Correlation between Surface Tension and Void Fraction in Ionic

Aug 12, 2008 - The existence of a relation between γ and xv is rationalized with a simple capillary model minimizing the energy. Our success in corre...
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J. Phys. Chem. B 2008, 112, 12401–12407

12401

Correlation between Surface Tension and Void Fraction in Ionic Liquids Carlos Larriba,*,† Yukihiro Yoshida,‡ and Juan Ferna´ndez de la Mora† Yale UniVersity, 9 Hillhouse AVenue RM M7, New HaVen, Connecticut 06511, and DiVision of Chemistry, Graduate School of Science, Kyoto UniVersity, Sakyo-ku, Kyoto 606-8502, Japan ReceiVed: April 01, 2008; ReVised Manuscript ReceiVed: June 22, 2008

An effort to systematize published and new data on the surface tension γ of ionic liquids (ILs) is based on the hypothesis that the dimensionless surface tension parameter γVv2/3/kT is a function of the void fraction xv ) Vv/Vm. The void volume Vv is defined as the difference between the liquid volume Vm occupied by an ion pair (known from cationic and anionic masses and liquid density measurements) and the sum V+ + V- of the cationic and anionic volumes (known from crystal structures), while kT is the thermal energy. Our hypothesis that γVm2/3/kT ) G(xv) is initially based on cavity theory. It is then refined based on periodic lattice modeling, which reveals that the number N of voids per unit cell (hence the dimensionless surface tension) must depend on xv. Testing our hypothesis against data for the five ILs for which surface tension and density data are available over a wide range of temperatures collapses all of these data almost on a single curve G(xv), provided that slight (4%) self-consistent modifications are introduced on published crystallographic data for V+ and V-. An attempt to correlate the surface tension vs temperature data available for inorganic molten salts is similarly successful, but at the expense of larger shifts on the published ionic radii (8.8% for K; 3.3% for I). The collapsed G(xv) curves for ILs and inorganic salts do not overlap anywhere on xv space, and appear to be different from each other. The existence of a relation between γ and xv is rationalized with a simple capillary model minimizing the energy. Our success in correlating surface tension to void fraction may apply also to other liquid properties. 1. Introduction The exploitation of the great promise of the large number of potentially synthesizable ionic liquids (ILs) for concrete goals1,2 requires an ability to predict how specific desired properties are related to the choice of anion and cation. There is hence a need to develop practical tools to anticipate this relation. The present article reports initial progress in the difficult task of predicting the surface tension of ionic liquids. It is motivated by the fact that ILs having singularly high surface tension and electrical conductivity are particularly promising as high quality ion sources in a vacuum.3-5 Our studies have led us into several loosely connected directions guided by several points of view from different authors,6-8 which have resulted in the following structure of the article. Section 2 compiles a number of previously available surface tension data and complements them with new ones. Section 3 considers the insights provided by Fu¨rth’s formulation of cavity theory,9 which appears to suggest that the group 4γ〈r〉2/kT is a constant close to unity, where γ is the surface tension of the liquid, 〈r〉 Fu¨rth’s mean hole radius, T the liquid temperature, and k Boltzmann’s constant. We first adopt the provisional hypothesis that 〈r〉 can be based on the hole volume defined as the difference between the volume Vm of an ion pair (the total mass of the ion pair divided by the experimental liquid densities) and the sum V+ + V- of anion and cation volumes (4/3π〈r〉3 ) Vv ) Vm - V+ - V-), from which it follows that the group γVv2/3/kT is a constant close to 0.674. Near the melting point, this prediction is satisfied approximately for inorganic molten salts but is too low by a constant factor of about 3.3 for most ionic liquids. In spite of this initial failure, the fact that γVv2/3/kT is almost constant for † ‡

Yale University. Kyoto University.

ILs still provides an excellent correlation for surface tension at room temperature ((12%). Some insights on the origin of the variation of γVv2/3/kT by a factor of 3.3 from molten salts to ionic liquids follow from the fact that the void fraction xv ) Vv/Vm is very different in both cases. This observation alone suggests a substitution of the prior evidently invalid result γVv2/3/kT ) 0.674 by a generalization of the form γVv2/3/kT ) f(xv). Larriba10 has explored possible forms of the function f(xv) by modeling the IL as a periodic lattice matching the experimental void fractions of real ILs. This modeling reveals something that should have been clear from the beginning. Fu¨rth’s theory determines the mean hole radius but does not specify the number of holes per unit volume. There is therefore no grounds to assume that the molar volume contains exactly one cavity, as done in the hypothesis that Vv ) Vm - V+ - V-. The modeling reveals also that the number N of holes per unit cell depends considerably on the lattice, hence on the void fraction. For instance, in the compact hexagonal lattice, which (remarkably enough) matches closely the observed void fraction of ILs at room temperature, there are six cavities associated with an ion pair (section 2.4.1). Consequently, the mean void volume is 1/6 of that previously assumed, and the hole theory surface tension increases by a factor of 62/3 ) 3.3, in excellent agreement with room temperature experiments. Similarly, a lattice model appropriate for molten inorganic salts yields approximately one cavity per lattice, rationalizing the success of the previous correlation. In view of these qualitative insights, we set aside both cavity theory and lattice models of ILs, and simply adopt the hypothesis (already implicit in these models) that the group ξm ) γVm2/3/kT is a function G(xv), primarily of the void fraction:

10.1021/jp8027929 CCC: $40.75  2008 American Chemical Society Published on Web 08/12/2008

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ξm ) γVm2/3/kT ) G(xv)

TABLE 1: Surface Tension and Other Physical Properties for Imidazolium and Ammonium Based ILs

This hypothesis, if correct, and once the function G is obtained from empirical data, would reduce the effort of determining γ to that of finding V+, V- 11-13 (taken to be temperature independent), and Vm (inferred from the temperature dependence of the density). Using available data for the ionic volumes and a large number of data on surface tension and molar volumes at room temperature, the hypothesis γVm2/3/kT ) G(xv) leads to no substantial improvement in the correlation over that found before. Suspecting that this dispersion may be due to imprecise measurement of the ionic radii, the same hypothesis is then tested for the few ILs for which experimental surface tensions and molar volumes (densities) are available over a wide range of temperatures. This leads to the discovery that the various functions G(xv) resulting for each IL at varying temperatures are quite similar to each other, and can in fact be matched quite closely by slight modification ((2%) of the published ionic radii. This manipulation is in fact justified in principle by the limited precision with which these volumes are known. Only one IL (C8MI-Br) fails this test. However, the mismatch is traced to the incorrect V- reported for the molar volume C8MI-Br,14 and is resolved by use of a corrected value. A similar manipulation is carried out with a similar outcome for the large number of ILs for which surface tension and density data are known at only one or a few temperatures. The same exercise is performed with comparable success for inorganic molten salts, though the ionic radius of K+ needs in this case to be changed by 9% with respect to published data. 2. Results 2.1. Compilation of New and Old Data on Surface Tension. In view of the significance of the surface tension, we have examined a relatively large number of ionic liquids synthesized by Yoshida et al.15,16 expected to have a high surface tension.17 In order to be able to do so with minimal samples, we have used the improvements previously introduced on the capillary rise method aimed at using liquid samples smaller than 0.1 cm3.17 Larriba et al.18 introduced improvements in the accuracy of the technique, and used them to obtain better surface tensions for previously studied liquids. A compendium of room temperature surface tensions from imidazolium based ILs is shown in Table 1. These values include also revisions to earlier results,17 included in parentheses for reference. 2.2. Insights from Cavity Theory. In Fu¨rth’s hole theory,19,20 the probability of finding a hole with a radius between r and r + dr in a liquid is

Prdr )

16 7/2 6 -ar2 a re , 15π1/2

with a ) 4πγ/kT

(1)

The average hole radius 〈r〉 obtained by integrating Prrdr from 0 to ∞ is 〈r 〉

) (8/5π)(kT/γ)1/2

(2)

Equation 2 yields the surface tension if one has a rational way of determining the average hole radius. A first guess presumes that 〈r〉 is the radius of a sphere whose volume is the difference between the volume Vm of an ion pair in the bulk liquid and the sum of the ionic volumes V+ and V- of the anion and the cation:

Vv)Vm - V+ - VThe hypothesis that 〈r〉 ) that the group

(3Vv/4π)1/3

(3)

then leads to the conclusion

ionic liquid DMI-N(CN)2 EMI-NbF6 EMI-TaF6 EMI-F(HF)2.3 EMI-SbF6 EMI-(C2F5SO2)2N EMI-GaCl4 EMI-FeCl4 EMI-TfO EMI-BF4 EMI-Tf2N BMI-GaCl4 BMI-FeCl4 BMI-FeBr4 BMI-N(CN)2 C6MI-FeBr4 C6MI-N(CN)2 C6MI-FeCl4 C6MI-Au(CN)2 C8MI-FeBr4 C8MI-FeCl4 C8MI-Au(CN)2 MPI-(C2F5)3PF3 EtNH3-COOH Et3NH-Tf2N

γ (at 23 °C) (dyn/cm) 54.94 51.74 51.29 47.98 47.76 28.75c 48.64 (50)a,b 47.3 (47.7)a,b 38.31 (39.2)a 45.21 (44.3)a 35.80 (35.2)a 43.57 (41.5)a 45.91 (44.9)a,c 47.12 46.56 42.01 40.55 39.37 39.31 38.11 37.21 36.00 30.33 37.35 30.19

density K (S/m) Tm (°C) (g/cm3) 1.14 1.67 2.17 1.135 1.85 1.60 1.53 1.42 1.39 1.29 1.52 1.43 1.38 1.98 1.06 1.86 1.04 1.33 1.65 1.74 1.28 1.59 1.59 1.04 1.40

3.6 0.85 0.71 10.0 0.62 0.34 2.2 2.0 0.86 1.36 0.88 0.95 0.89 0.55 1.1 0.28 0.51 0.47 0.14 0.22 0.229 2.61 0.39

34 -1 2 -65 10 d 11 18 d d d Na Na -2 -10 Na Na Na 13 Na Na Na Na Na Na

a Values in brackets are from ref 17. b 21 °C. c 20 °C. d There is disagreement on the published melting points of these ionic liquids.

ξv)γVv2/3/kT

(4)

should be a constant, equal to ξ* ) ) 0.674. This result is reasonably successful in predicting the surface tension of inorganic molten salts not far from the melting point.9 The stage to apply these ideas to room temperature molten salts has been set by a number of previous studies reporting densities and surface tension data for a number of ionic liquids, as well as by compilations of data for V+ and V-. For the latter, we will for the time being use those reported by Jenkins et al.11,12 and Watanabe and co-workers,13 whose accuracy will be reconsidered in section 2.4. Following this idea, one can also infer a theoretical surface tension, γcalcd, based on the provisional assumption that ξ* ) 0.674: (8/5π)2(4π/3)2/3

γcalcd ) ξ*kT/Vv2/3

(4′)

Table 2 shows published values of the volumes V+, V- of the cations and anions, the molar volume Vm ) F/m of the liquid obtained from its measured density F and its formula mass m. Also shown are the void fraction xv ) Vv/Vm, the experimental (γ) and theoretical (γcalcd) surface tensions, their ratio γ/γcalcd, and ξv, defined in eq 4. The table shows that ξ* differs from 0.674 by an average ξv/ξ* ) γ/γcalcd ) 3.38 ( 0.32 ((10% maximum scatter) (the opaque FeBr4 salts are not included because their surface tension data have large errors18). In spite of the obvious failure of the model for ILs, the constancy of this ratio is noteworthy. This level of collapse of the data was sufficient to spot an anomalous point for the surface tension of EMI-N(CN)2 from Martino et al.17 This led to a new measurement and a corrected value falling within the cloud of data points. The rest of the remeasured liquids show values similar to those obtained by the previous method.

Surface Tension and Void Fraction in Ionic Liquids

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Figure 1. Correlation between the calculated (eq 4′) and experimental surface tensions.

2.3. Role of the Void Fraction. The questions to be addressed now are why the group ξv varies so little for ILs, why its average differs so much from the model value ξ*, and why ξv is so much closer to ξ* for inorganic molten salts. The first thing to note is that, near the melting point, the void fraction

xv)Vv/Vm

(5)

is relatively small for ILs (∼25%), and is considerably larger for molten salts (>50%). xv can in fact be seen in Table 2 to be rather constant for all the ILs for which we have data. This suggests that the relation between mean hole radius and void volume cannot be of the simple form 〈r〉 ) (3Vv/4π)1/3 so far assumed, and must include a dependence on the void fraction. In other words, we expect that 〈r 〉

) Vm1/3F(xv)

(6)

from which it follows that the group ξv ) γVv would also be a function of xv. The same holds for the closely related but more convenient group 2/3/kT

ξm ) γVm2/3/kT ) G(xv)

(7)

The reasonability of this hypothesis will be explored in the following sections. 2.4. Lattice Models. 2.4.1. Room Temperature Ionic Liq-

Figure 2. Illustration of the structure of voids in tetrahedral (a) and octahedral (b) spaces in the packed hexagonal lattice (c). The colorless sphere in the middle just shows the possible void volume in each array. (d) The possible density cell concept including the cation, the anion, and the void.

uids. We have noted that the void fraction xv of ionic liquids near their melting points is approximately 0.26. This corresponds closely to the maximum packing density possible for identical spheres, η ) 1 - (Vv/Vm) ) 1 - xV ) (π/32) ) 0.740 48, which holds both for the face centered cubic (FCC) lattice and the (equivalent) hexagonal closed packing (HCP) lattice. This coincidence suggests a model of a liquid where cations and anions are identical spheres organized in a closed packed hexagonal lattice (Figure 2). Each atom is coordinated with 12 neighboring atoms, and is also surrounded by 6 octahedral and 8 tetrahedral volumes. Since each octahedron is shared by 6 atoms, and each tetrahedron by 4 atoms, it follows that each atom is associated to two full tetrahedra and one full octahedron. There are therefore three cavities per atom, and six cavities per ion pair. The mean cavity volume is therefore Vv/6 rather than Vv, and the appropriate prediction for γ is 62/3 ()3.302) times

TABLE 2: Room Temperature Molar and Ionic Volumes, Surface Tension, and Corresponding ξv from eq 4 ionic liquid

γ (dyn/cm)

Vm (nm3)

V+ (nm3)

V- (nm3)

xv ) Vv/Vm

γcalcda (dyn/cm)

γ/γcalcd

ξv

DMI-N(CN)2 EMI-BF4 EMI-NbF6 EMI-TaF6 EMI-GaCl4 EMI-SbF6 EMI-FeCl4 EMI-Tf2N EMI-N(CN)2 BMI-FeBr4 BMI-N(CN)2 BMI-FeCl4 BMI-GaCl4 C6MI-FeBr4 C6MI-N(CN)2 C6MI-FeCl4 C6MI-Au(CN)2 C8MI-FeBr4 C8MI-FeCl4 C8MI-Au(CN)2

59.00 52.00 51.74 51.29 48.64 47.76 47.27 35.80 49.00 47.12 46.56 45.91 43.5 42.01 40.55 39.37 39.31 38.11 37.23 35.99

0.238 0.256 0.319 0.313 0.353 0.314 0.364 0.429 0.274 0.435 0.323 0.408 0.410 0.487 0.375 0.458 0.406 0.547 0.512 0.445

0.099 0.117 0.117 0.117 0.117 0.117 0.117 0.117 0.117 0.151 0.151 0.151 0.151 0.184 0.184 0.184 0.184 0.217 0.217 0.217

0.082 0.073 0.125 0.128 0.145 0.121 0.155 0.199 0.082 0.175 0.082 0.155 0.145 0.175 0.082 0.155 0.092 0.175 0.155 0.092

0.240 0.257 0.240 0.217 0.257 0.241 0.252 0.266 0.273 0.250 0.279 0.250 0.278 0.262 0.287 0.259 0.319 0.283 0.273 0.305

18.504 16.968 15.290 16.542 13.655 15.412 13.559 11.692 16.654 12.118 13.704 12.621 11.722 10.857 12.178 11.403 11.044 9.541 10.216 10.417

3.188 3.065 3.384 3.101 3.562 3.099 3.486 3.062 2.942 3.889 3.397 3.638 3.711 3.869 3.330 3.452 3.559 3.994 3.644 3.456

2.15 2.07 2.28 2.09 2.43 2.09 2.37 2.06 2.15 2.62 2.29 2.48 2.53 2.61 2.24 2.33 2.40 2.69 2.46 2.33

a

The formula used is γcalcd ) ξ*N2/3kT/Vv2/3, where N ) 6.

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the value given in eq 4′. This factor corrects quite accurately to the γ/γcalcd ) 3.38 ( 0.32 resulting from eq 4′, suggesting the following reformulation factoring in the average number of holes N per ion pair:

γcalcd ) ξ*N2/3kT/Vv2/3

(4″)

Its predictions agree with the data of Table 2, with errors in most cases well below 7%. 2.4.2. Molten Salts (Inorganic Salts). Molten salts (inorganic salts) require a rather different treatment than ILs, particularly given their much larger void fraction and the comparable size of ions and holes near the melting point.9 This by itself calls for at least a one-to-one coordination between the hole and the ions, which is paradoxical given that inorganic salt crystals also group in FCC (perhaps simple cubic) coordination. This anomaly is related to the large thermal expansion of inorganic salt crystals (and perhaps partly also to the rather different role of Coulombic forces in inorganic versus organic salts). The anomalous thermal expansion of inorganic salts above 780 K is due to so-called “thermally generated Schottky defects”.21,22 These defects are known to be “vacancies” of ions in the lattice, whose importance increases exponentially with temperature. This trend is clear in the larger void fraction of salts melting at higher temperatures (i.e., NaF) and the smaller values of those with lower melting points (i.e., NaI).23 One way of combining a “space filling” closed packed coordination on the one hand and a one-to-one coordination between holes and ions on the other is by using a HCP coordination between ions and holes. This means that just before the salt melts we have a very poor FCC coordination, which still remains crystal due to strong Coulombic forces, but with many displaced ions and full of new interstices formed where the ions were supposed to be. This scenario is illustrated visually in Figure 3. This does not mean that the whole liquid is modeled in this way. However, it seems logical that a percentage of the liquid will be arranged similarly to fulfill the large void requirement. In this picture, the space is filled by octahedra and cuboctahedra in a 1/1 proportion. Each atom is then surrounded by 4 cuboctahedral and 2 octahedral volumes. Since each cuboctahedron is shared by 12 atoms and each octahedron by 6 atoms, there are 2/3 cavities per atom and 4/3 cavities per ion pair. Inserting the value N ) 4/3 (N2/3 ) 1.211) into eq 4′ leads to the excellent agreement between γ and γcalcd shown in Table 3 near the melting point. The largest discrepancies observed correspond to potassium salts. We shall see in section 2.6 that the mismatch is halved when increasing the ionic radius of the potassium atom by 10%. 2.5. Correlations Involving the Void Fraction. Let us now test the hypothesis (eq 7) against experimental data. The modest scattering of the data seen in Figure 1 could in principle be due to the slight variation of their respective void fractions. However, a plot of these data in terms of the surface tension group ξm versus the void fraction (empty circles in Figure 4) does not reduce noticeably the scatter with respect to that seen in Figure 1. Figure 4 includes also data on a few ILs for which surface tensions24 and densities (C8MI-Cl,25 BMI-PF6,26 and C8MI-BF4, C8MI-PF6, and BMI-BF427) have been measured over a wide range of temperatures. The circles in Figure 4 correspond to discrete values of the surface tension obtained at a single temperature from Table 2, or at just a few temperatures (Table 4).14 The scatter in the new series of data is comparable to that of the discrete points ((10%). However, when plotting ξm versus the logarithm of Vv/Vm, one is struck by the fact that

Figure 3. (a) Cuboctahedra and (b) cuboctahedra-to-octahedra assembly. Voids are, as before, the spheres in the middle of the polyhedra. Note the approximate 1 to 1 relation between the size of the void and the ions.

the five curves obtained for a fixed IL at varying temperatures are almost exactly identical to each other, except for a horizontal displacement. This means that the various curves would collapse if their corresponding horizontal variables were multiplied by properly selected constants. Because this horizontal rescaling is equivalent to a modification of the sum of ionic volumes (V+ + V-), one is then led to wonder if the variation from liquid to liquid seen in Figure 4 is genuine, or if it is due to inaccuracies in the ionic volumes used. A first hint about the reasonableness of the second interpretation is obtained by observing that three ILs containing the large C8MI+ cation are grouped to the left of the figure, while the two ILs containing the BMI+ cation are quite close to each other but relatively distant from the other group. An adjustment of the ionic radii for either of these cations would then evidently lead to a much better collapse of the data. 2.6. Is the Scatter of the Data due to Inaccuracies in the Ionic Volumes? The ionic volumes used so far and collected in several tables have been obtained by a thermochemical method using the Born-Haber cycle (Jenkins).11,12 Using experimental X-ray structures, these authors have compiled a vast set of data on ion sizes, including most of the anions commonly used for IL synthesis. However, their anion volumes are obtained by neglecting the void of the cell. This error is probably not large given that the cations paired with these anions were generally small, leaving little separation for free cell space. In particular, Jenkins states that anion volumes tend to be overestimated while cation volumes tend to be underestimated.11 Thus, one should expect little error for the sum of the ionic radii from Jenkins, but an error is to be expected for the anions. Watanabe and co-workers13 have also reported cation volumes using van der Waals volumes calculated by Ue et al.28 These volumes tend to be smaller than those reported by Chemlab-II or Cerius 2,28 probably halving the error from Jenkins. In view of these circumstances, it is not unreasonable to expect errors of a few percent in the ionic radii, implying relative errors 3 times as large in the ionic volumes. We have therefore selected optimally corrected values V+ and V- (shown in Table 5) in order to minimize the separation between the five ξm(xv) curves for these five ILs. The process has been carried out with certain reasonable constraints detailed in Appendix A1 of the Supporting Information. The results are presented in Figure 5, showing an excellent fit for all data at the expense of changing the radii by as much as 4.4% (less than 4% for most liquids). Whether or not these changes are justifiable remains to be established. Still, even if they are not, one could interpret the corrected ionic volumes simply as fitting parameters only loosely related to ion dimensions. In view of this correlating success for the ILs for which data are available over a broad temperature range, we have performed a similar determination of corrected ionic volumes for all the ILs for which data exist only at one or a few temperatures. The

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TABLE 3: Hexagonal and Cubic Coordination for Molten Salts molten salts

density (g/cm3)

V- (nm3)

V+ (nm3)

amu

T (K)

γexptl (dyn/cm)

γcalcda (dyn/cm)

1.530 2.360 1.551 2.750 1.947 2.343

0.025 0.042 0.025 0.042 0.010 0.031

0.010 0.010 0.004 0.004 0.004 0.004

74.550 166.003 58.442 149.894 41.998 102.894

1044.0 996.15 1073.95 924.15 1269.0 1020.0

99.99 75.58 114.02 88.59 185.55 99.60

90.78 69.18 114.76 85.41 180.32 101.77

KCl KI NaCl NaI NaF NaBr a

The formula used is γcalcd ) ξ*N2/3kT/Vv2/3, where N ) 2/3.

Figure 4. Experimental ξm(xv) for the data points of Figure 1 and Table 4 and for five additional ILs for which surface tensions and densities are known over a wide range of temperatures.

corrected data are shown also in Figure 5, with the corresponding corrected volumes reported in Table 5. We were puzzled by the fact that all liquids were well correlated except for two data points for C8MI-Br. This anomaly suggested a 2% error in the densities reported by Rebelo et al.14 This suspicion has been confirmed by subsequent measurements,29 which bring the two anomalous C8MI-Br data to the proper place in Figure 5, and provide additional evidence on the predictive power of our correlation. A similar exercise has been carried out for inorganic salts for which data exist over a wide temperature range, as also shown in Figure 5. For these salts to fit into a single curve, changes to the published ionic radii are also required, as described in Appendix B of the Supporting Information. These changes are smaller than 4% except for the potassium salts (9% enlargement). This peculiarity of potassium might be due to a tendency to be organized in simple cubic cells. Table 6 reports the corrected volumes for the inorganic salts. Figure 5 shows that the trends followed by the organic and inorganic salts are quite different. This is more clearly seen in the logarithmic representation of Figure 6, where both liquid types obey the following power laws:

γ ) 0.0856kT(Vm/Vv)-3.101/(π1/3Vm2/3) -4.836

γ ) 0.0693kT(Vm/Vv)

1/3

/(π

Vm2/3)

for organic salts for inorganic salts

with the rather different exponents -3.1 and -4.84 for organic and inorganic salts, respectively. The lack of an overlapping region for the two data sets leaves open the question of whether or not these two distinct lines are two pieces of a single curve. What is clear is that, if the answer were to be positive, that single curve would have to undergo a fairly abrupt transition. 3. Discussion We conclude that a correlation of the form of eq 7 is excellent for ILs, even when one accepts uncritically published values for the ionic volumes. The drastically reduced scatter of the data resulting from adjustment of the ionic volumes proposed must be considered provisional until more accurate values

become available through more direct determinations. However, given the large number of constraints associated to the numerous repetitions of a limited number of anions and cations in the many salts studied, and in the many temperatures used, it is difficult to ascribe the fit obtained to the ionic volume manipulation. The least that our findings suggests is the potential fruitfulness of carrying out more refined determinations of ionic volumes, and densities, since slight shifts in these quantities lead to large variations in the ξv variable. The success of the hypothesis that the surface tension parameter ξm depends primarily on void fraction suggests also that there must be some theoretical basis for this assumption, so far based mainly on experimental observations and on lattice models. One way to construct a more quantitative theoretical model is to use the capillary theories underlying Born’s model for the evaporation energy of an ion, Thomson’s model for ion-induced nucleation, or Kelvin’s model for particle nucleation.30,31 In these models, the energy of a solvated ion or neutral particle is determined as if it were a macroscopic drop whose potential energy includes a Coulombic term and a surface energy term. Something similar can be done for an IL lattice by supposing it is composed of ions and empty spaces, with a certain average geometrical configuration. The corresponding average free energy F of a unit cell (treated as a macroscopic object) will then involve a surface term (proportional to γ times the square of a characteristic lattice length l), a Coulombic term (inversely proportional to l), and a third thermal term (cubic in l): F ) F(γ, l, etc.). If the same configuration is maintained while shrinking the characteristic dimension l, the Coulombic (negative) and surface (positive) energies will both decrease, while the thermal contribution will increase as a result of the compression work. Minimization of the free energy with respect to l will fix a certain characteristic cell size l ) l(γ, etc.). This fixes the packing fraction xv ) xv(γ, etc.). This is exactly what we have hypothesized in eq 7, except that other variables besides γ may appear (as indicated by the etc. symbol) due to their presence in the expression for the free energy. Such extra terms include the valences of the ions (z1, z2) and the dielectric constants (ε1 and ε2) of the anion and cation. Consequently,

xv)xv(γ, kT, z1,z2, ε1, ε2, e2/ε0, etc.) where e is the elementary charge and ε0 is the permittivity of a vacuum, both arising in the Coulombic potential. There may be an additional surface energy parameter γ2 associated with the contact area between the anion and the cation. For all of our data, z1 ) z2 ) 1, so that the above equation states indeed that a dimensionless surface tension formed with γ, kT, and the lattice length l (essentially the variable ξm) is a function of xv, and possibly other dimensionless variables such as the dielectric constants and perhaps a second dimensionless surface tension parameter ξ2 defined as ξmγ2/γ

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TABLE 4: Data for Liquids at Two Different Temperatures ionic liquid

temperature (K)

γ (dyn/cm)

amu

density (g/cm3)

Vm (cm3/mol)

anion size (nm3)

cation size (nm3)

Vv/Vm (%)

ξm (nm)

N2/3

BMI-PF6 BMI-BF4 C8MI-PF6 C8MI-BF4 C8MI-Br C8MI-Cl BMI-PF6 C6MI-PF6 DMI-Tf2N BMI-Tf2N C6MI-Tf2N

313-343 314-344 313-343 313-343 313-343 313-333 298-338 299-338 302-338 303-338 304-338

44.7-42.3 39.7-37.9 34.3-32.4 30.6-29.0 34.0-31.4 31.9-30.7 42.9-40.3 37.4-35.1 34.7-32.8 31.5-29.6 30.6-28.6

283.97 225.81 339.97 281.81 274.90 230.453 283.97 311.97 391.00 419.00 447.00

1.35-1.32 1.19-1.17 1.21-1.19 1.09-1.07 1.19-1.16 1.01-1.00 1.37-1.33 1.29-1.26 1.51-1.48 1.43-1.40 1.36-1.33

210.7-214.7 189.3-192.6 280.3-285.4 258.1-262.9 230.90-237.40a 229.1-230.8 207.8-213.0 241.6-247.4 258.2-264.5 292.8-299.8 329.3-337.0

0.109 0.073 0.109 0.073 0.032 0.025 0.109 0.109 0.199 0.199 0.199

0.151 0.151 0.217 0.217 0.217 0.217 0.151 0.184 0.117 0.151 0.184

26.24-27.62 29.29-30.50 30.28-31.52 32.63-33.86 35.45-37.21 36.69-37.16 25.21-27.04 27.32-29.02 26.90-28.64 28.58-30.25 30.36-31.95

5.155-4.508 4.250-3.746 4.785-4.175 4.040-3.537 4.168-3.578 3.890-3.537 5.149-4.335 4.947-4.172 4.750-4.077 4.674-3.999 4.894-4.178

3.135-2.836 2.781-2.519 3.201-2.869 2.841-2.550 3.097-2.747 2.958-2.712 3.049-2.690 3.090-2.714 2.936-2.628 3.008-2.674 3.280-2.897

a

The molar volume of C8MI-Br has been corrected from 208.4-215.2 cm3/mol14 to 230.90-237.40 cm3/mol.29

TABLE 5: Original and Corrected Ionic Radii Used in Figure 6 (The Underlined Ions Are Those Whose ILs’ Surface Tension Are Known over a Wide Range of Temperatures and Are Used for the Primary Fit) ionic liquid

ionic radii (nm)

corrected radii size (nm)

change (%)

DMI+ EMI+ BMI+ C6MI+ C8MI+ N(CN)2BF4NbF6TaF6GaCl4SbF6FeCl4FeBr4Au(CN)2Tf2NPF6-

0.287 0.303 0.330 0.353 0.373 0.270 0.259 0.310 0.313 0.326 0.307 0.333 0.347 0.280 0.362 0.296

0.289 0.306 0.338 0.364 0.387 0.260 0.249 0.303 0.299 0.324 0.297 0.322 0.341 0.275 0.359 0.285

+0.662 +0.870 +2.316 +3.174 +3.666 -3.630 -3.772 -2.321 -4.413 -0.553 -3.068 -3.264 -1.776 -1.730 -0.791 -3.933

TABLE 6: Original and Modified Ionic Volumes for Molten Salts

FClBrINa+ K+

Vi (nominal) (nm3)

Vi (corrected) (nm3)

ionic radii ratio corrected/uncorrected

0.0105 0.0248 0.0311 0.0422 0.0038 0.0099

0.0100 0.0241 0.0290 0.0377 0.0036 0.0127

0.984 0.990 0.977 0.963 0.979 1.088

take relatively small or large asymptotic values. There is in fact ground to expect that γ2 , γ, making ξ2 relatively irrelevant. Not much is known on the dielectric constant of ionic liquids, but the few for which data are available are generally larger than 10-15,32,33 whereby one could speculate that the function H could be approximated by its triple asymptote H(ξm, 0, ∞, ∞), providing a rationalization for the hypothesis of a single curve of the form of eq 7. 4. Conclusions

based on γ2. This would then imply a law of the form xv) H(ξm, ξ2, ε1, ε2). We recover eq 7 but with three additional parametric dependences. Since the values of these extra parameters could differ from one IL to another, there is no rigorous ground to expect that the corresponding curves will collapse exactly unless the three extra variables identified

A semiempirical effort to predict the surface tension of ionic liquids has been undertaken on the basis of Fu¨rth’s hole theory and numerous data available on surface tensions and densities, with the following results: (i) Fu¨rth’s theory works very well near the melting point provided void volume is defined based on the total void space

Figure 5. Representation of the data of Figure 4 and additional data for molten salts based on the modified ionic volumes shown in Tables 5 and 6, respectively. The two displaced filled squares to the right of the figure are for uncorrected molar volumes of C8MI-Br,14 which readily shift to fit the trend when corrected values30 are used.

Figure 6. Logarithmic representation of the data from Figure 5, showing quite different power laws.

Surface Tension and Void Fraction in Ionic Liquids in the unit cell Vv divided by the number N of cavities in the unit cell. For ionic liquids near the melting point, N ) 6. (ii) The hypothesis ξm ) π1/3γVm2/3/kT ) G(xv) fits all available data very well over the whole range of ILs and temperatures over which they are available, provided that modest shifts in published ionic volumes are introduced. (iii) The excellent fit obtained relies on the arbitrary assumption that a single ξm(xv) curve should collapse all data. However, the possible level of arbitrary shift of published ionic volumes is much smaller than the number of data collapsed close to a single curve. This fact strongly suggests that the assumption of a single curve is not as arbitrary as it appears, a point confirmed by the limited accuracy of many of the available ionic volumes, and by the very strong dependence of the variable ξm on V+ + V- through the variable xv. (iv) Point iii suggests the potential fruitfulness of an independent reevaluation of the ionic volumes and densities of ILs with higher accuracy than previously achieved. (v) Although this study has been directed mainly at ionic liquids, most of the previous conclusions apply similarly to inorganic molten salts, with a few variations. First, far fewer data have been analyzed. Second, the shift in ionic radius required for a good fit is larger for inorganic salts, including a necessary 9% shift in the ionic radius of potassium. Third, the form of the correlating curve G(xv) is quite different for inorganic molten salts than for ILs in their respective nonoverlapping regions of xv space. If both curves were to coincide, a sharp transition from one to the other would need to arise in the intermediate region xv ∼ 0.35-0.53. Acknowledgment. This study has been supported by Grantsin-Aid for Scientific Research (nos. 17750126 and 20750107) from Japan Society for the Promotion of Science (JSPS), and by the US AFOSR, directly through Yale grant FA9550-06-10104, as well as via subcontracts with the companies Busek and Connecticut Analytical Corporation. Acknowledgment is made to the Donors of the American Chemical Society Petroleum Research Fund, for partial support of this research. Note Added after ASAP Publication. This paper was published ASAP on August 12, 2008. The Acknowledgment and Table 4 footnote have been corrected in addition to some minor typographical errors throughout the paper. The updated paper was reposted on August 27, 2008. Supporting Information Available: Supporting Information is available which shows how the ion volume sizes were optimized. This material is available free of charge via the Internet at http://pubs.acs.org. References and Notes (1) Rogers, R. D.; Seddon, K. R. Ionic liquids-Industrial Applications to Green Chemistry; ACS Symposium Series 818; American Chemical Society: Washington, DC, 2002. (2) Seddon, K. Ionic liquids: designer solVents for green synthesis; Tce(730): 33-35; 2002. (3) Gamero-Castano, M.; de la Mora, J. F. Direct measurement of ion evaporation kinetics from electrified liquid surfaces. J. Chem. Phys. 2000, 113 (2), 815–832. (4) Larriba, C.; Castro, S.; et al. Monoenergetic source of kilodalton ions from Taylor cones of ionic liquids. J. Appl. Phys. 2007, 101 (8), 1–6. (5) Romero-Sanz, I.; Bocanegra, R.; et al. Source of heavy molecular ions based on Taylor cones of ionic liquids operating in the pure ion evaporation regime. J. Appl. Phys. 2003, 94 (5), 3599–3605. (6) Abbott, A. R.; Capper, G.; et al. Design of improved deep eutectic solvents using hole theory. ChemPhysChem 2006, 7 (4), 803–806.

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