Effects of an Applied Electric Field on the Vapor−Liquid Equilibria of

Mar 4, 2010 - Gibbs ensemble Monte Carlo simulations are employed to examine the influence of moderately strong electric fields on the vapor−liquid ...
89 downloads 9 Views 2MB Size
Download PDF
J. Phys. Chem. B 2010, 114, 4261–4270

4261

Effects of an Applied Electric Field on the Vapor-Liquid Equilibria of Water, Methanol, and Dimethyl Ether Katie A. Maerzke and J. Ilja Siepmann* Departments of Chemistry and of Chemical Engineering and Materials Science, UniVersity of Minnesota, 207 Pleasant Street SE, Minneapolis, Minnesota 55455 ReceiVed: October 23, 2009; ReVised Manuscript ReceiVed: January 24, 2010

Gibbs ensemble Monte Carlo simulations are employed to examine the influence of moderately strong electric fields on the vapor-liquid coexistence curves and on structural and energetic properties of the saturated phases of water, methanol, and dimethyl ether. The application of an electric field of 0.1 V/Å increases the critical temperature and normal boiling point by approximately 3% compared to the zero field case for all three compounds, whereas the critical density is found to decrease by 1% for methanol and dimethly ether and by 3% for water. For the special case of an electric field applied in only the liquid phase, these effects are magnified with a 4% increase in TC and a 13% decrease in FC. For the case of an electric field in only the vapor phase, the opposite effect is seen with a 4% decrease in TC and a 12% increase in FC. Structural analysis shows very little change in the radial distribution functions, but greatly increased orientational ordering with the application of an electric field. The orientational ordering effect is stronger in the liquid phase than in the vapor phase. An examination of the energetics reveals that, in the presence of an electric field, the interactions with the first and second solvation shells become less favorable but these are outweighed by a larger increase in the favorable long-range interactions with more distant molecules and the field. 1. Introduction The influence of an electric field on chemical processes is an important area of study in fields as diverse as ion-induced nucleation1–5 and patterning in polymer films.6–13 Electric fields are also useful in many separation techniques, from the widely used electrophoresis to the more specialized electroextraction,14 electrofiltration,15 and micellar electrokinetic chromatography (MEKC) techniques.16–18 Microfluidic separation techniques, such as micro free-flow electrophoresis19 and electromembrane extraction20 also utilize electric fields. A separation technique of great importance for the chemical industry is distillation. Experimental work has shown that the application of an electric field can enhance the distillation efficiency of certain polar mixtures; however, the mechanism responsible for this behavior remains unclear.21–23 In recent years, much research has been focused on understanding the behavior of liquids in electric fields, especially the vapor-liquid and liquid-liquid equilibria behavior of singlecomponent systems and simple mixtures. Experimental studies of the effects of an electric field on the vapor-liquid and liquid-liquid equilibria have led to contradictory results. Debye and Kleboth24 performed experiments using a high-voltage pulse generator to obtain electric fields up to 4.5 × 10-4 V/Å and light scattering to determine the critical temperature for the binary system 2,2,4-trimethylpentane/nitrobenzene. These experiments indicated a decrease in the critical temperature, in agreement with their theoretical predictions. These results were confirmed by Orzechowski,25 who used measurements of the nonlinear dielectric effect to estimate the change in critical temperature. Beaglehole26 studied the effects of an electric field on adsorption at the vapor-liquid interface of a binary system of cyclohexane/aniline and found that the sign of the shift in the * Corresponding author. E-mail: [email protected].

critical temperature depends on the orientation of the field relative to the interface. Cyclohexane/aniline mixtures were also studied by Early,27 who built a capacitor from a set of electrodes immersed in a spherical sample cell and used an oscillator to measure the dielectric constant as a function of temperature for several different aniline concentrations. Although no critical behavior in the dielectric constant was observed for these cyclohexane/aniline mixtures, this result is consistent with theoretical predictions of the droplet model28 for this system. This agreement lends credibility to the prediction of the droplet model of an increase in TC, which contradicts earlier experimental results.27 The discrepancy between these measurements was explained as likely being due to local heating of the sample due to the electric field in the experiments of Debye and Kleboth. Wirtz and Fuller29 developed a mean-field theory for polymer-solvent mixtures and polymer blends that predicts an electric field induces mixing by lowering the upper critical solution temperature and/or raising the lower critical solution temperature. They performed light scattering experiments on the mixtures of poly(p-chlorostyrene)/ethylcarbitol, poly(styrene)/ cyclohexane, and nitrobenzene/n-hexane and discovered that the electric field does indeed induce mixing. However, more recent experiments by Lee et al. for the polymer blends poly(vinylidene fluoride)/poly(1,4-butylene adipate)30 and poly(vinylidene fluoride)/poly(methyl methacrylate)31 show that an electric field induces demixing of these polymer blends. Tsori et al. have also observed field-induced demixing in the binary systems poly(methylphenylsiloxane)/squalane and poly(isobutylene)/ poly(methylsiloxane) for nonuniform electric fields.32 Hegseth and Amara have recently studied the effects of an electric field on the critical temperature of a single-component system.33 They applied an alternating electric field to a spherical capacitor filled with SF6 at its critical density and above the critical temperature, then slowly decreased the temperature while

10.1021/jp9101477  2010 American Chemical Society Published on Web 03/04/2010

4262

J. Phys. Chem. B, Vol. 114, No. 12, 2010

monitoring the light transmission through the fluid. Due to critical opalescence, the critical temperature is the temperature of the minimum light intensity. Hegseth and Amara observed an increase in TC of several mK at field strengths on the order of 10-5 V/Å, with larger increases for larger field strengths. More recent measurements by Mohsen-Nia et al. using the cloud point method with a flat capacitor for the ternary system (water + propionic acid + dichloromethane) show that the application of an electric field on the order of 10-8 V/Å increases the two phase region.34 Purely theoretical work most often predicts that the application of an electric field causes an increase in the critical temperature,35–39 although the recent statistical-mechanical theory of Stepanow and Thurn-Albrecht predicts an increase in the critical temperature for polarizable liquids and a decrease in the critical temperature for polar liquids.40 However, recall that the theories of Debye and Kleboth24 and Wirtz and Fuller29 predict a decrease in the critical temperature, which they do in fact observe experimentally, although there were possible problems with temperature control in their experiments.27,33 Researchers have also studied the effects of electric fields on the boiling points of liquids. An early study by Katti and Chaudhri41 showed that the boiling point of methanol, ethanol, and 2-propanol decreases in the presence of an electric field; however, these results could not be reproduced, and problems in the original experiment were discovered.42 Other experiments have shown no change in the boiling point for methanol, ethanol, 2-propanol, chloroform, carbon tetrachloride, and benzene.42–44 Theoretical work, on the other hand, predicts that the application of an electric field will cause an increase in the boiling point of dipolar fluids.39,45–47 Of course, it is possible that the field strength needed to observe a measurable change in the boiling point is outside the range of experimentally accessible fields. Given the varying experimental and theoretical results, a more complete understanding of vapor-liquid and liquid-liquid equilibria under the influence of an applied electric field is needed. The interactions in these systems are complex and may depend on the polarity and symmetry of the molecules, as well as the amount of aggregation present in the system. Monte Carlo (MC) molecular simulation is a powerful tool for exploring these systems. Not only do MC simulations provide molecular-level detail, but also the difficulties of controlling the heating effects of strong electric fields are completely avoided. Furthermore, a much stronger electric field can be used in a simulation than in an experiment, resulting in largersand more easily observables changes in the vapor-liquid equilibrium properties. Simulation studies of either Stockmayer fluids46,48 or water3,49 find that the application of an electric field increases the critical temperature. However, these studies do not fully explain the reasons for this behavior, simply stating that the electric field enhances the attracive interactions between dipolar molecules without providing a more detailed analysis.3 Other simulation studies have revealed that a strong electric field can induce crystallization of liquid water, an effect known as electrofreezing,50 or induce demixing in nitrotoluene/decane mixtures.51 However, many simulation studies with an electric field do not examine the effects of the field on phase equilibria but instead examine the changes induced in the structure of the liquid.52–57 In the present work, the vapor-liquid coexistence curves of water, methanol, and dimethyl ether are computed for applied electric field strengths ranging from 0.0 to 0.1 V/Å; i.e., the highest field strength used in the simulations is about a factor of 200-1000 larger than those used in experiments.24,33 The details of these simulations are described in section 2. Changes

Maerzke and Siepmann in the vapor-liquid equilibrium properties are reported in section 3.1. The effects of the electric field on the structure of these systems are discussed in section 3.2. In section 3.3, the effects of the electric field on the energetics of these systems is discussed. The conclusions of this work can be found in section 4. 2. Simulation Details For this work, the TraPPE-UA force field was employed for methanol58 and dimethyl ether,59 and the TIP4P model60 was used for water. For increased computational efficiency, the TraPPE-UA force field61 employs pseudoatoms for all CHx groups (where 0 e x e 4), which are located at the position of the carbon atom. Nonbonded interactions are described by pairwise-additive Lennard-Jones (LJ) and Coulomb potentials:

[( ) ( ) ]

unonbonded(rij) ) 4εij

σij rij

12

-

σij rij

6

+

qiqj 4πε0rij

(1)

where rij, εij, σij, qi, and qj are the separation, LJ potential well depth, LJ diameter, and partial charges, respectively. The unlike LJ interactions are determined using the Lorentz-Berthelot combining rules:62

1 σij ) (σii + σjj) 2

and

εij ) (εiiεjj)1/2

(2)

Rigid bond lengths are used for bonded interactions. Flexible bond angles for methanol and dimethyl ether are treated using a harmonic potential:

ubend(θ) )

kθ (θ - θeq)2 2

(3)

where θ is the bond angle, θeq is the equilibrium value for that bond angle, and kθ is the force constant. The energy from an applied electric field is given by

ufield(θ) ) -µ · E ) -|µ||E| cos θ

(4)

where µ is the molecular dipole moment and E is the applied electric field. To simplify the calculation, we chose to apply the field in the z-direction only

ufield(ri,z) ) µi,zEz ) -qiri,zEz

(5)

which eliminates the need to calculate an angle and hence reduces the computational complexity. Gibbs ensemble Monte Carlo63,64 (GEMC) simulations in the canonical (NVT) ensemble were carried out to compute the vapor-liquid coexistence curves (VLCC), saturated vapor pressures, and heats of vaporization at applied electric field strengths of 0.00, 0.01, 0.03, and 0.10 V/Å. The NVT-Gibbs ensemble63 allows for the efficient simulation of phase equilibria by employing two separate periodic simulation boxes in thermodynamic contact, but without an explicit interface. To equilibrate the chemical potential, molecules are swapped between the boxes with the aid of coupled-decoupled configurational-bias Monte Carlo (CBMC).65–68 The pressure is equilibrated through volume exchanges between the boxes, and

Vapor-Liquid Equilibria of Water, Methanol, and DME

J. Phys. Chem. B, Vol. 114, No. 12, 2010 4263

translational and rotational moves along with CBMC regrowths for flexible molecules allow the system to reach thermal equilibrium. System sizes of 1000, 500, and 400 molecules were used for water, methanol, and dimethyl ether, respectively. These system sizes were chosen to maintain a liquid box length of at least 30 Å. A spherical cutoff of rcut ) 14 Å with analytic tail corrections69 was employed for the LJ interactions. The Coulombic interactions were computed using the Ewald summation technique with tinfoil boundary conditions.70–72 The real-space cutoff was equal to rcut and the Ewald sum convergence parameter was set to 3.5/rcut. Vapor-phase box sizes were adjusted so that on average at least 40 molecules were present in the vapor phase. In cases where the saturated vapor pressure is low the vapor-phase box must be quite large for 40 molecules to be present. For large simulation boxes the Ewald sum becomes quite expensive; hence, once the vapor box length exceeds 50 Å, we set rcut to be equal to 40% of the box length. As there are few molecules in the vapor phase, this increases the efficiency by reducing the number of reciprocal-space vectors used in the Ewald sum. Volume moves were accepted on average once every 10 Monte Carlo (MC) cycles (1 MC cycle ) N moves, where N is the number of molecules in the system). The fraction of swap moves was adjusted so that on average one swap move was accepted every 10 MC cycles. Of the remaining moves, approximately one-third were CBMC regrowths (for flexible molecules), with the rest divided equally between translational and rotational moves. At every state point, 1.5 × 105 MC cycles of equilibration were followed by 2.5 × 105 MC cycles of production. To reduce the statistical uncertainty, four independent simulations were performed at every state point. Calculated uncertainties are given as the standard error of the mean. The critical temperature (Tc) and density (Fc) were determined using the scaling law73

Fliq - Fvap ) B(T - Tc)β

(6)

where β ) 0.325 and the law of rectilinear diameters74

1 (F + Fvap) ) Fc + A(T - Tc) 2 liq

(7)

where Fliq and Fvap are the liquid and vapor densities, respectively. For all systems, the five highest temperatures were considered in the determination of the critical properties. Since the aim of this work is to investigate the effect of the electric field on the vapor-liquid equilibrium properties, the same fitting procedure is applied to all three compounds, whereas an analysis of experimental data shows larger deviations from the universal Ising behavior for methanol than for dimethyl ether.58,59 Heats of vaporization are computed on-the-fly during the GEMC simulations according to75

∆Hvap ) Uvap - Uliq + Psat∆V

(8)

where Uvap, Uliq, Psat, and ∆V are the instantaneous values of the molar internal energy of the vapor and liquid phases, the saturated vapor pressure, and the difference in molar volume between the vapor and liquid phases, respectively. The normal boiling points (Tboil) were calculated using a linear fit of the logarithm of the saturated vapor pressures versus the

Figure 1. Vapor-liquid coexistence curves for water (top curves), methanol (middle curves), and dimethyl ether (bottom curves) with various applied electric field strengths: E ) 0 (black up triangles), 0.01 (red left triangles), 0.03 (green right triangles), and 0.10 V/Å (blue circles). The stars of the corresponding colors indicate the critical point.

inverse temperature (i.e., the Clausius-Clapeyron equation) for the two temperatures surrounding Tboil. It is likely that the systematic error from the use of the Clausius-Clapeyron equation would not depend on field strength, and hence, this procedure should allow for a precise determination of the field effect on the normal boiling point. Applying the same procedure (two points spread by about 25 K) to experimental saturated vapor pressures of water indicates that the systematic error introduced by this procedure is smaller than 0.05 K. 3. Results and Discussion 3.1. Vapor-Liquid Equilibria. The VLCCs for all three compounds show a small increase in liquid density and a small decrease in vapor density in the presence of an applied electric field (see Figure 1). For a field strength of 0.1 V/Å, the critical temperature increases by 3.3, 3.0, and 3.2% for water, methanol, and dimethyl ether, respectively, whereas the critical density shows a decrease of 3, 1, and 1%, respectively (see Table 1). An increase in Tc with a smaller change in Fc is in agreement with previous simulation results,3,46,48,49 as well as some theoretical36–39,46 and experimental33 work. The present simulations show that field strengths of 0.01 and 0.03 V/Å have relatively little effect on the vapor-liquid equilibria (e.g., an increase of less than 1% in Tc), thereby indicating that relatively large field strengths, which are difficult to access experimentally, are required to observe substantial changes in the vapor-liquid equilibria. A recent statistical mechanical theory predicts that Tc increases in nonpolar but polarizable systems and decreases in polar systems;40 however, the present simulations yield an increase for all three polar compounds studied using nonpolarizable force fields. Polarization is an important consideration when the effects of an applied electric field are examined. Extracting the critical parameters from a multiparametric equation of state fitted to the results of molecular dynamics simulations at over 50 state points in the near-critical region, Svishchev and Hayward49 found that a field strength of 0.1 V/Å leads to an increase in Tc of approximately 5% for the polarizable PPC water model,76 which is somewhat larger than

4264

J. Phys. Chem. B, Vol. 114, No. 12, 2010

Maerzke and Siepmann

TABLE 1: Critical Temperature, Critical Density, and Normal Boiling Point for Water, Methanol, and Dimethyl Ether at Various Electric Field Strengthsa

b

Fc [g/mL]

E [V/Å]

Tc [K]

0.00 0.01 0.03 0.10 0.1/0.0 0.0/0.1b

583.16 586.69 587.08 602.47 605.73 560 ( 25

0.00 0.01 0.03 0.10

517.69 518.89 519.89 533.35

Methanol 0.2662 0.2652 0.2652 0.2631

339.72 340.13 341.21 349.63

0.00 0.01 0.03 0.10

397.05 396.53 397.85 410.03

Dimethyl Ether 0.2751 0.2751 0.2762 0.2721

243.12 243.72 244.71 252.13

Water 0.3162 0.3122 0.3132 0.3072 0.2751 0.355 ( 0.015

Tboil [K] 362.52 362.73 364.42 372.51 380.62 353.82

a Subscripts indicate statistical uncertainties in the final digit. Uncertainties in critical point reflect fits with β ) 0.325 and 0.2.

Figure 3. Clausius-Clapeyron plots for water (top), methanol (middle), and dimethyl ether (bottom) with various applied electric field strengths: E ) 0 (black up triangles), 0.01 (red left triangles), 0.03 (green right triangles), 0.10 (blue circles), 0.1/0.0 (orange diamonds), and 0.0/0.1 V/Å (purple down triangles).

Figure 2. Close up of the vapor-liquid coexistence curve for water with various applied electric field strengths: E ) 0 (black up triangles), 0.01 (red left triangles), 0.03 (green right triangles), 0.10 (blue circles), 0.1/0.0 (orange diamonds), and 0.0/0.1 V/Å (purple down triangles).

the increase observed here for nonpolarizable models. In GEMC simulations (for smaller systems and shorter runs than used here), Gao et al.3 found a somewhat larger upward shift of the VLCC for the nonpolarizable SPC/E77 water model than observed here for the TIP4P water model. However, neither Svishchev and Hayward nor Gao et al. provided an estimate of the statistical uncertainties of their simulations, and it is not obvious whether the relative changes found for the PPC and SPC/E models are statistically different from those determined here for the TIP4P model. Thus, although polarization may increase the effects of an applied electric field, the effect of polarization is likely relatively small. To further probe the effects of an applied electric field on the vapor-liquid equilibria, GEMC simulations were performed for water with an applied electric field in only one of the two simulation boxes, and the corresponding VLCCs are depicted in Figure 2. For the system with a field strength of 0.1 V/Å in the liquid box and 0.0 V/Å in the vapor box (referred to as 0.1/0.0), a slight increase in Tc of 0.5% is observed above the value for the system with E ) 0.1 V/Å in both boxes (i.e., the total increase is 3.9% above the zero-field value).

Although the effect of removing the field in the vapor phase on Tc is relatively small, the saturated vapor densities are significantly reduced, and a 10% decrease in Fc is found compared to the system with the same field strength applied to both phases. The simulations for the reverse case, i.e., a system with the field applied only in the vapor box (referred to as 0.0/0.1), proved particularly challenging because above a certain temperature most of the molecules tended to swap into the simulation box with the field and, hence, exchanged the simulation boxes for the vapor and liquid phases. For temperatures below this threshold, the saturated vapor densities are significantly enhanced even when compared to the zero field simulations, whereas the saturated liquid densities are close to the data for the zero field case. This enhancement of the vapor densities leads to a “flat” VLCC and the scaling law with the Ising exponent does not describe the VLCC as well as for the other compounds/fields investigated here (see also Supporting Information). Thus, there is considerable uncertainty in the extrapolation of the GEMC simulation data to the critical point (a grand canonical Monte Carlo simulation with a single box would not work here because it does not allow for the use a field on only the vapor phase) and the numerical data in Table 1 reflect an average for critical points extrapolated using β ) 0.325 and 0.2 (see also Supporting Information). Nevertheless, it appears likely that this special case results in a substantial decrease of Tc (≈4%) and increase of Fc (≈12%) compared to the zero field situation. Upon examining the Clausius-Clapeyron plot (see Figure 3), one can see that the presence of the largest field (0.1 V/Å) leads to a substantial decrease in the saturated vapor pressure for all three compounds. Concomitantly, the normal boiling point is increased by 2.8, 3.2, and 3.6% for water, methanol, and dimethyl ether, respectively (see Table 1). Even for the

Vapor-Liquid Equilibria of Water, Methanol, and DME

J. Phys. Chem. B, Vol. 114, No. 12, 2010 4265

Figure 4. Heat of vaporization for water (top curves), methanol (middle curves), and dimethyl ether (bottom curves) as a function of temperature for various applied electric field strengths: E ) 0 (black up triangles), 0.01 (red left triangles), 0.03 (green right triangles), 0.10 (blue circles), 0.1/0.0 (orange diamonds), and 0.0/0.1 V/Å (purple down triangles).

intermediate field (0.03 V/Å), a statistically significant increase of about 0.5% is observed. Again for water with an applied field in only one of the two phases, one observes a greater increase in Tboil for the system with the field in the liquid phase, and a decrease in Tboil for the system with the field in the vapor phase (see Figure 3). Zhu et al. predicted that in the presence of an electric field, the more favorable interactions in the liquid phase will result in a lower vapor pressure, provided that the lowering of the entropy is not too large.52 A lower vapor pressure in the presence of an electric field is also in agreement with previous theoretical and simulation work.39,45–47 Furthermore, the application of an electric field in both phases leads to substantial increases in the heat of vaporization with increasing field strength (see Figure 4). For a field of 0.1 V/Å, ∆Hvap increases by about 4, 7, and 6% for water, methanol, and dimethyl ether compared to the zero field case at low reduced temperatures. As should be expected from the increase in Tc, the differences increase with increasing temperature. Again, larger effects are found for the simulations with the field applied to only one phase and the opposite behavior (decrease in ∆Hvap) for the system with an applied field in only the vapor phase. The energetics for these systems are discussed in more detail in section 3.3. 3.2. Structural Properties. The observed changes in the VLE properties indicate that, for these three compounds, the liquid phase becomes more favorable relative to the vapor phase upon the application of an electric field. To elucidate some of the reasons for this behavior, several structural properties are examined. Radial distribution functions (RDFs) were computed for all three compounds. For brevity, only the RDFs at one temperature (375 K for water, 350 K for methanol, and 250 K for dimethyl ether) are discussed. These low temperatures were chosen to examine the systems where the liquid phases are more structured; moreover, at these temperatures the systems are all at similar reduced temperatures, T* ) T/TC, which makes for easier comparisons. Upon examination of the RDFs for all three compounds (see Figure 5), one finds very little change with the application of an electric field. The largest relative change in peak height is found for the oxygen-oxygen RDF of dimethyl

Figure 5. Radial distribution functions for the saturated liquid phases of water (375 K), methanol (350 K), and dimethyl ether (250 K) with various applied electric field strengths: E ) 0 (black), 0.01 (red), 0.03 (green), and 0.10 V/Å (blue). Note that the lines overlap for the different field strengths and the difference in scale on the y-axis.

ether, where the height of the first peak increases from 1.71 for zero field to 1.76 for a field of 0.1 V/Å. Previous simulations of water and methanol with an applied electric field yielded significant changes in the RDFs only for fields in the range 0.5-1.0 V/Å,52–54,56,57 which are much stronger than the fields studied here. To further examine the structure of these systems, we computed the orientational order parameter, S, defined as

1 S ) 〈3 cos2 θ - 1〉 2

(9)

where θ is the angle between the electric field and the dipole vector. This order parameter approaches 1 for dipoles aligned with the field, -0.5 for dipoles perpendicular to the field, and 0 for dipoles with no preferential orientation (or at the magic angle). As can be seen in Figure 6, the liquid-phase order parameter for all three compounds is zero at all temperatures in the absence of an electric field. Even for the smallest field used here (E ) 0.01 V/Å), the order parameter changes to slightly positive values (0.01, 0.02, and 0.03 for water, methanol, and dimethyl ether) at the lowest temperatures. For E ) 0.03 V/Å, S increases to approximately 0.1 for water and slightly larger values for the other two compounds at the lowest temperatures and then decreases with increasing temperature and reaches a value of 0.02 close to TC. At the highest field strength (E ) 0.1 V/Å), S becomes as large as 0.44 for dimethyl ether, 0.39 for water, and 0.35 for methanol and then decreases with increasing temperature; however, even at the highest temperatures S remains larger than the highest values for the

4266

J. Phys. Chem. B, Vol. 114, No. 12, 2010

Figure 6. Order parameter S for the liquid phase of water, methanol, and dimethyl ether as a function of temperature for varying field strengths: E ) 0 (black up triangles), 0.01 (red left triangles), 0.03 (green right triangles), and 0.10 V/Å (blue circles).

Figure 7. Distribution of the dipole vector orientation for water (375 K), methanol (350 K), and dimethyl ether (250 K) with respect to the z-direction (direction of the applied field) for varying field strengths: E ) 0 (black), 0.01 (red), 0.03 (green), and 0.10 V/Å (blue).

lower field strengths. These changes in S indicate that although the RDFs do not show a perceptable change upon the application of an electric field, the orientational ordering of the molecules does change. The field dependence of the orientational structure of these systems can be further examined by computing the distribution of the angle between the electric field and the dipole vector (see Figure 7). Again, a uniform distribution is found in the absence of an electric field. This changes to a slight preference for parallel alignments at the lowest field strength of 0.01 V/Å, an increased preference for parallel alignments for E ) 0.03 V/Å, and a strong preference for parallel alignments at the highest field strength. More interestingly, one observes a greater preference for parallel alignments in the liquid phase than in the vapor phase, despite the fact that molecules are more free to rotate in the vapor phase. This condensed-phase enhancement is somewhat more pronounced at low field strength; e.g., the

Maerzke and Siepmann

Figure 8. Distribution of the mutual dipole vector orientation with respect to surrounding molecules within the first (black) and second (cyan) solvation shells and beyond (magenta) for the liquid phase of water (375 K), methanol (350 K), and dimethyl ether (250 K) for varying field strengths: E ) 0, 0.01, 0.03, and 0.10 V/Å.

excess fraction of water molecules with θ e 60° in the liquid phase exceeds that in the vapor phase by a factor of 3.2, 2.9, and 2.0 for E ) 0.01, 0.03, and 0.1 V/Å, respectively. Furthermore, at the highest field, the fraction of molecules with θ g 120° is vanishingly small for all three compounds in the liquid phase. This condensed-phase enhancement can be attributed to collective ordering favored by long-range energetic factors (see section 3.3, below), which are not as significant in the vapor phase. Moreover, aligning with the field reduces the rotational entropy, which is a larger factor in the vapor phase than in the liquid phase. To explore the distance dependence of the orientational ordering caused by the electric field on the liquid structure of these systems, we divided the surrounding of a given molecule into first solvation shell (1S), second solvation shell (2S), and long-range (LR) intervals using a distance criteron deduced from the center-of-mass RDFs (see Supporting Information). The boundaries of these shells are 3.65 and 5.80 Å for water, 3.85 and 6.29 Å for methanol, and 6.72 and 11.00 Å for dimethyl ether. As there is no significant difference between the RDFs at all field strengths studied here, these values are used to define the solvation shells regardless of the strength of the applied electric field. Using this division of the system, the differences between the short-range, intermediate-range, and long-range structure can be examined by calculating the orientation of dipoles with respect to other dipoles in the 1S, 2S, and LR intervals (see Figure 8). For water, even in the absence of an electric field there is a slight preference for the dipoles to align in a parallel arrangement within the first solvation shell. As the field strength increases, this preference becomes more pronounced. Beginning with E ) 0.03 V/Å, the water molecules within the second solvation shell and beyond also show a slight preference for parallel alignments. At the highest field strength, the dipoles show a marked preference for parallel alignments at all distances, which is not surprising given that at the highest field strength the dipoles tend to align with the field. Hence, the electric field reinforces the preference for the parallel alignment of the dipoles, in agreement with the previous findings of Kiselev and Heinzinger.53 Overall, the behavior for methanol

Vapor-Liquid Equilibria of Water, Methanol, and DME

J. Phys. Chem. B, Vol. 114, No. 12, 2010 4267

Figure 9. Tetrahedral order parameter q for the liquid phase of water as a function of temperature for varying field strengths: E ) 0 (black up triangles), 0.01 (red left triangles), 0.03 (green right triangles), and 0.10 V/Å (blue circles).

and dimethyl ether is quite similar, but these compounds exhibit significantly more alignment (as measured by the fraction of molecules with θ e 90°) in the first solvation shell than water. Furthermore, the maximum in the distribution for the first solvation shell of methanol and dimethyl ether is shifted to θ ≈ 45°. The position of this maximum shifts to smaller angles with increasing field strength. For the second solvation shell and beyond, a parallel alignment of the dipole vectors is preferred for methanol and dimethyl ether and this alignment becomes more pronounced with increasing field strength. It is quite remarkable that, for all three compounds, there is rather little difference in the mutual alignment with molecules in the second solvation shell and with molecules beyond the second solvation shell. In addition, the differences between the three compounds are rather small for the second solvation shell and the long-range mutual orientation. Given the influence of the applied electric field on the orientational ordering of these compounds, it is interesting to investigate whether the field also leads to change in the tetrahedral order (for water) and the number of hydrogen bonds (for water and methanol). To measure the extent of tetrahedrality, the following tetrahedral order parameter, q,78,79 is used 3

q)1-

4

∑ ∑

3 8 j)1

k)j+1

(cos φ

jk

+

1 3

2

)

(10)

where φjk is the angle between an oxygen atom and its nearest neighbors j and k. The tetrahedral order parameter is unity for a perfect tetrahedral network. As can be seen in Figure 9, q is close to 0.57 for the saturated liquid phase at T ) 350 K and decreases monotonically for all field strengths. At elevated temperatures, an increase in the field strength leads to slightly larger q values, e.g., at T ) 550 K q ) 0.358, 0.359, 0.362, and 0.383 for E ) 0, 0.01, 0.03, and 0.10 V/Å, respectively. The low sensitivity of q to the field strength is in agreement with the previous simulation results by Kiselev and Heinzinger,53 who found little change in the deviation from tetrahedrality (estimated via a different order parameter than used here) for liquid water and an aqueous chloride solution at E ) 0.5 V/Å and T ) 300 K.

Some previous simulation53,56,57 and theoretical80,81 results indicate that the number of hydrogen bonds present in the liquid phase of water and methanol increases with increasing field strength. To examine the hydrogen bonding for the saturated liquid and vapor phases, a distance-distance-angular hydrogen bond definition is used that has been previously employed for water, alcohols, and their mixtures;82–84 i.e., a hydrogen bond exists between two molecules when the O · · · O distance is less than 3.3 Å, the O · · · H distance is less than 2.5 Å, and the cosine of the O · · · H-O angle is less than -0.1. The numerical data resulting from the hydrogen bond analysis are reported in Table 2. For both water and methanol, there is a slight increase in the number of hydrogen bonds in the liquid phase and a slight decrease in the number of hydrogen bonds in the vapor phase with the application of an electric field. In general, the relative increase in the liquid phase appears to increase with temperature but is always much smaller than the relative decrease in the vapor phase, with the latter being quite insensitive to changes in the temperature. Specifically, for water at T ) 350 we find an increase by a factor of 1.004 for the liquid phase and a reduction by a factor of 1.6 for the vapor phase. These factors change to 1.05 and 1.5, respectively, at T ) 550 K, and quite similar factors are found for methanol. For the special cases of an electric field applied only in one of the two phases, one observes a slightly different behavior. For the system 0.1/0.0, the number of hydrogen bonds in the liquid phase is statistically indistinguishable from the system with E ) 0.1 V/Å applied to both phases, but the number of hydrogen bonds in the vapor phase shows a further decrease by a factor of ≈1.4 from the systems with E ) 0.1 V/Å applied to both phases; i.e., it is smaller by a factor of ≈2.1 than for the zero field system. For the system 0.0/0.1, the number of hydrogen bonds in the liquid phase is consistent with the zero field system, but here an increase in the number of hydrogen bonds in the vapor phase by a factor of ≈1.4 is found. Although these results are statistically signficant, the very small increases in the liquid phase and somewhat larger decreases in the vapor phase (for situations with the same field applied to both phases) may quite possibly reflect only the changing coexistence densities rather than being the cause for the change in any of the vapor-liquid equilibria properties of these systems. Furthermore, similar changes in VLE behavior are observed for dimethyl ether as for water and methanol, even though dimethyl ether does not form hydrogen bonds. Hence, it appears unlikely that hydrogen bonding plays a pivotal role for these changes. 3.3. Energetics. To more completely explain the effects of the electric field on the VLE properties, we looked beyond the structural analysis and examined the energetics of these systems. The analysis of the dipole alignment showed that in the presence of the field there is some long-range orientational order in these systems (see Figure 8). Using the same definitions discussed in section 3.2, the intermolecular potential energy can be divided into contributions from the first solvation shell (1S), the second solvation shell (2S) and the more distant molecules (LR). We define the long-range energy as

ULR ) UCoulomb + ULJ - U1S - U2S

(11)

where UCoulomb and ULJ are the total Coulomb and LennardJones (including the tail correction) energies of the system, respectively. These two quantities are obtained directly from the simulation trajectory (and are computed at evey MC step). The solvation shell contributions, U1S and U2S, are not available on the fly and are computed through analysis of the movie files

4268

J. Phys. Chem. B, Vol. 114, No. 12, 2010

Maerzke and Siepmann

TABLE 2: Average Number of Hydrogen Bonds (HB) in the Liquid and Vapor Phases at Variouos Temperatures for Water and Methanola Water E ) 0 V/Å

E ) 0.01

E ) 0.03

E ) 0.1

E ) 0.1/0.0

E ) 0.0/0.1

T [K]

HBliq

HBvap

HBliq

HBvap

HBliq

HBvap

HBliq

HBvap

HBliq

HBvap

HBliq

HBvap

350 400 450 500 550

3.6251 3.3912 3.1151 2.7792 2.3333

0.0522 0.1291 0.2622 0.522 0.923

3.6251 3.3931 3.1192 2.7822 2.341

0.0523 0.1314 0.2763 0.491 0.986

3.6291 3.3991 3.1232 2.7893 2.3543

0.0513 0.1242 0.2546 0.4717 0.844

3.6381 3.4212 3.1571 2.8452 2.4586

0.0321 0.0861 0.1843 0.3355 0.6218

3.6381 3.4211 3.1581 2.8432 2.4531

0.0231 0.0641 0.1431 0.2523 0.4237

3.6261 3.3952 3.1171

0.0691 0.1784 0.3826

Methanol E ) 0 V/Å

a

E ) 0.01

E ) 0.03

E ) 0.1

T [K]

HBliq

HBvap

HBliq

HBvap

HBliq

HBvap

HBliq

HBvap

325 375 425 475

1.8821 1.7391 1.5361 1.2612

0.132 0.251 0.3868 0.542

1.8821 1.7401 1.5391 1.2653

0.1299 0.2326 0.3516 0.573

1.8831 1.7451 1.5431 1.2713

0.193 0.2005 0.3229 0.543

1.8765 1.7591 1.5721 1.3191

0.0348 0.151 0.2403 0.3633

Subscripts indicate statistical uncertainties in the final digit.

TABLE 3: Average Coulomb, LJ, First Solvation Shell (1S), Second Solvation Shell (2S), Electric Field, and Long-Range (LR) Energies (in Units of kJ/mol) in the Vapor and Liquid Phases of Water (375 K), Methanol (350 K), and Dimethyl Ether (250 K) at Various Electric Field Strengthsa Water liquid

vapor

E [V/Å]

UCoulomb

ULJ

U1S

U2S

Ufield

ULR

UCoulomb

ULJ

U1S

0.00 0.01 0.03 0.10 0.1/0.0 0.0/0.1

-42.4745 -42.4667 -42.5278 -42.8528 -42.8474 -42.4592

5.6682 5.6573 5.6533 5.6572 5.6553 5.6603

-29.1697 -28.9508 -27.732 -23.9837 -23.9818 -29.1625

-5.0982 -4.9834 -4.3444 -2.4769 -2.4715 -5.0905

-0.0621 -0.4872 -3.0702 -3.0701

-2.5399 -2.8764 -4.802 -10.7369 -10.7407 -2.5484

-1.061 -1.053 -0.982 -0.721 -0.511 -1.472

0.1764 0.1786 0.1605 0.1161 0.0842 0.2385

-0.851 -0.862 -0.802 -0.592 -0.411 -1.192

Ufield -0.02041 -0.18214 -1.8172 -1.8251

ULR -0.0346 -0.0137 -0.0156 -0.01510 -0.0185 -0.0407

Methanol liquid

vapor

E [V/Å]

UCoulomb

ULJ

U1S

U2S

0.00 0.01 0.03 0.10

-29.352 -29.371 -29.511 -30.051

-5.2944 -5.2927 -5.2668 -5.2555

-21.003 -20.984 -20.761 -20.111

-9.342 -9.273 -8.763 -7.491

Ufield

ULR

UCoulomb

ULJ

U1S

-0.0612 -0.5157 -3.1264

-4.302 -4.421 -5.252 -7.701

-2.82 -2.42 -2.21 -1.187

0.152 0.112 0.111 0.0499

-2.52 -2.22 -2.01 -1.106

Ufield

ULR

-0.0231 -0.2041 -2.0054

-0.092 -0.124 -0.073 -0.0344

Dimethyl Ether liquid

vapor

E [V/Å]

UCoulomb

ULJ

U1S

U2S

0.00 0.01 0.03 0.10

-4.5152 -4.5301 -4.6353 -5.0062

-14.677 -14.7404 -14.7856 -14.9525

-13.6925 -13.6352 -13.3308 -12.4885

-4.1913 -4.1801 -4.0726 -3.8401

a

Ufield

ULR

UCoulomb

ULJ

U1S

-0.05624 -0.4502 -2.77508

-1.307 -1.4556 -2.0192 -3.6294

-0.07369 -0.07128 -0.06967 -0.05915

-0.1351 -0.1312 -0.1271 -0.0961

-0.1954 -0.1874 -0.1842 -0.1416

Ufield

ULR

-0.02341 -0.20744 -1.9441

-0.0144 -0.0166 -0.0133 -0.0146

Note that there is no second solvation shell in the vapor phase. Subscripts indicate statistical uncertainties in the final digit(s).

(i.e., less frequently). The long-range energy, ULR, can then be calculated for each of the independent simulations, and these are used to estimate the statistical uncertainty for ULR (instead of propagating the uncertainties for the terms on the right-hand side of eq 11). The numerical data obtained from this energetic analysis for the saturated phases of all three compounds at low reduced temperature are listed in Table 3. The change in the total intermolecular energy (Uinter ) UCoulomb + ULJ, but excluding Ufield, the interaction with the external field) caused by increasing the field strength from 0 to 0.1 V/Å is relatively small for all three compounds. Specifically, Uinter becomes more favorable in the liquid phase with changes of -0.39, -0.66, and -0.77

kJ/mol for water, methanol, and dimethyl ether, respectively, where Uinter becomes less favorable in the vapor phase with changes of 0.28, 1.47, and 0.02 kJ/mol, respectively. Thus, the intermolecular contribution to the heat of vaporization changes by 0.67, 2.13, and 0.79 kJ/mol, respectively. Overall, these intermolecular contributions are similar in magnitude as the effect from direct interactions with the field. As can be inferred from the distribution of the dipole vectors with the field (see Figure 7), the values of Ufield are quite similar for all three compounds but differ between the phases. The contributions of Ufield to the change in the heat of vaporization upon increasing the field strength from 0 to 0.1 V/Å are 1.25, 1.12, and 0.83 kJ/mol for water, methanol, and dimethyl ether, respectively.

Vapor-Liquid Equilibria of Water, Methanol, and DME The fact that Ufield is more favorable for the liquid than the vapor phase reflects the cooperative orientational ordering in the condensed phase. However, there is some retardation in the increase of Ufield with E for the liquid phase caused by intermolecular interactions; i.e., the liquid-phase Ufield increases only by a factor of 50 upon increasing E by a factor of 10, whereas the increase in the vapor phase is close to the quadratic dependence expected for isolated dipoles.85 Examination of the distance dependence in the changes of Uinter for the liquid phase shows that U1S and U2S become significantly less favorable upon an increase in the field strength; i.e., the reorganization caused by the field strongly weakens the interactions with the first and second solvation shells. It is quite surprising that these energetic changes are not picked up by the analyses of the tetrahedral order parameter (see Figure 9) and the number of hydrogen bonds (see Table 2). For water and dimethyl ether, the change in U1S is much larger than the change in U2S, whereas the opposite is true for methanol. The total changes in the contribution from the first and second solvent shells upon increasing E from 0.0 to 0.1 V/Å are +7.81, +2.75, and +1.56 kJ/mol for water, methanol, and dimethyl ether, respectively. However, these changes are more than compensated by changes in ULR that become much more favorable with increasing field strength, with the corresponding changes being -8.20, -3.40, -2.33 kJ/mol, respectively. As one should expect, the changes in ULR for the vapor phase are insignificant. For the special water cases where the field is applied to only one of the phases, the energetics for the liquid phase are entirely consistent with the corresponding simulations with fields applied to both phases; i.e., all individual contributions to the energy for system 0.1/0.0 agree with those for E ) 0.1 V/Å, and the same holds true for system 0.0/0.1 and the zero field system. However, there are some differences for the vapor phases with Uinter being -0.43 and -0.88 kJ/mol for systems E ) 0.1/0.0 and 0.0 V/Å, respectively, and -1.23 and -0.60 kJ/mol for systems E ) 0.0/1.0 and 0.1 V/Å, respectively. It is likely that these changes simply reflect the differences in the saturated vapor densities compared to the regular cases. 4. Conclusions The application of a strong electric field is found to increase the critical temperature, normal boiling point, and heat of vaporization for water, methanol, and dimethyl ether. For the special cases of an electric field in only one of the two phases, a greater increase in Tc and Tboil is seen when the field is applied to only the liquid phase, and the opposite behavior (i.e., a decrease in Tc and Tboil) when the field is applied in just the vapor phase. Field strengths up to at least 0.1 V/Å have very little effect on the radial distribution functions, the number of hydrogen bonds, and the tetrahedral order parameter in the liquid phase at low reduced temperature; however, structural changes still occur. Even at lower field strengths, the dipoles tend to align with the field and each other. The alignment of the dipoles is stronger in the liquid phase than in the vapor phase due to cooperative effects and leads to favorable long-range interactions that more than compensate for the less favorable interactions with molecules in the first and second solvation shells. This overcompensation by the enhanced long-range interactions and the stronger interactions with the field in the liquid phase result in the observed changes in the heats of vaporization and, hence, the upward shifts in the vapor-liquid coexistence curves for water, methanol, and dimethyl ether in the presence of moderately strong fields.

J. Phys. Chem. B, Vol. 114, No. 12, 2010 4269 Acknowledgment. Financial support from the National Science Foundation (CBET-756641) and a Graduate School Doctoral Dissertation Fellowship (K.A.M.) is gratefully acknowledged. Part of the computer resources were provided by the Minnesota Supercomputing Institute. Supporting Information Available: Three tables listing the numerical data for the vapor-liquid coexistence properties of water (Table S1), methanol (Table S2), and dimethyl ether (Table S3) at all field strengths; a brief discussion with two figures of the applicability of a different critical exponent for the special water system 0.0/0.1; and a figure showing the centerof-mass radial distribution functions used in determining the boundaries for the first and second solvation shells. This material is available free of charge via the Internet at http://pubs.acs.org. References and Notes (1) Rabeony, H.; Mirabel, P. J. Phys. Chem. 1987, 91, 1815–1818. (2) He, R.; Hopke, P. K. J. Chem. Phys. 1993, 99, 9972–9978. (3) Gao, G. T.; Oh, K. J.; Zeng, X. C. J. Chem. Phys. 1999, 110, 2533– 2538. (4) Fisenko, S. P.; Kane, D. B.; El-Shall, M. S. J. Chem. Phys. 2005, 123, 104704. (5) Kathmann, S. M.; Schenter, G. K.; Garrett, B. C. Phys. ReV. Lett. 2005, 94, 116104. (6) Amundson, K.; Helfand, E.; Quan, X.; Smith, S. D. Macromolecules 1993, 26, 2698–2703. (7) Amundson, K.; Helfand, E.; Quan, X.; Hudson, S. D.; Smith, S. D. Macromolecules 1994, 27, 6559–6570. (8) Xiang, H.; Lin, Y.; Russell, T. P. Macromolecules 2004, 37, 5358– 5363. (9) Olszowka, V.; Hung, M.; Kuntermann, V.; Scherdel, S.; Tsarkova, L.; Boker, A. ACS Nano 2009, 3, 762–774. (10) Schobert, H. G.; Schmidt, K.; Schindler, K. A.; Boker, A. Macromolecules 2009, 42, 3433–3436. (11) Bandyopadhyay, D.; Sharma, A.; Thiele, U.; Reddy, P. D. S. Langmuir 2009, 25, 9108–9118. (12) Arun, N.; Sharma, A.; Pattader, P. S. G.; Banerjee, I.; Dixit, H. M.; Narayan, K. S. Phys. ReV. Lett. 2009, 102, 254502. (13) Schaffer, E.; Thurn-Albrecht, T.; Russell, T. P.; Steiner, U. Europhys. Lett. 2001, 53, 518–524. (14) Stichlmair, J.; Schmidt, J.; Proplesch, R. Chem. Eng. Sci. 1992, 47, 3015–3022. (15) Ptasinki, K. J.; Kerkhoff, P. J. A. M. Sep. Sci. Technol. 1992, 27, 995–1021. (16) Terabe, S.; Otsuka, K.; Ichikawa, K.; Tsuchiya, A.; Ando, T. Anal. Chem. 1984, 56, 111–113. (17) Terabe, S.; Otsuka, K.; Ando, T. Anal. Chem. 1985, 57, 834–841. (18) Welsch, T.; Michalke, D. J. Chromatogr. A 2003, 1000, 935–951. (19) Turgeon, R. T.; Bowser, M. T. Anal. Bioanal. Chem. 2009, 394, 187–198. (20) Petersen, N. J.; Jensen, H.; Hansen, S. H.; Rasmussen, K. E.; Pedetersen-Bjergaard, S. J. Chromatogr. A 2009, 1216, 1496–1502. (21) Blankenship, K. D.; Shah, V. M.; Tsouris, C. Sep. Sci. Technol. 1999, 34, 1393–1409. (22) Tsouris, C.; Blankenship, K. D.; Dong, J.; DePaoli, D. W. Ind. Eng. Chem. Res. 2001, 40, 3843–3847. (23) Luo, G. S.; Pan, S.; Liu, J. G.; Dai, Y. Y. Sep. Sci. Technol. 2001, 36, 2799–2809. (24) Debye, P.; Kleboth, K. J. Chem. Phys. 1965, 42, 3155–3162. (25) Orzechowshi, K. Chem. Phys. 1999, 240, 275–281. (26) Beaglehole, D. J. Chem. Phys. 1981, 74, 5251–5255. (27) Early, M. D. J. Chem. Phys. 1992, 96, 641–647. (28) Goulon, J.; Greffe, J. L.; Oxtoby, D. W. J. Chem. Phys. 1979, 42, 4742–4750. (29) Wirtz, D.; Fuller, G. G. Phys. ReV. Lett. 1993, 71, 2236–2239. (30) Lee, J. S.; Prabu, A. A.; Kim, K. J.; Park, C. Macromolecules 2008, 41, 3598–3604. (31) Lee, J. S.; Prabu, A. A.; Kim, K. J. Macromolecules 2009, 42, 5660–5669. (32) Tsori, Y.; Tournilhac, F.; Leibler, L. Nature 2004, 430, 544–547. (33) Hegseth, J.; Amara, K. Phys. ReV. Lett. 2004, 93, 057402. (34) Mohsen-Nia, M.; Jazi, B.; Amiri, H. J. Chem. Thermodyn. 2009, 41, 1081–1085. (35) Landau, L. D.; Lifshitz, E. M. Electrodynamics of Continuous Media; Pergamon Press: Oxford, U.K., 1960. (36) Onuki, A. Europhys. Lett. 1995, 29, 611–616.

4270

J. Phys. Chem. B, Vol. 114, No. 12, 2010

(37) Boda, D.; Szalai, I.; Liszi, J. J. Chem. Soc., Faraday Trans. 1995, 91, 889–894. (38) Tsori, Y.; Leibler, L. Proc. Natl. Acad. Sci. U.S.A. 2007, 104, 7348– 7350. (39) Gabor, A.; Szalai, I. Mol. Phys. 2008, 106, 801–812. (40) Stepanow, S.; Thurn-Albrecht, T. Phys. ReV. E 2009, 79, 041104. (41) Katti, P. K.; Chaudhri, M. M. Nature 1961, 190, 80. (42) Ramakrishna, V.; Pillai, P. K. C.; Katti, P. K. Nature 1963, 1963, 181. (43) Anderson, R. A.; Infirri, S. S. Nature 1962, 196, 267. (44) Sharma, R. C. J. Appl. Phys. 1971, 42, 1234–1235. (45) Lyon, R. K. Nature 1961, 192, 1285–1286. (46) Boda, D.; Winkelmann, J.; Liszi, J.; Szalai, I. Mol. Phys. 1996, 87, 601–624. (47) Kronome, G.; Szalai, I.; Liszi, J. J. Chem. Phys. 2002, 116, 2067– 2074. (48) Stevens, M. J.; Grest, G. S. Phys. ReV. E 1995, 51, 5976–5983. (49) Svishchev, I. M.; Hayward, T. M. J. Chem. Phys. 1999, 111, 9034– 9038. (50) Svishchev, I. M.; Kusalik, P. G. J. Am. Chem. Soc. 1996, 118, 649– 654. (51) Maerzke, K. A.; Siepmann, J. I. J. Phys. Chem. B 2009, 113, 13752– 13760. (52) Zhu, S.-B.; Zhu, J.-B.; Robinson, G. W. Phys. ReV. A 1991, 44, 2602–2608. (53) Kiselev, M.; Heinzinger, K. J. Chem. Phys. 1996, 105, 650–657. (54) Jung, D. H.; Yang, J. H.; Jhon, M. S. Chem. Phys. 1999, 244, 331– 337. (55) Vegiri, A.; Schevkunov, S. V. J. Chem. Phys. 2001, 115, 4175– 4185. (56) Sun, W.; Chen, Z.; Huang, S.-Y. Mol. Simul. 2005, 31, 555–559. (57) Sun, W.; Chen, Z.; Huang, S.-Y. Fluid Phase Equilib. 2005, 238, 20–25. (58) Chen, B.; Potoff, J. J.; Siepmann, J. I. J. Phys. Chem. B 2001, 105, 3093–3104. (59) Stubbs, J. M.; Pottoff, J. J.; Siepmann, J. I. J. Phys. Chem. B 2004, 108, 17596–17605. (60) Jorgensen, W. L.; Chandrasekhar, J.; Madura, J. D.; Impey, R. W.; Klein, M. L. J. Chem. Phys. 1983, 79, 926–935. (61) Martin, M. G.; Siepmann, J. I. J. Chem. Phys. 1998, 102, 2569– 2577.

Maerzke and Siepmann (62) Maitland, G. C.; Rigby, M.; Smith, E. B.; Wakeham, A. Intermolecular Forces: Their Origin and Determination; Pergamon Press: Oxford, U.K., 1987. (63) Panagiotopoulos, A. Z. Mol. Phys. 1987, 61, 813–826. (64) Panagiotopoulos, A. Z.; Quirke, N.; Stapleton, M.; Tildesley, D. J. Mol. Phys. 1988, 63, 527–545. (65) Siepmann, J. I.; Frenkel, D. Mol. Phys. 1992, 75, 59–70. (66) de Pablo, J. J.; Laso, M.; Siepmann, J. I.; Suter, U. W. Mol. Phys. 1993, 80, 55–63. (67) Vlugt, T. J. H.; Martin, M. G.; Smit, B.; Siepmann, J. I.; Krishna, R. Mol. Phys. 1998, 94, 727–733. (68) Martin, M. G.; Siepmann, J. I. J. Phys. Chem. B 1999, 103, 4508– 4517. (69) Wood, W. W.; Parker, F. R. J. Chem. Phys. 1957, 27, 720–733. (70) Ewald, P. Ann. Phys. 1921, 64, 253–287. (71) Madelung, E. Phys. Z. 1918, 19, 524–532. (72) Allen, M. P.; Tildesley, D. J. Computer Simulation of Liquids; Oxford University Press: New York, 1987. (73) Rowlinson, J. S.; Widom, B. Molecular Theory of Capillarity; Oxford University Press: New York, 1989. (74) Rowlinson, J. S.; Swinton, F. L. Liquids and Liquid Mixtures; Butterworth: London, 1982. (75) Atkins, P. W.; de Paula, J. Physical Chemistry, 7th ed.; W. H. Freeman and Company: New York, 2002. (76) Svishchev, I. M.; Kusalik, P. G.; Wang, J.; Boyd, R. J. J. Chem. Phys. 1996, 105, 4742–4750. (77) Berendsen, H. J. C.; Grigera, J. R.; Straatsma, T. P. J. Phys. Chem. 1987, 91, 6269–6271. (78) Chau, P.-L.; Hardwick, A. J. Mol. Phys. 1998, 93, 511–518. (79) Errington, J. R.; Debenedetti, P. G. Nature 2001, 409, 318–321. (80) Suresh, S. J.; Satish, A. V.; Choudhary, A. J. Chem. Phys. 2006, 124, 074506. (81) Suresh, S. J.; Prabhu, A. L.; Arora, A. J. Chem. Phys. 2007, 126, 134502. (82) Stubbs, J. M.; Siepmann, J. I. J. Am. Chem. Soc. 2005, 127, 4722– 4729. (83) Chen, B.; Siepmann, J. I. J. Phys. Chem. B 2006, 110, 3555–3563. (84) Sun, L.; Siepmann, J. I.; Schure, M. R. J. Phys. Chem. B 2006, 110, 10519–10525. (85) Hill, T. L. An Introduction to Statistical Thermodynamics; Dover: Mineola, NY, 1986.

JP9101477