Critical Parameters and Surface Tension of the Room Temperature

Feb 5, 2010 - The surface tension γ of the room temperature ionic liquid ... The estimates of ρc obtained from our simulation data are lower by 13âˆ...
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J. Phys. Chem. C 2010, 114, 3599–3608

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Critical Parameters and Surface Tension of the Room Temperature Ionic Liquid [bmim][PF6]: A Corresponding-States Analysis of Experimental and New Simulation Data Volker C. Weiss,*,† Berit Heggen,‡,§ and Florian Mu¨ller-Plathe‡ Bremen Center for Computational Materials Science, UniVersita¨t Bremen, Am Fallturm 1, 28359 Bremen, Germany, Eduard-Zintl-Institut fu¨r Anorganische und Physikalische Chemie, Technische UniVersita¨t Darmstadt, Petersenstrasse 20, 64287 Darmstadt, Germany, and Max-Planck-Institut fu¨r Kohlenforschung, Kaiser-Wilhelm-Platz 1, 45470 Mu¨lheim an der Ruhr, Germany ReceiVed: NoVember 17, 2009; ReVised Manuscript ReceiVed: January 13, 2010

The surface tension γ of the room temperature ionic liquid 1-butyl-3-methylimidazolium hexafluorophosphate, [bmim][PF6] in short, has been evaluated in the framework of Guggenheim’s corresponding-states analysis using experimental data taken from the literature as well as new simulation data. The critical temperature is estimated to be Tc ) 1100 K ( 100 K from the surface-tension data, which are analyzed in terms of the empirical expressions of Guggenheim and of Eo¨tvo¨s. Adopting this value of Tc, a critical density of Fc ) 0.35 g cm-3 ( 0.04 g cm-3 is obtained from the experimental coexistence curve using the linear-diameter rule. The estimates of Fc obtained from our simulation data are lower by 13-25%. According to Guggenheim, -2/3 the reduced surface tension γred ) cγ/(TcF2/3 ), where Mr denotes the molar mass and c a constant, is c Mr supposed to be a universal function of the reduced temperature T/Tc for all pure fluids. While many nonpolar, weakly polar, and moderately polar liquids adhere to this principle of corresponding states, deviations from Guggenheim’s master curve are seen for strongly polar, hydrogen-bonding substances and for simple inorganic molten salts, such as NaCl and KCl. We find that, despite its ionicity, the data for [bmim][PF6], both from experiment and from simulation, follow the master curve for at most moderately polar liquids, thereby indicating significant differences from the physicochemical properties of simple inorganic molten salts. 1. Introduction Room temperature ionic liquids (RTILs) are molten salts whose normal melting points are below 100 °C. These substances have received a lot of attention within the past decade due to their unusual properties, which open up new applications as solvents and catalysts in synthesis, extraction, and electrochemical processes.1-3 In addition to offering many desirable features, such as chemical stability and low vapor pressure, their properties can be varied to meet exactly the needs of the process of interest. This tailoring of properties can be achieved by combining different cations and anions as well as by fine-tuning the respective ions, e.g., by adjusting the length of an alkyl side chain on one of the ions. Due to the vast number of different RTILs that can be made in this way and the impossibility to measure all of their different properties, it is of fundamental importance to study not only the individual details of each compound but also the general behavior of this class of substances. The corresponding-states idea has proved to be of great help in developing a unifying view of the thermodynamic properties of different fluids.4 Moreover, any deviation from the corresponding-states behavior of simple fluids yields valuable information regarding the intermolecular forces that determine the properties of a specific substance4 and, thereby, augments our general understanding of the behavior of fluids. In most corresponding-states approaches, critical parameters play a central role. Before analyses of this kind can be applied to * Corresponding author. Phone: +49 421 218 7762. Fax: +49 421 218 4764. E-mail: [email protected]. † Universita¨t Bremen. ‡ Technische Universita¨t Darmstadt. § Max-Planck-Institut fu¨r Kohlenforschung.

RTILs, their respective liquid-vapor critical points need to be located. The difficulties arising in this task will, therefore, have to be addressed first. In general, pure ionic substances tend to be solids at ambient temperatures due to the strong Coulomb forces acting among the charged entities. While neutral molecules interact via dispersion (London) forces and, in case the molecules possess a permanent dipole moment, polar (Keesom and Debye) forces, the strong Coulomb forces in ionic systems are expected to dominate over the before-mentioned types of interactions.5 Just like other compounds, however, ionic substances may be present in the liquid state, but the melting points of simple inorganic salts, such as alkali halides, may be located well above 800 K under atmospheric pressure.6,7 In more complex salts, chemical decomposition of the compound may occur before the melting point is reached. This is the case for many, if not all, RTILs which have been described recently.6,8-11 For thermally stable compounds, the upper limit to the temperature range in which a liquid phase may exist is set by the critical temperature, which is estimated to be above 3000 K for alkali halides.7,12-15 Accurate measurements of the physical properties of molten salts near their critical points are, therefore, extremely difficult, if not impossible. In view of their lower melting points compared to inorganic salts, RTILs are expected to have lower critical temperatures. The near-critical region would, therefore, be more readily accessible if decomposition did not take place. Designing thermally stable RTILs is, however, very challenging due to the conditions that are imposed by requiring the liquid phase to be stable at room temperature. Low-melting salts have been known since 1914 when Walden described the synthesis and properties of ethylammonium nitrate, which melts at 12 °C.1 The general strategy of designing RTILs

10.1021/jp9109282  2010 American Chemical Society Published on Web 02/05/2010

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Figure 1. Graphical representation of the molecular structure of 1-butyl-3-methylimidazolium hexafluorophosphate ([bmim][PF6]). The unlabeled whitish spheres represent hydrogen atoms.

is to destabilize the crystalline phase with respect to the melt by introducing bulky ions1 and by reducing the symmetry of the particles.16 The large separation of charges weakens the Coulomb interaction among ions, while a low symmetry of the particles hampers efficient packing in the crystal. Both goals are achieved by introducing alkyl chains on the ions, possibly in an asymmetric substitution pattern, and by using complex ions, such as BF4- or PF6-. The covalent bonds thus introduced make RTILs susceptible to thermal decomposition at temperatures well below their critical points. The chemical diversity which is opened up by combining different cations and anions and by modifying side chains allows one to tailor the properties of the RTIL according to one’s needs but, at the same time, prohibits the accurate measurement of all relevant thermodynamic properties for each salt. Corresponding-states approaches, which focus on the generic properties of a class of substances rather than their specific differences, may, therefore, prove particularly valuable in predicting and correlating thermodynamic data of RTILs. It is essential to be able to obtain the critical parameters of the substance of interest, as they provide a natural scale for the respective state variable to be used in the corresponding-states analysis. Due to the thermal instability of the RTILs that are in use today at elevated temperatures,6,8-11 actual measurements in the critical region of the pure fluid are not possible; instead, suitable extrapolation schemes for the data taken at lower temperatures have to be applied. Rebelo et al.6 described a method to estimate the critical temperature for a number of RTILs, among them 1-butyl-3-methylimidazolium hexafluorophosphate (abbreviated as [bmim][PF6]), on which we focus in this work and the molecular structure of which is shown in Figure 1. The approach of Rebelo et al. uses the empirical expressions proposed by Guggenheim4 and by Eo¨tvo¨s17 to describe the behavior of the liquid-vapor interfacial tension γ as a function of temperature T. Both approaches involve the critical temperature Tc, at which γ vanishes. An estimate of Tc can therefore be obtained by extrapolating γ(T) to the temperature at which γ ) 0 if surface-tension data at different temperatures are available. For simple fluids, both expressions work very well and give accurate estimates of Tc,6 even if the experimental data to be extrapolated are taken at temperatures far from Tc. On the other hand, they are known to fail for hydrogen-bonding substances, such as water and short-chain alcohols. These substances exhibit an inflection point in γ(T),18-20 the occurrence of which is captured neither in Guggenheim’s nor in Eo¨tvo¨s’ approach. The direct application of either expression to determine the critical temperature of RTILs is, therefore, hampered by two problems.

Weiss et al. First, it is by no means certain that the surface tension of an RTIL as a function of temperature does not exhibit an inflection point and can be described by the equations valid for simple fluids. Indeed, simulations of simple models of ionic systems, such as the restricted primitive model (RPM), which consists of charged hard spheres, and closely related models, indicate that there is such an inflection point for purely ionic systems.21,22 Whether or not it occurs also for simple real molten salts, such as NaCl or KCl, is yet unknown due to the problem of carrying out reliable surface-tension measurements at sufficiently high temperatures. Second, due to the high critical temperatures of salts or the decomposition of the ionic substance, experimental data can only be obtained at relatively low reduced temperatures, T/Tc, which means that the extrapolation to the temperature at which the surface tension vanishes, the critical one, is more error-prone even if there is no inflection point in γ(T), the occurrence of which would invalidate the approach altogether. The second problem is therefore not a matter of principle but concerns the accuracy of the extrapolation to γ ) 0, by means of which Tc is determined. The decomposition of [bmim][PF6] is reported to begin near 400 K;8,9,11 the estimate of the critical temperature by Rebelo et al.6 is 1100-1180 K, making T/Tc ≈ 0.36 the highest reduced temperature at which reliable measurements of γ can be expected. Huddleston et al.10 reported a decomposition temperature as low as 350 K, which would restrict the accessible range of reduced temperatures even further. At this point, molecular simulations based on a reliable force field can be most helpful. In a classical force field, the bonds between atoms do not break, so the compound cannot decompose, and the behavior at higher temperatures may be studied directly. This procedure is, of course, only meaningful if a reliable force field exists, which has to be capable of describing the properties of the fluid at ambient temperatures. For [bmim][PF6], this is fortunately the case.23,24 Being one of the most-studied RTILs, there is not only a considerable body of experimentally determined thermodynamic properties (within the accessible temperature range 273-393 K) but also a number of well-tested force fields for molecular simulations,23,25-28 of which the one of Bhargava and Balasubramanian23 is currently favored by us. The application of a force field that has been developed and validated using data for various thermophysical properties in the experimentally accessible region at higher temperatures is expected to be more reliable than a direct extrapolation of the data for just one property, such as the surface tension. The present study serves four purposes: (1) We provide simulation data of the coexistence curve and the surface tension extending to much higher temperatures (400-800 K) than previously studied.23,24,29 An important question to be addressed in this context is whether or not there is an inflection point in the function γ(T). An affirmative answer would invalidate any attempt to deduce Tc from Guggenheim’s or Eo¨tvo¨s’ expression for γ(T). (2) Once the absence of an inflection point is confirmed, an estimate of Tc can be obtained based on the available experimental surface-tension data and on the new simulation data for a wider range of temperatures by subjecting these data to the analyses of Guggenheim and of Eo¨tvo¨s. By combining experimental and simulation data, a more reliable estimate of the critical temperature of [bmim][PF6] is obtained, reducing the uncertainty of this quantity. (3) Using the refined value of Tc, the experimental and new simulation data for the density of the liquid phase are analyzed in terms of the lineardiameter rule for the coexistence curve, and an estimate of the critical density Fc is obtained as well. Knowledge of the critical

Critical Parameters and Surface Tension of [bmim][PF6]

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parameters Tc and Fc enables us to analyze the surface-tension data in terms of Guggenheim’s corresponding-states approach, which correlates the reduced surface tension γred with the reduced temperature T/Tc; γred is defined according to

γred ) c

γM2/3 r

(1)

TcF2/3 c

-1

where c is a constant that equals 1/(10 ≈ 1.016 J K mol2/3 if γ is measured in mN m-1, the molar mass Mr in g mol-1, Tc in K, and Fc in g cm-3; here, kB denotes Boltzmann’s constant and NA Avogadro’s number. For simple fluids, which include nonpolar, weakly polar, and even moderately polar fluids, γred(T/Tc) is found to be a nearly universal function.4 Significant deviations from this master curve are seen for hydrogen-bonding fluids and for simple inorganic ionic fluids (molten salts).19,20 (4) On the basis of the corresponding-states analysis of the surface tension, we investigate whether the behavior of γred for [bmim][PF6] is similar to that of simple inorganic salts (NaCl and KCl), to that of weakly and moderately polar substances (such a “regular” behavior would be most useful for correlation purposes),19,20 or, perhaps, entirely different from either class of fluids. These four tasks have to be addressed in the logical sequence outlined above. Therefore, the remainder of the paper is organized as follows. In the next section, the general methodology of obtaining the desired corresponding-states surface tension γred(T/Tc) from given data of the temperature-dependent surface tension and of the densities of the coexisting phases is explained. In section 3, we give the details of our simulation of the interfacial properties of [bmim][PF6]. Section 4 is dedicated to the specific evaluation of the surface-tension and density data for [bmim][PF6], both for experimental data taken from the literature as well as for our new simulation data, and for the simple inorganic salts NaCl and KCl. In section 5, we compare the results of the corresponding-states surface tension γred obtained for simple fluids, for inorganic salts, and for the RTIL. Section 6 concludes the paper by summarizing the main results and the significance of our findings for the research on RTILs and on the liquid state in general. 7

kBNA2/3)

tension was determined. For Guggenheim’s approach, the surface-tension data alone suffice. It should be mentioned that there are additions and modifications to Eo¨tvo¨s’ formula, a commonly applied one being30

γ(T)[Fl(T)]-2/3 ) b(Tc - T - 6 K)

(4)

which accounts for the fact that the product γF-2/3 is no longer l a linear function of the temperature close to the critical point. Applying the original formula, eq 3, to a given set of surfacetension and density data taken at temperatures reasonably below the critical one would lead to an underestimation of Tc. For the present purpose, the shift of 6 K, however, is insignificant in view of the high critical temperature of RTILs and the large uncertainty associated with its value. The values of the coefficients a and b are immaterial for the task of determining Tc. Once Tc is known, the critical density Fc may be obtained using another empirical rule of thermodynamics, the CailletetMathias “law” of linear diameter. In fact, the diameter of the coexistence curve, Fdia(T) ) (Fl(T) + Fv(T))/2, where Fv denotes the density of the vapor phase, is often found to be a linear function over a very wide temperature range. Very close to Tc, however, there are singularities which cause deviations from the linear-diameter rule,31 but their small effect on the numerical value of Fc is of no concern here. The approximate validity of this simple linear relationship over a wide range of temperatures allows one to get a reasonably accurate estimate of Fc even if the available data of Fl and Fv are taken at temperatures that are far from Tc (in these cases, Fv will usually be very close to zero). To a good approximation, Fc will thus be located on the line Fdia(T), with the result that the function Fdia(T) ) mT + h may be interpreted as a correlation between the estimates of Fc and Tc according to

Fc ) mTc + h

(5)

Equipped with the so-obtained values of Tc and Fc, the corresponding-states surface tension γred(T/Tc) can now be evaluated from γ(T) according to eq 1.

2. General Methodology

3. Simulation Details

In order to evaluate the reduced surface tension γred(T/Tc) defined by Guggenheim (cf. eq 1), estimates of the critical parameters Tc and Fc are needed in addition to the surface-tension data γ(T). In cases in which the critical region cannot be probed experimentally, e.g., because the substance of interest decomposes below Tc or simply because Tc is too high to allow accurate measurements, the critical temperature may be estimated from the surface-tension data directly. Two empirical approaches, which are frequently used,6 are the ones by Guggenheim (Gug)4 and by Eo¨tvo¨s (Eot).17 A given data set γ(T) (originating from experiment or from simulation) is fitted to the following expressions:

Two simulation packages, YASP32 and Gromacs33 (version 3.3), were used to carry out molecular dynamics simulations of [bmim][PF6] at liquid-vapor coexistence and at different temperatures. We employed two different programs to be able to investigate the influence of the way in which the long-range electrostatic interactions are accounted for. While YASP is restricted to the reaction-field (RF) method,34 Gromacs allows one to use the particle-mesh Ewald (PME) summation.35,36 Furthermore, different methods of enforcing the constraints of having fixed bond length in the empirical force field for [bmim][PF6] can be employed; while YASP uses SHAKE,37 LINCS38 was employed within Gromacs. To model the interactions in liquid [bmim][PF6], the force field of Bhargava and Balasubramanian23 was chosen. This model is a refinement of the force field developed by Canongia Lopes et al.,28 which, by readjusting the partial charges, leads to a better agreement of the density, the self-diffusion coefficient, and the surface tension with experimental data23 than the original force field. It is noteworthy in this context that particularly the improvement of dynamical properties was achieved by reducing the ionic charges from (1e to (0.8e. While the original all-

γ(T) ) a(Tc - T)11/9 γ(T)[Fl(T)]-2/3 ) b(Tc - T)

(Gug) (Eot)

(2) (3)

where a and b are constants, while Fl denotes the density of the liquid phase. To apply Eo¨tvo¨s’ rule, the density of the liquid phase is required at the same temperature at which the surface

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TABLE 1: Surface Tension γ, Density of the Liquid Phase Gl, and Density of the Vapor Phase Gv at Different Temperatures T for [bmim][PF6] Obtained Using the YASP Simulation Package and the Conditions Specified in Section 4.2; the Last Column Shows the Simulation Time t at the Respective Temperature T (K) a

300 320a 340a 360a 380a 500 600 700 800 a

γ (mN m-1)

Fl (g cm-3)

Fv (g cm-3)

t (ns)

37.28 ( 2.57 35.59 ( 2.88 34.89 ( 3.87 32.51 ( 3.90 31.80 ( 2.50 21.13 ( 3.57 20.26 ( 0.84 14.53 ( 0.74 13.18 ( 1.95

1.35 1.32 1.30 1.28 1.26 1.14 1.04 0.93 0.82

0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00026 0.001

20 18 11.4 12.8 14 9.8 9.8 12.6 10.4

Taken from Heggen et al.29

TABLE 2: Surface Tension γ, Density of the Liquid Phase Gl, and Density of the Vapor Phase Gv at Different Temperatures T for [bmim][PF6] Obtained Using the Gromacs Simulation Package and the Conditions Specified in Section 4.2; the Last Column Shows the Simulation Time t at the Respective Temperature T (K) 300 340 380 450 600 800 a

a

γ (mN m-1)

Fl (g cm-3)

Fv (g cm-3)

t (ns)

45.55 ( 4.23 33.09 ( 3.48 31.11 ( 5.07 30.78 ( 3.18 20.17 ( 1.56 11.08 ( 2.99

1.35 1.32 1.28 1.22 1.07 0.85

0.00 0.00 0.00 0.00 0.00 0.04

6 8 5 8 11 4

Taken from Heggen et al.29

of 1.2 nm. The reaction-field dielectric constant was fixed at the experimental value for [bmim][PF6] of εr ) 11.4.39 A time step of 2 fs was chosen. The temperature was held constant using a Berendsen thermostat40 with a coupling time of 0.2 ps. The simulated systems comprised 512 ion pairs. These systems were obtained by doubling a system consisting of 256 ion pairs, placed in a cubic box of side length L ) 4.46 nm and pre-equilibrated at 300 K and 1 atm under NPT conditions, in the z-direction. Two liquid-vapor interfaces were created by increasing the length of the box in the z-direction (i.e., by adding a vacuum part of the same size) and applying periodic boundary conditions; the lateral dimensions of the simulation box were therefore Lx ) Ly ) 4.46 nm, while the length of the box in the direction perpendicular to the interfaces was Lz ) 17.84 nm. The simulations were carried out under NVT conditions to ensure two-phase coexistence at the chosen overall density. At 300 K, there is a vapor (vacuum) layer of 9 nm thickness intervening the slabs of liquid; at 800 K, the thickness of the vapor layer between the liquid slabs has shrunk to 4.5 nm. At all temperatures, an equilibration period of 1.5 ns for the simulations using YASP and of 1 ns for Gromacs runs was followed by a production period of varying lengths; the simulation time spent in each case is listed in Table 1 for the simulations using YASP and in Table 2 for those using Gromacs. The surface tension γ was calculated as a time (ensemble) average of the instantaneous surface tension γ(t), which results from the difference between the normal and lateral components of the pressure tensor via the Kirkwood-Buff formula:18

γ(t) ) 23

atom force field of Bhargava and Balasubramanian treats all bonds as flexible by introducing harmonic potentials between the atoms, we keep the lengths of all bonds constant by imposing holonomic bond constraints. Doing so allows us to use a single and relatively large time step and helps us to save computation time by neglecting the vibrational details of bond stretching. We expect no significant change of the surface tension by this modification of the force field; this anticipated indifference is indeed found in simulations at 300 K using Gromacs, but within YASP, a small shift of the surface tension toward smaller values is seen.29 The reason for this behavior is believed to be of technical nature and may be related to the way the virial is calculated.29 The magnitude of the shift (about 10-20% at 300 K) is, however, no cause for concern in the present context; furthermore, our present simulation data indicate that Gromacs and YASP results agree to within their uncertainties at higher temperatures. In addition to constraining all bond lengths, our simulations apply two modifications compared to the treatment of Bhargava and Balasubramanian:23 First, a slightly different cutoff distance for the long-range interactions is adopted (1.2 vs 1.3 nm, see below), and second, the intramolecular 1-4 nonbonded interactions are scaled by a factor of 1/2, as is usual in OPLS-AA force fields, while Bhargava and Balasubramanian neglect this term completely. Possible consequences of these minor changes are discussed in section 4.2. To evaluate the Lennard-Jones interactions between nonbonded atoms, standard Lorentz-Berthelot mixing rules were applied. The long-range cutoff for Lennard-Jones interactions was 1.2 nm. Long-range electrostatic interactions were accounted for by the reaction-field method34 within YASP using a cutoff range of 1.2 nm and by the particle-mesh Ewald summation35,36 within Gromacs employing a real-space cutoff

1 n

∫0L

z

[

]

Pzz(z, t) -

Pxx(z, t) + Pyy(z, t) dz 2

(6)

〈

〉

(7)

γ)

Lz Pxx + Pyy Pzz n 2

where n is the number of interfaces (n ) 2 in our case), Lz is the box length in the z-direction, i.e., perpendicular to the interfaces, while Pxx, Pyy, and Pzz are the diagonal components of the pressure tensor, and the angular brackets denote an ensemble average. Quantities such as short-range and long-range contributions to the energy, components of the pressure tensor, and/or boxsize changes were monitored every 1000th time step within YASP and every 100th time step within Gromacs. The respective simulation times for the production runs are given in Tables 1 and 2. The quoted uncertainty of the surface tension is the standard deviation of the instantaneous values γ(t) from the mean within a time series. Typically, such a time series comprises on the order of 10 000 data points. In cases in which these uncertainties appear unrealistically small (below 0.5 mN m-1), we have taken the liberty of doubling the error margin. For relatively simple model fluids, the surface tension, as obtained from simulations, has been found to be an oscillatory function of the surface area in the simulated systems.41 For very small surface areas, even a negative surface tension may be obtained. For Lennard-Jones and square-well fluids, however, Orea et al. showed that the oscillatory behavior is damped rapidly and the surface tension converges to its asymptotic value for Lx and Ly larger than 7 reduced units of length (molecular diameters).41 Our ionic diameters are roughly 0.6 nm (anion) and 0.7 nm (cation), so the lateral dimension of the simulated

Critical Parameters and Surface Tension of [bmim][PF6]

J. Phys. Chem. C, Vol. 114, No. 8, 2010 3603 TABLE 3: Estimates of the Critical Temperature Tc from Different Data Sets for the Surface Tension of [bmim][PF6] Based on Guggenheim’s (Gug) and Eo¨tvo¨s’ (Eot) Methods data set a

Freire et al. Ghatee and Zolghadrc Kilaru et al.d Pereiro et al.e lin. fit, eq 8f Rebelo et al.g YASPf Gromacsf Figure 2. Surface tension γ of [bmim][PF6] as a function of temperature T in the experimentally accessible range. Shown are four sets of experimental data: Ghatee and Zolghadr43 (filled circles), Kilaru et al.42 (filled squares), Freire et al.45 (filled triangles), and Pereiro et al.44 (filled diamonds). The continuous line represents our linear fit to the four experimental data sets, eq 8. The simulation data obtained in this temperature range using YASP (open squares) and Gromacs (open diamonds) are also shown; in addition, the simulation datum of Bhargava and Balasubramanian23 at 300 K (open circle) is indicated.

surface is about 7 times larger than the mean particle diameter, and a reliable estimate of the surface tension can be expected. 4. Evaluation of Surface-Tension and Density Data In this section, we first analyze the surface-tension data and the densities of the liquid phase which were obtained from actual experimental measurements on [bmim][PF6] in the accessible temperature range, i.e., below the decomposition temperature of 400 K. To extend this range to temperatures which are closer to the (hypothetical) critical one, we then evaluate the corresponding data obtained from our simulations. In order to compare the results for the RTIL with those for simple inorganic salts, we repeat the analysis using experimental data for NaCl and for KCl. 4.1. Experimental Data for [bmim][PF6]. First, the critical temperature is estimated from the available surface-tension data using Guggenheim’s and Eo¨tvo¨s’ methods.4,17 We evaluated four data sets,42-45 which cover the entire range of temperatures in which experiments can be carried out. For a given temperature, the data from these four sets agree with each other to within 15%. Other sources of experimental data, which we did not consider in our analysis, quote values of the surface tension that are in the same range.46,47 Of the four data sets we selected for our analysis, only the one of Ghatee and Zolghadr43 was measured at liquid-vapor coexistence. The other three sets were obtained at atmospheric pressure.42,44,45 Due to the low compressibility of the liquid,48,49 we consider these different experimental conditions to be a source of only minor errors. Uncertainties arising from different water contents of the samples10,45 or different methods to measure the surface tension are more likely to be relevant in this context. Differences among the four data sets are clearly noticeable in Figure 2, but they are of no relevance to the main conclusion of our study. An uncertainty of up to 20% in the values of the surface tension (amounting to an absolute uncertainty of about 8 mN m-1) is tolerable for our purpose as long as the temperature dependence of γ is captured correctly. We compromised by averaging the data of the four sets to obtain a representative correlation of γ(T). To this end, each of the four sets was subjected to a linear regression and the mean values of slope and intercept, respectively, were calculated. The result of this procedure is given by

γ ) 65.87 mN m-1 - 0.06819 mN m-1 K-1T

(8)

TGug (K) c

TEot c (K)

1153 ( 150 1092 ( 135 1193 ( 160 1015 ( 130 1104 ( 140 1102 1092 ( 85 1035 ( 70

(1269 ( 130)b (1168 ( 80)b (1335 ( 140)b 1070 ( 130 1186 ( 75 1187 1186 ( 140 1077 ( 270

a Reference 45. b Values in parentheses were obtained using density data for the liquid phase following from our correlation, eq 9; i.e., the densities were not measured in the same sample as the surface tension. c Reference 43. d Reference 42. e Reference 44. f This work. g Reference 6.

This fit reproduces all experimental data to within 3 mN m-1 and is shown as a continuous line in Figure 2. All four data sets and the representative linear correlation, eq 8, were subjected to the analyses of Guggenheim and of Eo¨tvo¨s to get an estimate of Tc. The results are compiled in Table 3. For the Guggenheim correlation, the quoted uncertainties of Tc were estimated from varying the value of the exponent, which is 11/9 in eq 2, between 1 (lower limit of “regular” behavior far from Tc) and 1.26 (upper limit of asymptotic behavior near Tc);18 in this sense, the quoted uncertainty is a worst-case estimate provided that the function γ(T) does not display an inflection point (otherwise, the error margin will be even larger but difficult to quantify). For the Eo¨tvo¨s approach, the error bars were obtained from the combined statistical uncertainties of the slope and the intercept in the linear regression and from the absolute uncertainties of the measured surface-tension data. Earlier attempts to obtain estimates of Tc from similar analyses of experimental data did not specify the uncertainties6 or quoted just the statistical uncertainty of the fit,45 but the overall uncertainty must be suspected to be of similar magnitude as ours. From the extrapolated data of Tc in Table 3, we conclude that the best estimate of the critical temperature of [bmim][PF6] is Tc ) 1100 ( 100 K. This range includes all individual estimates in Table 3 with only two exceptions. In general, we favor Guggenheim’s expression over the one of Eo¨tvo¨s because the former is known to yield accurate estimates of Tc even from very few data points for the surface tension that may be taken far from the critical point and, furthermore, it is capable of describing also the behavior in the near-critical region, in which Eo¨tvo¨s’ approach fails completely. Among the experimental sets, we tend to favor the data by Ghatee and Zolghadr,43 which, unlike the other sets, were taken at liquid-vapor equilibrium. The linear fit to all four experimental sets, eq 8, yields virtually the same estimate of Tc within Guggenheim’s approach and agrees also well with the datum of Rebelo et al.6 and with the result from our YASP simulation set, which includes more data points than the Gromacs set. In view of the combined evidence mentioned above, we regard Tc ) 1100 K as the best estimate and consider the assignment of an uncertainty of 100 K to be adequate. For the densities of the liquid phase, there are several data sets available,8,9,44,50 all of which were measured at atmospheric pressure, i.e., not at liquid-vapor coexistence. Discrepancies caused by these different conditions are thought to be small due to the low compressibility of the liquid.48,49 [Furthermore, the corresponding data for NaCl and KCl, which we evaluate in section 4.3, were measured and analyzed by Kirshenbaum et

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Figure 3. Density of the liquid phase, Fl, of [bmim][PF6] as a function of temperature T in the experimentally accessible range. The filled symbols denote experimental data taken at atmospheric pressure: Jacquemin et al.9 (circles), Pereiro et al.44 (triangles up), Tokuda et al.8 (squares), and Tariq et al.50 (triangles down). In addition, our fit to the available experimental data, eq 9, is represented by the continuous line. The open symbols are results for the density of the liquid phase at liquid-vapor coexistence obtained from simulations using YASP (open squares) and Gromacs (open diamonds) and for Fl at zero pressure (open circles) obtained from simulations by Bhargava and Balasubramanian;23 the dashed lines connecting the data points are just meant to guide the eye.

al.12 in the same way.] There is good agreement among the experimental data sets (cf. Figure 3). We obtained a representative equation by performing a linear regression to the data of Jacquemin et al.9 and of Pereiro et al.,44 according to which

Fl ) 1.6162 g cm-3 - 8.3636 × 10-4 g cm-3 K-1T

(9) Assuming the vapor density to be essentially zero for the temperature range considered here, an equation for the diameter of the coexistence curve can be obtained by approximating Fdia ≈ Fl/2. Neglecting any effects of critical fluctuations, the critical density is located on the coexistence-curve diameter, i.e., Fc ) Fdia(Tc). Using the recommended value of Tc ) 1100 ( 100 K (cf. Table 3), we obtain Fc ) 0.35 ( 0.04 g cm-3. Equipped with these estimates of the critical parameters, the surface-tension data can now be analyzed within Guggenheim’s corresponding-states approach for this quantity, eq 1. The results will be discussed in the next section. 4.2. Simulation Data for [bmim][PF6]. A system containing two liquid-vapor interfaces of [bmim][PF6] was simulated at different temperatures to obtain the surface tension and the densities of the coexisting phases as described in section 3. The temperature range in which simulations were performed (300-800 K) extends well beyond the thermal stability limit of the real substance (∼400 K), which enables us to get a more reliable estimate of the location of the hypothetical critical point. Table 1 summarizes the results obtained using the simulation package YASP32 in conjunction with the reaction-field (RF) method34 to account for the long-range part of the electrostatic interactions and SHAKE37 to impose constant bond lengths, while Table 2 contains those found from simulations using Gromacs,33 the particle-mesh Ewald (PME) summation,35,36 and the LINCS algorithm38 to keep the bond distances fixed. Before deducing the critical parameters and evaluating the reduced surface tension γred, we will briefly compare the absolute values of γ(T) and Fl(T) obtained within the two sets of our simulations to experimental data and to the simulation data of Bhargava and Balasubramanian23 who used essentially the same force field. It must be stressed that, although minor differences

Figure 4. Surface tension γ of [bmim][PF6] in the temperature range 300 K e T e 800 K. Shown are the simulation data obtained using YASP (open squares connected by a dashed line) and Gromacs (open diamonds connected by a dashed line). For comparison, four sets of experimental data are displayed: Ghatee and Zolghadr43 (continuous line), Kilaru et al.42 (dotted line), Freire et al.45 (dot-dashed line), and Pereiro et al.44 (long-dashed line). The real substance decomposes beyond 400 K.

between the two sets and small discrepancies to the experimental data as well as to the values reported by Bhargava and Balasubramanian exist, they are on the order of the experimental uncertainties and, more importantly, of no relevance to the main conclusion of this study, which concerns the behavior of the reduced surface tension γred. In Figure 4, the results for the surface tension in the temperature range 300-800 K are displayed. The datum at 300 K obtained within YASP is seen to be lower than the one resulting from Gromacs runs, but agreement (to within the indicated uncertainty) between the two sets is found at higher temperatures. The data of both sets are somewhat scattered, but there is no indication of an inflection point of γ(T). The experimental data sets are displayed, too, but a comparison is difficult on the scale of Figure 4. For this purpose, we return to Figure 2, which focuses on the surface tension in the temperature range 280-400 K. The experimental data (filled symbols), of which only the set of Ghatee and Zolghadr (filled circles) was measured at liquid-vapor coexistence, while the other sets were obtained at atmospheric pressure, agree on the temperature dependence of the surface tension, but the absolute values may differ by up to 6 mN/m (or 15%) at 300 K. The surface tension obtained within Gromacs (open diamonds) at 300 K is found within the range of the experimental data and agrees well with the value of 47 mN/m quoted by Bhargava and Balasubramanian23 (open circle). The surface-tension data calculated within Gromacs at higher temperatures and those obtained within YASP (open squares) are somewhat lower, but the temperature dependence of γ is generally well reproduced. In spite of the discrepancy at 300 K, it is noteworthy that the predictions of YASP and Gromacs are very similar at higher temperatures. The two programs differ in the way in which the components of the pressure tensor are computed. While YASP uses an atomic virial, Gromacs applies a molecular one. The latter definition appears to produce more stable pressure anisotropies (and, therefore, also surface tensions) when used in conjunction with constraints, such as fixed bond lengths. For unconstrained force fields, both programs have been found to yield very similar results. A detailed discussion is given by Heggen et al.29 In Figure 5, the simulation results for the density of the liquid phase, Fl, obtained from YASP and Gromacs runs in the temperature range 300-800 K are displayed along with the corresponding data of Bhargava and Balasubramanian.23 It is seen that the temperature dependence of Fl is similar in all three cases, but the YASP data are systematically lower than the ones

Critical Parameters and Surface Tension of [bmim][PF6]

J. Phys. Chem. C, Vol. 114, No. 8, 2010 3605 TABLE 4: Estimates of the Critical Parameters Tc and Gc and Coefficients of the Correlation for the Diameter of the Coexistence Curve, Gdia ) h + mT, for the Ionic Fluids

Figure 5. Density of the liquid phase, Fl, of [bmim][PF6] in the temperature range 300 K e T e 800 K. The open symbols are results for Fl at liquid-vapor coexistence obtained from simulations using YASP (open squares) and Gromacs (open diamonds) and at zero pressure (open circles) obtained from simulations by Bhargava and Balasubramanian;23 the dashed lines connecting the data points are meant to guide the eye. The continuous line represents our correlation of the experimentally measured liquid densities at atmospheric pressure, cf. eq 9; differences among the experimental sets8,9,44,50 cannot be seen on the scale of the figure. The real substance is thermally unstable beyond 400 K.

obtained within Gromacs, which in turn are below the values of Bhargava and Balasubramanian. The linear fit to the experimentally measured densities of the liquid phase, given by eq 9, is indicated by the continuous curve in the range 273-393 K; the experimental data are found in between the Gromacs results and the ones of Bhargava and Balasubramanian. Discrepancies among the different experimental data sets, from which eq 9 was obtained, cannot be seen on the scale of Figure 5. To facilitate a comparison with experimental data, we return to Figure 3, in which the densities of the liquid phase are displayed in the temperature range 270-400 K. All experimental data shown (filled symbols) were measured at atmospheric pressure, while our simulations using YASP (open squares) and Gromacs (open diamonds), respectively, were carried out at liquid-vapor coexistence; Bhargava and Balasubramanian (open circles) simulated their systems at zero pressure,23 which, in the specified temperature range, is virtually the same condition as two-phase coexistence because the vapor pressure is extremely low. Even an increase of the pressure to 1 atm is not expected to increase the density significantly due to the low compressibility of the ionic liquid.48,49 All data shown in Figure 3 should, therefore, be comparable. The experimental data agree with each other; the line represents the linear fit, eq 9. The values reported by Bhargava and Balasubramanian23 are seen to overestimate the density by about 1.5%. The results obtained within Gromacs, in turn, underestimate the density of the real fluid by about the same margin. Within YASP, the densities are still lower and approximately 3% below the experimental values. The comparatively low density of the liquid phase may, in part, account for the low surface tension obtained within the YASP simulations. In Figure 3 and, even more so, in Figure 5 (which covers the temperature range from 300 to 800 K), it can be seen that, despite the small differences in the absolute values of Fl, the simulations provide reasonable descriptions and, at elevated temperatures, predictions of the density in the liquid phase. To each data set for γ(T), we applied the expressions of Guggenheim, eq 2, and of Eo¨tvo¨s, eq 3, to estimate Tc; the results are listed in Table 3. As for the experimental data, we find that Tc ) 1100 ( 100 K includes all estimates of Tc obtained from our simulations. An almost perfectly linear coexistencecurve diameter (not shown) was found from the density data

substance

Tc (K)

Fc (g cm-3)

h (g cm-3)

m (g cm-3 K-1)

NaCl KCl [bmim][PF6]a exptl YASP Gromacs

3400 ( 200 3200 ( 200

0.222 ( 0.048 0.175 ( 0.054

1.0293 1.0308

-2.3736 × 10-4 -2.6741 × 10-4

1100 ( 100 1100 ( 100 1100 ( 100

0.35 ( 0.04 0.26 ( 0.05 0.30 ( 0.05

0.8081 0.8280 0.8169

-4.1818 × 10-4 -5.1881 × 10-4 -4.6559 × 10-4

a For this substance, three different sets of parameters were obtained from experimental data (exptl) and from simulations using YASP and Gromacs, respectively.

Figure 6. Corresponding-states surface tension γred as a function of the reduced temperature T/Tc for simple fluids, simple molten salts, and the RTIL [bmim][PF6]. Data for argon (dashed curve) and for CHClF2 (continuous curve) indicate the range for nonpolar to moderately polar fluids. Simple inorganic molten salts are represented by NaCl (filled triangles left) and KCl (filled triangles right). For [bmim][PF6], experimental data (filled circles) as well as simulated data using YASP (open squares) and Gromacs (open diamonds) are shown.

obtained within the simulations using YASP and Gromacs, which encouraged us to determine the critical density using the linear-diameter rule; the results of this procedure are compiled in Table 4. While the experimental results of section 4.1 and the two sets of simulation data essentially agree on a best estimate for Tc to within the indicated uncertainty, the different density data (Figures 3 and 5) force us to use three different estimates for the critical density (cf. Table 4). The value of Fc corresponding to each set of surface-tension data (experimental, simulated using YASP, simulated using Gromacs) was adopted in the corresponding-states analysis shown in Figure 6. The error bars indicated in Figure 6 imply that the uncertainties of the critical parameters Tc and Fc as well as of the absolute values of the surface tension γ do not affect the main conclusion of our study even in a worst-case scenario. A detailed discussion of Figure 6 is postponed to section 5. 4.3. Simple Inorganic Salts: NaCl and KCl. The normal melting points of sodium chloride and potassium chloride are at 1073 and 1063 K.12 Reliable experimental surface-tension data are only available in a temperature range from the melting point to about 200 K above it.7,51-53 Density data for the liquid phase were measured between the melting point and the normal boiling point of 1738 K for NaCl and of 1680 K for KCl.12 Kirshenbaum et al.12 extrapolated these density data with the help of the vapor densities measured by Barton and Bloom54 to obtain estimates of the critical temperature and the critical density using the linear-diameter rule. Their critical parameters are Tc ) 3400 ( 200 K and Fc ) 0.22 ( 0.05 g cm-3 for NaCl and Tc ) 3200 ( 200 K and Fc ) 0.175 ( 0.05 g cm-3 for KCl.12 We have repeated the analysis of Kirshenbaum et al.

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and, using the linear-diameter rule and the density data given by these authors,12 obtained a correlation of Tc and Fc, which reads Fc ) h + mTc. The values of the coefficients h and m are listed in Table 4 together with our estimates of the critical parameters, which confirm the values deduced previously; furthermore, the range of these values includes the estimates of other authors.7,14,15 The experimentally measured surface-tension data can be represented by straight lines in all cases,7,51 but the temperature range in which these values were obtained is narrow and, what is more, very far from the critical temperature; data are available only in the reduced-temperature range of 0.3 < T/Tc < 0.4. If one applies Guggenheim’s or Eo¨tvo¨s’ approach to estimate the critical temperature from the surface-tension data directly, values between 2600 and 3100 K are obtained, which are somewhat lower than the values of Tc deduced from extrapolation of the densities of the coexisting phases (cf. Table 4). It is, however, not known if the empirical correlations for the surface tension of simple fluids work for molten salts as well. Simulations of simple models of ionic fluids indicate an inflection point of the function γ(T),22,21 the presence of which makes Guggenheim’s and Eo¨tvo¨s’ approaches as inapplicable to these ionic fluids as to associating polar fluids, such as water or alcohol.6,19 In these cases, it is important to have an independent method of determining the critical parameters which does not rely on surface-tension data. Irrespective of the exact functional form of γ(T), the corresponding-states surface tension γred as defined by Guggenheim (cf. eq 1) can be evaluated. The results for NaCl and KCl are shown in Figure 6 and will be discussed in the context of the data for [bmim][PF6] in the next section. 5. Results and Discussion The main results of our study are summarized in Table 4 and in Figure 6. Estimates of the critical parameters Tc and Fc of the inorganic molten salts NaCl and KCl and the room temperature ionic liquid [bmim][PF6] were obtained from correlating experimental data for the surface tension and the liquid density as well as, in the case of [bmim][PF6], from molecular dynamics simulations. The best estimates resulting from our analysis are listed in Table 4 together with their respective uncertainties. The error margin quoted for Fc arises almost exclusively from the uncertainty in Tc; the experimental uncertainty of the measured liquid density is comparatively small. The coefficients of the linear regression of the coexistence-curve diameter Fdia ) h + mT are given in Table 4, too, because the critical point is assumed to lie (approximately) on this line; therefore, the coefficients h and m also establish the connection between Tc and Fc in this simple approach. Figure 6 shows the corresponding-states surface tension γred defined by eq 1 for simple inorganic salts (NaCl and KCl), for nonpolar (argon) and weakly polar fluids (CHClF2),55,56 and for the RTIL [bmim][PF6]. For the latter, three different sets, one for the experimental data and two for the simulation data obtained using YASP and Gromacs, respectively, are displayed. The data for the class of simple fluids can be found in a very narrow region which represents the master curve γred(T/Tc) originally found for nonpolar fluids by Guggenheim.4,19 To indicate the spread of the curves for simple fluids, two lines are shown in Figure 6, namely, for the nonpolar fluid argon (dashed line) and for the moderately polar fluid CHClF2 (continuous line). The latter is more typical of the vast majority of weakly and moderately polar fluids, while the former, being entirely nonpolar and composed of spherical particles, can be

Weiss et al. regarded as setting a kind of “lower limit” to γred at a given T/Tc for simple fluids.19 The experimental data for the inorganic salts NaCl (triangles left) and KCl (triangles right) are found at very much lower values of γred.19 Due to the high critical temperatures, only data points for T/Tc e 0.4 are usually available. This range of temperatures is insufficient to decide whether or not there is an inflection point in the function γred(T/Tc) (or, equivalently, in γ(T)) for these salts. Simulation results for simple models of ionic fluids, which incorporate only Coulomb interactions and short-range repulsions among spherical particles, indicate the presence of such an inflection point at 0.8 e T/Tc e 0.85.22,21 Guggenheim’s method to estimate the critical temperature from surface-tension data, eq 2, therefore, cannot be trusted for this class of substances. For the RTIL [bmim][PF6], the experimental data (filled circles) as well as simulation data obtained using YASP (open squares) and Gromacs (open diamonds), respectively, coincide with the master curve for at most moderately polar fluids to within their error margins. The main source of uncertainty is the critical temperature, the error of which propagates into the estimate of the critical density. The uncertainty of the absolute value of γ turns out to be of minor importance for the error in γred; this quantity was assumed to be 3 mN m-1 for all experimental surface-tension values analyzed here and should, thus, be comparable to the uncertainty of the values obtained from simulations. It is important to note that there is no overlap of the range of data for the RTIL [bmim][PF6] and that for the inorganic salts within their respective uncertainties! The RTIL rather behaves like a weakly or moderately polar fluid, which is very much in line with the experimentally determined dielectric permittivity of εr ) 11.4 at T ) 298 K39 and the simulated value of εr ≈ 9.5 at T ) 300 K.57 The ionic character is apparently not very pronounced in the RTIL, which may be explained by the increased importance of dispersion interactions compared to electrostatic interactions between ions.16,47,58 After all, bulky ions were chosen to lower the melting point down to room temperature for the RTILs; this is mainly achieved by weakening the electrostatic interactions due to the large spatial separation of the ionic charges. The polarity, as measured by the dielectric permittivity, is, however, not the only property influencing the corresponding-states surface tension γred. In order to observe significant deviations from the master curve, hydrogen bonds in addition to large dipole moments are required.19 The point of inflection in the function γ(T) that is observed for water and light alcohols is attributed to association in the liquid phase due to the formation of intermolecular hydrogen bonds.18-20 Typically, this inflection point is found at a reduced temperature of 0.8 < T/Tc < 0.9;19 this range is not covered by our simulation data for [bmim][PF6]. Nevertheless, from the available simulations, it can already be concluded that there is no indication of an inflection point. In all cases in which an inflection point is seen, the curvature of γ(T) is negative at low temperatures and turns positive at high temperatures (close to Tc). Our simulation data display a positive curvature already at low reduced temperatures, making the appearance of an inflection point at higher temperatures very unlikely. In addition to justifying the application of Guggenheim’s and Eo¨tvo¨s’ empirical expressions in the effort to determine Tc, the absence of an inflection point lends support to the view that the tendency to form strong hydrogen bonds is not very pronounced in this, and possibly also other, imidazolium-based

Critical Parameters and Surface Tension of [bmim][PF6] RTILs.16,57,59 Strong hydrogen bonds between anions and cations have also been suggested to lead to ion pairing, which may potentially explain why RTILs behave like moderately polar fluids in some respects, but detailed molecular dynamics simulations of imidazolium-based RTILs showed that these ion pairs are only very short-lived.59,60 One may also suspect that the reduction of the ionic charge in the force field used here from a full electronic charge to (0.8e is responsible for the weak influence of electrostatic interactions on the fluid behavior. The experiments on real [bmim][PF6], in which particles carry the full electronic charge, however, reveal the same behavior as the one shown by the model; it is therefore unlikely that the reduction of the ionic charge in the selected force field causes the agreement of γred(T/Tc) with the corresponding function for simple fluids. 6. Conclusion Using experimental data taken from the literature as well as our new simulation data for the room temperature ionic liquid [bmim][PF6], we obtained estimates of the critical temperature and the critical density of this fluid. Our recommended value of Tc ) 1100 ( 100 K agrees with the one of Rebelo et al.;6 our estimate of Fc ) 0.35 ( 0.04 g cm-3 is, to the best of our knowledge, the first one to be reported. The critical parameters are then applied to calculate the corresponding-states surface tension γred(T/Tc) defined by Guggenheim. It is found, maybe somewhat unexpectedly for an ionic substance, that the reduced surface-tension data of [bmim][PF6] coincide with the master curve for nonpolar and weakly polar fluids but are distinct from the corresponding data for simple inorganic molten salts (NaCl and KCl). Furthermore, our simulation data show no indication of an inflection point in the function γred(T/Tc); such an inflection point is found for hydrogen-bonding polar fluids as well as in simple models of ionic systems.19,21,22 The behavior of the latter systems is entirely governed by the Coulomb forces acting among the particles, which suggests that there are other significant contributions to the thermodynamic behavior of RTILs, such as dispersion forces, specific steric interactions, and, possibly, the formation of short-lived ion pairs,59,60 which reduce the ionic character of the fluid. The knowledge of the critical parameters of RTILs combined with the information that they, in a corresponding-states approach, behave like simple neutral fluids in certain respects, will help in devising correlation schemes to predict thermophysical properties of individual members of this large family of fluids. Such an approach would obviate the need for extensive measurements of the properties of each RTIL. In this way, the process of identifying the most suitable RTIL for a specific application can be made more efficient. Acknowledgment. We would like to thank Pia To¨lle, Wei Zhao, and Fre´de´ric Leroy for helpful discussions. This project has benefitted from the financial support of the German Research Foundation (Deutsche Forschungsgemeinschaft) through Priority Program SPP 1191 and other grants, which are gratefully acknowledged. References and Notes (1) Marsh, K. N.; Boxall, J. A.; Lichtenthaler, R. Fluid Phase Equilib. 2004, 219, 93. (2) Earle, M. J.; Seddon, K. R. Pure Appl. Chem. 2000, 72, 1391. (3) Wasserscheid, P.; Keim, W. Angew. Chem., Int. Ed. 2000, 39, 3772. (4) Guggenheim, E. A. J. Chem. Phys. 1945, 13, 253. (5) Israelachvili, J. N. Intermolecular and Surface Forces; Academic: London, 1985.

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