) with respect to the independent state variables β and V has the following form: mn =

(2)

N−1 N n k ∂ nU 1 k ∂ uij = a r , nk ij ∂V n 3n V n i−1 j =i+1 k=1 ∂rijk

(3)

where coefficients ank are constructed using recursive relations.18 The explicit expressions for several mn functions and the resulting thermodynamic state variables are summarized in the Appendix and Table I, respectively. TABLE I. Thermodynamic properties expressed state in terms of the derivatives of the partition function.8

Isothermal compressibility Adiabatic compressibility Thermal expansion coefficient Isochoric heat capacity Isobaric heat capacity

Joule-Thomson a

γV = k01 − (11 − 01 10 ) /T 1 2 βT = −V kT 02 − 01 1 βS

=

1 βT

coefficienta

+

2 T V γV N CV

αp = γV βT 20 −210 NkT 2 2 k01 −( 11 −01 10 ) /T

CV = CP = CV −

N 02 −201

μJ T =

=

V NβS M γV −1/(T βT ) N·CV /(V T βT )+γV2

ω02

Speed of sound

Despite its importance, no truly reliable ab initio potential is currently available for the methane-methane system.20 Numerous first principle calculations have been concerned mainly with mapping out the potential energy surface (PES) and finding the most stable configurations of methane dimers and trimers.21 Møller–Plesset (MP2) perturbation theory calculations often indicate a dimer configuration is the most stable with a C–C distance 3.8 Å and ground state energy of approximately −2.426 kJ/mol. However, Pratt and Chandler22 showed that in water, the solvent separated pair of non-polar particles is more energetically favourable than other configurations. Recently Li et al.,23 using fully quantum-mechanical molecular dynamics (MD) simulations, estimated the value of the optimal methane pair separation in water to be 4.8 Å. 2. Water-methane interactions

1 ∂ m+n . ∂β m ∂V n

The function mn has a complex dependence from ensemble averages of the volume derivatives (∂ i βU / ∂V i )β,N of the intermolecular potential energy function U(q) and is described in detail in Ref. 9(e). All thermodynamic properties are then expressed in terms of functions mn . In this work, we restrict ourselves pair interaction of atomic sysN to molecular tems U (q) = N−1 i=1 j =i+1 uij (rij ), where rij is the distance between atoms j and i. The volume derivatives of nth order are given by

Thermal pressure coefficient

1. Methane-methane interactions

In Ref. 8, the signs in the nominator of Eq. (20c) are reversed.

Due to the weak solubility of nonpolar solutes like methane in water, the exact form of the water-methane interaction potential is still not known.24 First principle investigations of water-methane have not yielded a commonly accepted result.25 Similar to methane-methane interaction, water-methane interaction also depends on the many possible mutual orientations of both molecules. A recent ab initio investigation by Mateus et al.25 discusses two general orientations schemes: methane as a proton acceptor (PA); and methane as a proton donor (PD). In the PA orientation, one of water’s hydrogens points towards the carbon atom, while in the PD orientation the methane hydrogen points towards the oxygen atom. The liquid state significantly distorts the alignment of the pair of atoms in both PA and PD cases compared to the gas phase. In the PA case, the optimized lowest energy is −3.809 kJ/mol and the C–O distance is 3.48 Å. In the PD case, the optimized lowest energy is −2.428 kJ/mol and the C–O distance is 3.65 Å. Mateus et al.25 showed that by adopting the above two general orientations, methane plays the role of PD in 56.4% cases of all selected configurations. On the other hand, for only 13.8% of the selected configurations methane plays the role of PA. In 7.8% of all configurations methane is both PA and PD. The ab initio potentials obtained for methane-methane and water-methane interactions are very computationally expensive and, thus, applicable to very small systems. In addition to their theoretical complexity, the common disadvantage of water-methane ab initio potentials is their inability to reproduce London dispersion interactions. In this work, for the sake of simplicity and computational economy, we have used the Lennard-Jones (LJ) potential to handle the methane-methane and water-methane interactions. It is well established that in the united-atom approximation adopted here, the LJ potential is a reasonable approximation for the


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intermolecular interactions within small methane clusters.28 Suitable values for the LJ size (σ = 3.73 Å) and well-depth (ε = 1.2263 kJ/mol) parameters are available in the literature for methane.6

In contrast, the investigation of water-water interaction has been arguably more successful.27 Existing water models reproduce liquid, vapor, and ice properties at ambient conditions with reasonable accuracy.28 However, it has been shown16, 28 that different models have different areas of applicability and specific limitations. Generally, the most popular rigid three-site simple point charge (SPC/E) and foursite transferable intermolecular potential (TIP4P/2005) models provide a good description for many properties. Flexible water models such as SPC/Fw, which account for the internal degrees of freedom, have improved the prediction of transport and polarization properties.29, 30 Polarizable models such as the Matsuoka-Clementi-Yoshimine non-additive31 (MCYna) or Baranyai-Kiss32 (BKd3) are superior for reproducing dielectric constants and dipole moments, and as was shown recently, thermodynamic properties.33, 34 In this work, we have used the SPC/E potetntial35 to calculate water-water interactions. The SPC/E potential is well known for computational efficiency and a very good representation of most liquid water properties at ambient conditions.28 Therefore, it is a good starting point for extending the investigation to a wider range of temperature and pressure. The SPC/E potential is a rigid three-site water model with an oxygen–hydrogen (O–H) distance of 1 Å and a H–O–H angle of 109.47◦ . Values of the parameters are σ = 3.166 Å and ε = 0.65 kJ/mol with a charge +0.4238 on both H atoms and a charge of −0.8427 on the O atom. The potential function combines contributions from both LJ interactions and an electrostatic term ⎧ 6 ⎫ 12 N N ⎨ ⎬ σ σ − u(r ) = 4ε ⎩ rijoo rijoo ⎭ +

σ1 + σ2 , √ 2 = ε1 · ε2 .

σ12 =

3. Water-water interactions

i

erties from solute particle size, potential energy well depth, and solute concentration (xS ). LJ parameters for water-solute interaction are defined by the usual Lorentz-Berthelot combining rules

ε12

(5)

Assigning zero values to ε and σ for hydrogen atoms means that only oxygen atoms participate in direct LJ interaction with solute particles.

C. Simulation details

Canonical NVT MD simulations using the Shake algorithm37 were performed for systems of N = 500 molecules with a density 0.998 g/cm3 and a temperature range of 278–650 K. The Ewald summation method was used to evaluate the long-range part of the Coulomb potential. The convergence parameter for the Ewald sum was α = 5.0/L, with summation over 5 × 5 × 5 reciprocal lattice vectors, where L is the box length. A cut-off of L/2 was applied. During the pre-equilibration stage, the temperature was held constant by rescaling the velocities every ten steps. The simulations were commenced from an initial face centered cubic lattice with a time step of 2 fs for thermodynamic properties and 1 fs for diffusion coefficients and velocity autocorrelation functions. The systems were equilibrated for 500 ps before any ensemble averages were determined. At each temperature, the total simulation time was at least 2 ns, which corresponds to 1.0 × 106 time steps. The equations of motion were integrated using a leap-frog algorithm.37 Ensemble averages were obtained by analysing post-equilibrium configurations at intervals of 100 time steps and standard deviations were determined. Error bars are not illustrated in Figs. 1–10 because, in most cases, the calculated statistical

j =i

N N 1 qi qj , 4π ε0 i j =i rij

(4)

where rij and rijoo are the distances between charged sites and oxygen atoms, respectively, and ε0 is the permittivity of the vacuum. Therefore, the combined SPC/E + LJ potential is used to describe all intermolecular interactions in water + methane mixtures. This potential model is probably the most widely used potential to describe hydrophobic interactions in dilute aqueous nonpolar solutions.4, 6, 7, 12, 13 Currently, most research efforts are still directed into investigating physical properties of very dilute water-methane mixtures, like solvation changes in thermodynamic potentials,12, 13 solvation changes of thermodynamic response functions,5, 12, 13 mass and size dependence of the hydrophobic interaction,36 diffusion coefficients,11 etc. The simplicity of this potential model allows us to quickly establish dependence of solvation prop-

FIG. 1. Pressure-temperature behavior of water + methane mixtures at 0.998 g/cm3 with methane concentrations of 0% (black ), 6% (blue ), 10% (red ●), and 15% (green ). IAPWS-95 reference data39 for water (—) are also illustrated. The lines through the data points are given only for guidance.


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FIG. 2. Isothermal compressibility of water + methane mixtures as a function of temperature at 0.998 g/cm3 with methane concentrations of 0% (black ), 6% (blue ), 10% (red ●), and 15% (green ). IAPWS-95 reference data39 for water (—) are also illustrated. The lines through the data points are given only for guidance.

uncertainty is similar to the size of the symbols used for the data points. III. RESULTS AND DISCUSSION A. Reference data for pure water

Water is likely to be the dominant contributor to interactions in most aqueous solutions, therefore, a thorough knowledge of its properties is a necessary precondition for understanding solution behavior. Most of the experimental data for water reported in the literature38 are at isobaric conditions whereas MD simulations in the NVT ensemble yield isochoric values. Therefore, we must either convert the data or find

FIG. 3. Adiabatic compressibility of water + methane mixtures as a function of temperature at 0.998 g/cm3 with methane concentrations of 0% (black ), 6% (blue ), 10% (red ●), and 15% (green ). IAPWS-95 reference data39 for water (—) are also illustrated. The lines through the data points are given only for guidance.

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FIG. 4. Thermal pressure coefficient of water + methane mixtures as a function of temperature at 0.998 g/cm3 with methane concentrations of 0% (black ), 6% (blue ), 10% (red ●), and 15% (green ). IAPWS-95 reference data39 for water (—) are also illustrated. The lines through the data points are given only for guidance.

an accurate alternative to the experimental values. For this purpose, we have used the International Association for the Properties of Water and Steam (IAPWS-95)39 software developed by Wagner to calculate pressure (p), isobaric (Cp ), and isochoric (CV ) heat capacities, Joule-Thomson coefficient (μJT ), and the speed of sound (w0 ) at isochoric conditions.26 IAPWS-95 is based on a highly accurate empirical equation of state.38 The remaining thermodynamic quantities, namely, the isothermal (β T ), and adiabatic compressibilities (β S ), pressure coefficient (γV ), and thermal expansion coefficient (α P ) are then calculated using the following well-known

FIG. 5. Thermal expansion coefficient of water + methane mixtures as a function of temperature at 0.998 g/cm3 with methane concentrations of 0% (black ), 6% (blue ), 10% (red ●), and 15% (green ). IAPWS-95 reference data39 for water (—) are also illustrated. The lines through the data points are given only for guidance.


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FIG. 6. Isochoric heat capacity of water + methane mixtures as a function of temperature at 0.998 g/cm3 with a methane concentration of 0% (black ), 6% (blue ), 10% (red ●), and 15% (green ). IAPWS-95 reference data39 for water (—) are also illustrated. The lines through the data points are given only for guidance.

relationships:40

⎫ V ⎪ ⎪ ⎪ ⎪ w02 M ⎪ ⎪ ⎪ ⎪ ⎪ βS CP ⎪ ⎪ βT = ⎬ CV , (6) −1 −1 ⎪ β − β C ⎪ V S T ⎪ ⎪ γV2 = ⎪ ⎪ TV ⎪ ⎪ ⎪ 1 ⎪ μJ T CP ⎪ ⎭ + αP = TV T where M is the total mass of the system. Experimental data for the thermodynamic properties of aqueous solution of nonpolar solutes are largely limited to very dilute solutions at only a few state points.5, 11–13 Experimental studies are mostly focused on the investiga-

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FIG. 8. Joule-Thomson of water + methane mixtures as a function of temperature at 0.998 g/cm3 with a methane concentration of 0% (black ), 6% (blue ), 10% (red ●), and 15% (green ). IAPWS-95 reference data39 for water (—) are also illustrated. The lines through the data points are given only for guidance.

tion of aqueous solutions of more soluble substances like alcohols.10, 41, 42

βS =

FIG. 7. Isobaric heat capacity of water + methane mixtures as a function of temperature at 0.998 g/cm3 with methane concentrations of 0% (black ), 6% (blue ), 10% (red ●), and 15% (green ). IAPWS-95 reference data39 for water (—) are also illustrated. The lines through the data points are given only for guidance.

B. One-phase region

Fig. 1 shows the pressure-temperature behavior of water + methane mixtures in the single liquid phase at temperatures between 298 K and 650 K. Results are given for mixtures with methane concentrations of xs = 0%, 6%, 10%, and 15% and IAPWS-95 reference data for pure water. It is apparent that pressure increases significantly with both temperature and solute concentration. The general trend is predominantly caused by the constraint imposed on the system volume (NVT ensemble). At constant volume and increasing temperature, water molecules experience significant LJ repulsion from the methane molecules, which increases with increasing methane concentration. The uniform dissolution of methane molecules

FIG. 9. Speed of sound coefficient of water + methane mixtures as a function of temperature at 0.998 g/cm3 with a methane concentration of 0% (black ), 6% (blue ), 10% (red ●), and 15% (green ). IAPWS-95 reference data39 for water (—) are also illustrated. The lines through the data points are given only for guidance.


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is more than 1000 times larger than reported by Zheng et al.46 and Errington et al.47 As a result of this confinement, the pressures observed in our simulations are much higher (up to 840 MPa), resulting in increased solubility.

C. Isothermal and adiabatic compressibilities

FIG. 10. Self-diffusion coefficients of (a) water and (b) methane in water + methane mixtures at methane concentrations of 0% (black ), 6% (blue ), 10% (red ●), and 15% (green ). Experimental self-diffusion coefficients for pure water (—) are given for comparison.65 The lines through the data points are given only for guidance. The density is 0.998 g/cm3 for all cases.

in water and absence of any stable methane clusters or assemblies of more than 1 nm in size at all solute concentrations was confirmed by both a careful analysis of radial distribution functions and visualization of the simulation box during the equilibration stage. These phase conditions obtained at constant volume in the NVT ensemble should not be confused with the very small solubility of methane in water at ambient conditions.6 Extensive experimental studies of various aqueous solutions of nonpolar solutes reported by Franck et al.43 confirm the existence of a single homogeneous phase at solute concentrations up to 30% and even higher concentrations at near critical and supercritical temperatures. For example, Fig. 10 from Franck44 and Fig. 2 from Wu et al.45 show phase equilibrium isopleths and isotherms at solute concentrations up to 30% at near critical temperatures for water + methane and water + argon systems, respectively. Although the existence of a single phase for water mixtures of relatively high non-polar solute concentrations (xs < 30%) has been observed experimentally at high temperatures and high pressures, the extent of miscibility at low temperatures (T < 500 K) and high pressures remains unclear. The very low methane solubility in water at T < 500 K and p < 100 MPa has been amply demonstrated by both experimental44–46 and simulation data.47 However, most measurements have been reported for methane in the gas state, e.g., a density of 0.00068 g/cm3 at T = 288 K, p = 0.1 MPa. In contrast, for the NVT simulations reported here, the density of the whole mixture is held constant at 0.998 g/cm3 , which

The values of β T and β S reflect the relative volume change of the system as a response to pressure at constant temperature and entropy, respectively. Simulation results for β T and β S as functions of temperature are presented in Figs. 2 and 3. Unlike the constant pressure data,48 β T and β S at a constant density of 0.988 g/cm3 keep gradually decreasing without any minima. The rate of decrease gradually slows down at elevated temperatures at all methane concentrations. This indicates smaller local density fluctuations of water-methane mixtures at elevated temperatures, where the pressure is higher. It is apparent from Figs. 2 and 3 that mixtures with higher concentrations of methane have smaller values of both β T and β S . The low values of β T and β S of the combined SPC/E + LJ potential at high temperatures can be largely attributed to the SPC/E potential. Almost up to the normal boiling temperature, compressibility values of pure SPC/E water (xs = 0%) are higher than the reference data, after which both β T and β S descend below the reference data for pure water. At 650 K, values of β T and β S are smaller than values from the IAPWS-95 data39 by 27% and 21%, respectively. Compressibility has a complex dependence on both local density and the H-bond network. In the case of pure water, both β T and β S start from experimental value 0.45 GPa−1 at ambient conditions, which is defined by the cohesive nature of the extensive H-bonding. Most theoretical and simulation studies2, 5, 49–51 have concluded that the reorganization of the H-bond network around hydrophobic solutes is responsible for many specific changes in solvation quantities. For example, MD simulations33, 50, 51 of dilute aqueous solutions of nonpolar solutes (xs < 1%) indicate a slight increase in water tetrahedrality and local density in the 1st hydration shell and a corresponding decrease in the 2nd shell. The rate of change of the structure is determined by the interplay between two components of the Gibbs free energy, namely, H and TS, as well as amount of solute particles and their size.2 Analysis of radial distribution functions of mixtures with higher solute concentrations (xs > 1%) indicates a negative effect of hydrophobic solutes on H-bonding.33 However, it is important to note that experimental studies have currently not detected any noticeable enhancement of water’s structure around nonpolar particles.52 In our case, due to strong thermal fluctuations at high temperature, water’s structure starts to collapse, opening up large cavities inside the H-bond network and shifting water’s structure closer to that of simple liquids. As a consequence, compressibility decreases with increasing temperature and pressure. In the absence of experimental data or similar MD results, we can only compare our predictions with theoretical calculations and other aqueous solutions. Analytical calculations by Zaitsev et al.53 for aqueous solutions of potassium chloride confirm the temperature and solute concentration


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dependence of both β T and β S . Isothermal compressibility of standard seawater with a salinity of 35.16504 g/kg is also smaller than that of pure water54 in the temperature range of 273.17–468.03 K.

D. Thermal pressure coefficient

The temperature dependence of γV for water + methane mixtures is shown in Fig. 4. In contrast to β T and β S from Figs. 2 and 3, γV for water + methane mixtures at solute concentrations xs = 6%, 10%, and 15% are higher than that of pure water. Such behavior is closely related to the higher internal pressures of the water + methane mixtures compared to pure water (see Fig. 1). The γV value of pure water and the water + methane mixtures start (xs = 6%) from values of 0.731 and 0.88 MPa/K, respectively, and increase almost linearly until the normal boiling temperature. Thereafter, the rate of change of γV at all methane concentrations starts slowing down peaking at around 600–630 K with a further tendency to decrease. This temperature dependence of γV at T > 500 K is in accord with the behavior of compressibilities β T and β S in the same temperature region. The almost constant values of γV indicate a linear increase in pressure, which is characteristic of an ideal gas or simple liquid. Our calculations of γV for SPC/E water are consistent with results reported in the reviews of Vega and Abascal28 and Wu et al.29 According to these reviews, all nonpolarizable water models overestimate pressure coefficient at 298 K. To the best of our knowledge, there are no previous MD simulations or analytical calculations of γV for aqueous nonpolar solutions. Higher γV values54 of standard seawater (0.7 MPa/K at ambient conditions) compared to pure water (0.436 MPa/K) can serve as an indirect confirmation of the general trend obtained here.

E. Thermal expansion coefficient

While β T usually measures volume contraction due to applied pressure, α p is the measure of the tendency of matter to expand in volume in response to a change in temperature. Unlike β T , α P is a non-monotonic function of temperature. It is well known that liquid water has a maximum density of 1 g/cm3 at 4 ◦ C, while ice Ih at approximately −4 ◦ C has the smallest density of 0.917 g/cm3 . Our data yield information for the temperature dependence of α p over a much wider temperature range. Thermal expansion coefficients for water + methane mixtures are presented in Fig. 5. The values of α p overestimate reference data for pure water in the region 298–400 K. Nevertheless, our calculations are superior to the results obtained from the other nonpolarizable water models using classical volume fluctuation formulas.28, 29 Peaking at around 425 K, the pure water curve starts to decrease, significantly deviating from the reference data. The values of α p and β T strongly depend on local density fluctuation. The initial increase in α p can be attributed to the temperature driven collapse of water’s structure resulting in large density fluctu-

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ations. The subsequent decrease of α p is caused by the constraint imposed on the system’s volume. The results in Fig. 5 show that the mixture curves repeat the general trend for pure water. However, they also exhibit significantly different temperature dependence of the α p peaks, which progressively shift to the left with increasing methane concentration. That is, α p increases with the addition of nonpolar solutes at temperatures up to 400 K. Analytical calculations of α p for urea55 (xs = 2.5%) and aqueous solutions of propylene + glycol56 (xs = 0%–60%) also yield higher values of α p than in pure water. Experimental curves for water + propylene + glycol mixtures reported for a smaller temperature interval also indicate a gradual shift to the left of the α p temperature maximum with increasing solute concentration.55

F. Isochoric and isobaric heat capacities

Values of CV and CP for water + methane mixtures are presented in Figs. 6 and 7, respectively. The behavior of the CP curve is especially interesting because it does not show any minimum. At constant pressure conditions, experimental CP behavior exhibit a minimum at ∼309 K, after which it keeps gradually increasing.38 However, at constant volume conditions, the reference values39 for CP monotonically decrease for the whole temperature range. While agreement with IAPWS-95 data is quite good for pure CV , with the exception of too high values at 298 K, the pure CP curve deviates from the reference data to a greater extent. Properties such as β T , β S , γV , and α P are largely defined by changes in spatial packing of molecules at different temperature and pressure whereas heat capacity depends on other factors. The introduction of nonpolar solutes not only disturbs the H-bond structure around the solute particles, but it also changes the energy levels of solvation shells. According to classical fluctuation formulas,29 CV and CP are defined by the fluctuations of internal energy (U) and enthalpy (H = U + pV ) with temperature. Comparison of CV and CP from Figs. 6 and 7 clearly show that the deviations of CP from the IAPWS-95 reference data is caused by the volume component (volume/density fluctuations). Possible reasons for this disagreement, such as internal degrees of freedom, quantum corrections, and polarization interactions are discussed in detail elsewhere.34 The values of CV (Fig. 6) and CP (Fig. 7) for water + methane mixtures are smaller than those of pure water (resulting in negative values of CP and CV ). There is no experimental data for the corresponding temperature range and methane concentrations. However, experimental data for water + alcohol mixtures41, 42 and seawater54 display negative CP values. The CP of the water + methanol system shows an initial increase, after which it starts decreasing and eventually attains values less than those of pure water.57 The solvation processes of methane and homologous alcohols are completely different and only qualitative conclusions can be made by comparing these cases. One possible unifying factor is the CP of the pure solute component. Methane and methanol have smaller CP


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values than pure water, thus CP of aqueous solutions of these substances tend to be smaller than in pure water. Currently, there is no clear understanding of solvation of either nonpolar solutes, polar or ionic species. Most simulation studies2, 5, 33, 50, 51 report that the hydration of a single nonpolar particle like methane or argon is accompanied by the short-range reorganization of water’s structure in the 1st and the 2nd hydration shells. As reported by Sharp and Madan,5 the 1st hydration shell contains smaller populations of H-bonded water molecule pairs with large H-bond angles and lengths, which indicates greater structural order. The hydration of a single polar or small ionic solute5 shows an increase in both the mean H-bond length and angle. The constant pressure solvation heat capacity of hydration of polar (CPS < 0) and nonpolar (CPS > 0) species, also have opposite signs.5 The signs of CP discussed in this work and in the works of Chandler,2 and Sharp and Madan5 (CPS ) should not be confused. The large positive values of CPS (∼209 J/mol K) for methane reported in the literature2, 3, 5 were calculated as the difference between CP of the 1st hydration shell around methane and CP of bulk water. In a sense, this is the excess heat capacity of the 1st hydration shell. This result may be interpreted in terms of a simplified twostate model, where the system has only two energy levels, activated (molecules in the hydration shell) and ground level (bulk molecules) separated by an energy gap (E).5 In this work, we simply compare the thermodynamic properties of pure water and water-methane mixtures at different concentrations (e.g., Figs. 6 and 7). In our simulations, CP is averaged over the whole ensemble because methane molecules at constant volume and high pressures are uniformly dissolved.

G. Joule-Thomson coefficient

The simulation results for the Joule-Thomson coefficient (μJT ) of water + methane mixtures are presented in Fig. 8. The simulation data fail to correctly reproduce the temperature dependence of μJT . Results for pure water overestimate reference data at temperatures up to 420–500 K, after which μJT start to decrease almost linearly. Using the following thermodynamic relationships,40 we can rewrite the formula for μJT from Table I in the following form: μJ T =

∂T ∂p

= H

V (αP · T − 1) . Cp

(7)

It becomes immediately apparent that the observed temperature dependence of μJT is consistent with the characteristic trend for the thermal expansion coefficients α P (see Fig. 5). The μJT peaks on Fig. 8 simultaneously decrease and shift towards smaller temperatures with increasing methane concentrations. In the temperature region up to the normal boiling temperature, the curves for all methane concentrations appear to merge into one line. The disparity increases with increasing temperature. Being proportional to α P and inversely proportional to CP , μJT incorporates the uncertainties from all of these quantities.

H. Speed of sound

Values of w0 in water58, 59 as a function of temperature are illustrated in Fig. 9. Similar to many other thermodynamic properties, w0 at isobaric conditions goes through a peak at around 348 K and then decreases with temperature.38 However, at isochoric conditions experimental sonic speed does not have a minimum and simply decreases with temperature. It is commonly assumed that w0 is related to the propagation of an adiabatic pressure wave because the density fluctuations are too high for any heat flow to take place.60 According to the formula from Table I, w0 is inversely proportional to the square root of β S , and, therefore, keeps freely increasing with temperature and pressure. In practice, the adiabatic approximation can be used even for ultrasound (2 MHz). Simulation results for water + methane mixtures are shown in Fig. 9. Starting from a value of 1460 m/s at 298 K, simulation results cross the reference data at approximately 373 K. The curves for all methane concentrations keep increasing almost linearly with temperature and occur considerably above the reference curve. The presence of methane molecules significantly increases w0 in the mixture. Numerous experimental measurements performed for aqueous solutions of potassium chloride, sodium citrates, metal halides, etc., confirm the general trend of increased sonic speed.61, 62 High simulation values of w0 for all mixtures could be partly attributed to specific structure and local density behavior of the SPC/E water model. The SPC/E model is known for strong oxygen-oxygen attraction that is responsible for excessively stable oxygen-oxygen solvation shells.33(b) I. Diffusion coefficient

The most common quantity to describe the dynamical behavior of a system is its diffusion coefficient (D). The diffusion coefficient is calculated from the mean-square displacement of the center of mass of the molecule37 |ri (t) − ri (0)|2 , (8) D = lim t→∞ 6t where ri (t) is the position of the ith molecule at time t and symbol ··· means ensemble average. Previous simulations of the diffusion coefficients for nonpolar molecules in water have been limited to very dilute mixtures at ambient conditions.11, 63, 64 In the present work, we calculate self-diffusion coefficients of water + methane mixtures at methane concentrations up to 15% and temperatures up to 650 K. Fig. 10 illustrates the temperature dependence of D for water + methane mixtures at solute concentrations of xs = 0%, 6%, 10%, and 15%. Fig. 10(a) compares the selfdiffusion coefficients of pure SPC/E water with experimental data65 over the temperature range of 278–650 K. Values of D for pure SPC/E water are in fairly good agreement with experimental values almost up to the normal boiling temperature. At higher temperatures, deviations start to increase yielding values of D about 35% smaller than the experimental ones. It is important to stress that the experimental data of Krynicki et al.65 shown on Fig. 10 can be used only for


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qualitative comparison with simulation results. Direct comparison is impossible because of nonmatching pressure values. The reason why simulation D values are lower than experimental ones is partly due to much higher pressures of the constant volume NVT simulations compared to constant pressure experimental values of diffusion coefficients. It is well known30, 65 that the diffusion coefficients decrease with increasing pressure, especially at near critical temperatures. Taking this into consideration, in the appropriate pressure range we can expect better agreement between simulations and experiment. For instance, recent experimental data of normal and heavy water66 give values of D up to 8% smaller than the data of Krynicki et al.65 Nevertheless, the data of Krynicki et al.65 cover a wide enough pressure-temperature region to estimate the general trend between experiment and simulations. The comparison given in Fig. 10(a), indicates that the presence of nonpolar particles slows down the dynamics of water molecules in the aqueous solutions (see curves for xs = 6%, 10%, 15%) compared to pure water (xs = 0%). This deceleration is proportional to the solute concentration. Methane in water acquires a large first hydration shell which contains approximately 18–20 water molecules.33(a) At ambient conditions, water molecules spend in average 23 ps inside such a solvation shell.67 As was shown by Koneshan et al.,67 the residence time of water molecules in the nonpolar solute solvation shell increases linearly as a function of nonpolar solute size. Data from Fig. 10, supported by other similar studies67, 68 indicate that the diffusion of the whole system is defined by the slowest solute component. Solute particles heavier than water not only diffuse much more slowly but also, in sufficiently large quantities (>2%) decrease the average diffusion coefficient of surrounding water molecules.

J. Chem. Phys. 140, 104505 (2014)

respectively, are significantly affected by the changes in the H-bond network and spatial packing of molecules at high T and p. By contrast, CV and CP strongly depend on temperature driven fluctuations of U and H. Accurate prediction of CP and μJT requires use of intermolecular potential models that fully account for configuration effects (solvation shell structure) and different energy contributions (polarization, many-body interactions, and quantum corrections) over a wide range of state points. D is determined by the dynamics of water and solute molecules and the stability of solvation shells.

ACKNOWLEDGMENTS

I.S. thanks Swinburne University of Technology for support through a postgraduate scholarship. We thank the National Computational Infrastructure (NCI) for an allocation of computing time.

APPENDIX: EVALUATION OF PARTITION FUNCTION DERIVATIVES mn

Explicit calculation of the derivatives8 on the right-hand side of Eq. (2). They are used to evaluate the thermodynamic quantities in Table I, βU β −1 1+ , 10 = − F /2 F /2

βU 2 1 βU β −2 1+ +2 + 20 = − , F /2 F /2 F /2 F /2 (A2)

IV. CONCLUSIONS

Our investigation of water + methane mixtures in a single liquid phase reveals a complex dependence of the thermodynamic properties on methane concentration. For example, properties such as β T , β S , CV , CP , and D clearly decrease throughout the whole 298–650 K temperature range with addition of methane. At the same time, α P and w0 of the water + methane mixtures show higher values compared to pure water. Furthermore, γV and μJT exhibit distinct maxima points, which progressively shift towards smaller temperatures and simultaneously decrease with increasing methane concentration. The α P , μJT , and w0 deviate significantly from the IAPWS-95 reference data for pure water starting from the normal boiling temperature. Comparison with the few available experimental measurements of other aqueous binary mixtures supports the general trends observed in our simulations. It is difficult to give simple explanations for such a wide range of phenomena. Aqueous solutions of hydrophobic solutes like methane exhibit very complex dependence from thermodynamic potentials and internal solvation structure. Properties such as β T, S , α P , γV , and w0 , which are largely determined by ∂V /∂p, ∂V /∂T , ∂p/∂T , and ∂p/∂V ,

(A1)

01

∂βU N + − , = V ∂V

N ∂βU 1 N 2 +2 − 1− = V N V ∂V 2 ∂βU 2 ∂ βU − + − + , ∂V 2 ∂V

(A3)

02

(A4)

βU β −1 N 1+ − F /2 V F /2 ∂βU βU ∂βU 1 − + − , + 1− F /2 ∂V F /2 ∂V

11 =

(A5) where U is intermolecular energy and F is the total number of degrees of freedom of the system of molecules. 1 L. 2 D.

R. Pratt and A. Pohorille, Chem. Rev. 102, 2671 (2002). Chandler, Nature (London) 437, 640 (2005).


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