Effect of Pressure on Transport Properties of the Ionic Liquid 1-Butyl-3

Feb 3, 2007 - Department of Molecular Science and Technology, Faculty of Engineering, Doshisha UniVersity, Kyo-Tanabe,. Kyoto 610-0321, Japan...
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J. Phys. Chem. B 2007, 111, 2062-2069

Effect of Pressure on Transport Properties of the Ionic Liquid 1-Butyl-3-methylimidazolium Hexafluorophosphate Mitsuhiro Kanakubo,*,†,‡ Kenneth R. Harris,*,† Noriaki Tsuchihashi,§ Kazuyasu Ibuki,§ and Masakatsu Ueno§ School of Physical, EnVironmental and Mathematical Sciences, UniVersity College, UniVersity of New South Wales, Australian Defence Force Academy, Canberra, ACT 2600, Australia, National Institute of AdVanced Industrial Science and Technology (AIST), 4-2-1 Nigatake, Miyagino-ku, Sendai 983-8551, Japan, and Department of Molecular Science and Technology, Faculty of Engineering, Doshisha UniVersity, Kyo-Tanabe, Kyoto 610-0321, Japan ReceiVed: NoVember 7, 2006

The self-diffusion coefficients (D) of the cation and anion in the ionic liquid 1-butyl-3-methylimidazolium hexafluorophosphate ([BMIM]PF6) have been determined together with the electrical conductivity (κ) under high pressure. All three quantities strongly decrease with increasing pressure to ∼20% of their atmospheric pressure values at 200 MPa. D(PF6-) is always less than D([BMIM]+), despite the larger van der Waals volume of the cation. The pressure effect on the transport coefficients is discussed in terms of velocity correlation coefficients (VCCs or fij), the Nernst-Einstein equation (ionic diffusivity-conductivity), and the fractional form of the Stokes-Einstein relation (viscosity-conductivity and viscosity-diffusivity). It was found that the VCCs for the cation-cation, anion-anion, and cation-anion pairs are all negative and strongly pressure-dependent, increasing (becoming less negative) with increasing pressure. However, when the values of the VCCs for a given isotherm are normalized relative to the corresponding atmospheric pressure values, they collapse onto a single curve, as might be expected because the pressure should affect the interionic velocity correlations in the same way for each type of interaction. These isothermal curves can be represented by the form exp(Rp + βp2). The Nernst-Einstein deviation parameter, ∆, which depends on the differences between the like-like ion and unlike ion VCCs (f++ + f-- - 2f+-), is very nearly constant under the conditions examined. The diffusion and molar conductivity (Λ) data are found to fit fractional forms of the StokesEinstein relationship with the viscosity, (ΛT) ∝ (T/η)t and Di ∝ (T/η)t , with t ) (0.92 ( 0.05), independent of both temperature and pressure within the ranges studied and common to the three independently determined properties.

Introduction Room temperature ionic liquids (ILs) are a promising class of “green” media for a wide range of chemical processes because of their negligibly small vapor pressures, high thermal and chemical stabilities, and broad electrochemical windows.1 Much effort has been made to investigate the thermodynamic and transport properties of neat ILs and IL-containing mixtures, in which Coulombic interactions, Van der Waals forces, and frequently, H-bonding play important roles. Naturally, most of the studies have focused on the temperature-dependence at atmospheric pressure, and there are yet only a few reports on the effect of pressure on liquid properties of room temperature ILs.2-7 Due to the difficulty of such studies, a similar situation applies to high-temperature molten salts.8 Yet, high-pressure studies are important in that they can, in principle, allow the separation of the effects of temperature from changes in density. The “hydrophobic” 1-butyl-3-methylimidazolium hexafluorophosphate [BMIM]PF6 can be regarded as a typical member of the class of imidazolium-based ILs. It has been used * To whom correspondence should be addressed. E-mails: m-kanakubo@ aist.go.jp, [email protected]. † University of New South Wales. ‡ National Institute of Advanced Industrial Science and Technology. § Doshisha University.

extensively as a reaction and electrochemical medium. Fortunately, the pVT data required in the manipulation of highpressure properties are available.4, 5a Monte Carlo simulations6 have also been performed as a function of temperature as well as pressure. Both experiment and simulation show that the molar volume decreases with increasing pressure more moderately than those of molecular organic liquids presumably due to Coulombic interactions; the isothermal compressibility also approaches a constant value at pressures of ∼200 MPa, suggesting a closedpacked state. To better understand the effect of pressure on interionic interactions in this IL, we have now determined the selfdiffusion coefficients (Di) of both the cation and the anion in [BMIM]PF6 at 50 and 70 °C to maximum pressures of 225 MPa (D+) and 150 MPa (D-), together with the electrical conductivity, κ (formerly called the specific conductance),9 at pressures to 200 MPa, at 25, 50 and 70 °C. Atmospheric pressure values have been determined for D+ between 30 and 80 °C, for Dbetween 35 and 80 °C, and for κ between 25 and 80 °C. These transport properties are closely linked through the ionic velocity cross-correlation functions because the transport of charge requires the transport of mass. We also compare these properties with the viscosity, which we have also measured at high pressures.2 This is possibly the first time that these four

10.1021/jp067328k CCC: $37.00 © 2007 American Chemical Society Published on Web 02/03/2007

J. Phys. Chem. B, Vol. 111, No. 8, 2007 2063

Transport Properties of [BMIM]PF6 transport properties have been determined at high pressure for a particular molten salt. These experiments provide an opportunity to determine the pressure and temperature-dependence of the velocity correlation coefficients, time and ensemble averages of the velocity correlation functions for the cation-cation, anion-anion, and cation-anion pairs in a molten salt and to examine the applicability at high pressure of both the Nernst-Einstein relation linking the conductivity with the diffusion coefficients and the Stokes-Einstein relation linking the viscosity with the diffusion coefficients. Experimental Section Samples of [BMIM]PF6, prepared and purified as described previously,2,10 were sealed into glass ampoules and then opened and transferred to high-pressure cells inside a dry glove box just prior to use. It was confirmed that the water contents of our samples were of the order of 30‚10-6 mass fraction as determined by Karl Fischer titration, and the chloride contents of aqueous solutions in contact with the samples are lower than the detection limit of AgNO3 testing. Self-diffusion coefficient (D) measurements were carried out by the NMR spin-echo technique at 20 MHz using steady gradients. This avoids many of the problems associated with pulsed-gradient measurements,11 particularly because the standard calibration fluids (water and benzene) are much less viscous than [BMIM]PF6. However the T, p range accessible with our apparatus and techniques was limited by very short T2 relaxation times. The sample was contained in a Teflon cell 35 mm in length by 5 mm in diameter connected by a capillary to a thin-walled Teflon chamber that acted as a bellows when compressed. A Viton O-ring was used to seal the join between the two halves of the cell. This cell was inserted in a Macor cylinder on which were wound both helical rf and quadrupolar magnetic-field gradient coils and contained within a Be-Cu pressure vessel. This, in turn, was mounted in a glass Dewar with the temperature regulated by circulating fluid from a Julabo FP40HC thermostat through brass tubing coiled around the pressure vessel. Fully fluorinated (3M FC-75 electronic fluid) or hydrogenated (Plexol) hydraulic fluids were used with the cation (1H resonance) and anion (19F resonance), respectively, to change the pressure on the cell within the pressure vessel. The pressure was measured with a Heise CM 500 MPa Bourdon-tube gauge and the temperature with a Pt resistance thermometer, both calibrated. The general techniques used to determine the diffusion coefficients have been described previously.12-14 Due to the short T2 relaxation times, which limit signal strength at the long 90180o pulse intervals (τ) required, experiments were carried out with the magnetic field gradient as the varied parameter, rather than the more usual 90-180° spacing, τ. (Normally, we carry out both types of experiment and average the result.) Both positive and negative gradients were used in a given experiment over the range (100 mT/(m rad). The signal strength observed increases with increasing temperature in the temperature range studied and decreases with increasing pressure due to the sensitivity of T2 to these two variables, and this determined the range of state points studied. Results obtained at 0.1 MPa were checked against results from a second probe employing a different gradient coil used only at atmospheric pressure. The sample used here was prepared separately from the first and was sealed under vacuum in a normal 5 mm NMR tube and, hence, acted as a stable control for the high-pressure cell samples. (Under some circumstances, due to cycling of T and

p and the thinness of the bellows, Teflon cells may leak. KelF, though stiffer, is somewhat better in this respect.) The uncertainties (coverage factor, k ) 1) of T, p, and D are estimated at 20 mK, 1 MPa, and 3%, respectively. The value for D is based on the calibration (1%), the fit to the spin-echo equation, and the reproducibility. Electrical conductivity measurements were carried out in a syringe-type cell using procedures previously given in detail.15,16 A pair of bright platinum electrodes with an area of 0.9 cm2 was fixed at a distance of 6 mm inside a glass cylinder. A movable glass piston, through which pressure was transmitted to the sample, was attached to the cylinder filled with 10 cm3 of IL sample. After pushing out gas bubbles in the cell, the upper hole in the syringe was covered with a Teflon cap. The conductance cell was installed in a high-pressure vessel connected to a hand pump and a calibrated Bourdon gauge (Heise or Nagano Keiki) with a maximum operational pressure of 400 or 500 MPa. The high-pressure vessel was immersed in a temperature-controlled oil bath and monitored by a calibrated thermistor thermometer (Takara, D632). The electrical resistances (R) were measured with a conductance bridge (Fuso, 362B), employing a sine wave oscillator (Fuso, 361B) and a standard resistor (Fuso, 363B). To eliminate the polarization effect at the bright platinum electrodes, resistances were measured at four different frequencies, f ) 0.5, 1, 2, and 5 kHz. Extrapolation to the limit of f ) ∞ is problematic for molten salts, and the linear fits in 1/xf generally used, for example, with aqueous electrolytes,17 can give rise to systematic errors (particularly with low R at high temperature). The magnitude of these errors can depend on the experimental conditions (use of bright or platinized electrodes, whether the cell capacitance is balanced in addition to the resistance, the frequency range, the number of frequencies examined, and T and p). This problem for molten salts has been analyzed by Robbins18 and by Hoover.19 As Robinson and Stokes17 also say with regard to aqueous electrolytes, one must “extrapolate to infinite frequency according to the kind of frequency dependence observed.” Hoover19 suggested employing the form R ) R∞ + b/(1 + cf2). We have found both this and a quadratic in 1/f2 to work well for ILs in our apparatus, the two methods giving similar values for R∞ and fits commensurate with the reproducibility of the resistance readings. The cell constants for high pressure measurements were determined to lie between 0.31 and 0.35 cm-1 at atmospheric pressure at 25 and 50 °C by using a 0.01 mol dm-3 KCl aqueous solution as the standard.20 The cell constant at 70 °C was found by correcting for the differential thermal expansion of the glass and the platinum,17 but it was assumed to be unaffected by changes in pressure at a fixed temperature. The temperature dependence of the conductivity at atmospheric pressure was separately determined between 25 and 80 °C in a second cell with a constant of 1.05 cm-1 at 25 °C; this was also corrected for thermal expansion at the other temperatures. The estimated uncertainties (coverage factor, k ) 1) of T, p, and κ are 30 mK, 0.5 MPa, and 3%, respectively. Results The self-diffusion coefficients of the cation [BMIM]+ (D+) and the anion PF6- (D-) have been determined at atmospheric pressure over a temperature range between 30 and 80 °C and 35 and 80 °C, respectively, and at pressures to 125 and 50 MPa, respectively, at 50 °C and to 225 and 150 MPa, respectively, at 70 °C. The electrical conductivity of [BMIM]PF6 (κ) has been measured at atmospheric pressure from 25 to 80 °C and as a

2064 J. Phys. Chem. B, Vol. 111, No. 8, 2007 function of pressure to 200 MPa at 25, 50, and 70 °C. The numerical data are tabulated in the Supporting Information document. The self-diffusion coefficients and the conductivity at atmospheric pressure can be represented by the following polynomials:

D+/(10-12 m2 s-1) ) -1.20809 + 2.05683‚10-1 (t/°C) + 2.02503‚10-3 (t/°C)2 + 7.76601‚10-5 (t/°C)3 (1a) D-/(10-12 m2 s-1) ) -2.95201 + 3.50288‚10-1 (t/°C) 4.25102‚10-3 (t/°C)2 + 1.19980‚10-4 (t/°C)3 (1b)

Kanakubo et al. TABLE 1: Self-Diffusion Coefficients D+ and D- and Electric Conductivity K in [BMIM]PF6 at Atmospheric Pressure at 25, 50, and 70 °C Compared with the Literature Results in Parentheses t/°C

D+/10-12 m2 s-1 6.92e (7.20,a

25 26.85 7.69e (7.10d) 50 23.8 (24.9a) 70 51.9 (51.8a)

19.7b)

D-/10-12 m2 s-1 5.15e

κ/S m-1

(5.24a)

0.144 (0.150,a 0.146c)

5.77e (5.40d) 19.1 (19.0a) 42.0 (41.0a)

0.440 (0.476,a 0.477c) 0.863 (0.938a)

a Ref 26; calculated from VFT parameters. b Ref 7. c Ref 22. d Ref 27. e Extrapolated using Litovitz equation.

κ/(S m-1) ) 9.3826‚10-2 - 2.9608‚10-3 (t/°C) + 1.9882‚10-4 (t/°C)2 (1c) with standard uncertainties of fit of 2.0, 2.4, and 1.6%, respectively. From the conductivities and the previously determined densities,2 one can obtain the molar conductivity (Λ ) κ/c),

Λ/(µS m2 mol-1) ) 21.8567 - 0.781543 (t/°C) + 4.42478‚10-2 (t/°C)2 (1d) with a standard uncertainty of fit also of 1.6%. For comparison with the results of other workers, the analogue of the Litovitz equation21 we have previously used for the viscosity of ionic liquids2,3 is convenient because it is better for extrapolation

ln[D+/(10-12 m2 s-1)] ) (-1.5246 ( 0.0085) 108/ (T/K)3 + (7.686 ( 0.024) (2a) ln[D-/(10-12 m2 s-1)] ) (-1.6155 ( 0.0086) 108/ (T/K)3 + (7.735 ( 0.025) (2b) ln[κ/(S m-1)] ) (-1.3784 ( 0.0076) 108/(T/K)3 + (3.2637 ( 0.024) (2c) ln[Λ/(µS m2 mol-1)] ) (-1.3998 ( 0.0074) 108/(T/K)3 + (8.6803 ( 0.023) (2d) with standard uncertainties of fit of 2.0, 2.4, 1.6, and 1.7%, respectively. For comparison, the temperature coefficient of the viscosity2 is (1.5966 ( 0.0039)‚108 K3. The transport properties can be very sensitive to the presence of small amounts of water, even in hydrophobic ILs.22-25 Our atmospheric pressure values of D+ and D- are in general good agreement with the results of Tokuda et al.26 and Nama et al.27 The mass fraction of water (w(H2O)) in Tokuda’s sample was less than 40‚10-6; that in Nama’s sample was not determined, but because the viscosity reported for 20 °C is 13% lower than that obtained by us,2 there may well be some water contamination in that case. D+ is much smaller than the values reported by Palmer et al.,7 in which w(H2O) was not given. In this IL, the cation [BMIM]+ diffuses more rapidly than the anion PF6-, though the Van der Waals volume of [BMIM]+ is larger than that of PF6-. This suggests caging of the anion in the liquid. On the other hand, preliminary results for [HMIM]PF6, [OMIM]PF6, and [OMIM]BF4 show the anion to diffuse more rapidly than the cation in these cases, so [BMIM]PF6 appears to be an exception in this group of ILs. There is good agreement for κ at 25 °C between our results and those of Widegren et al.22 (w(H2O) ≈ 10‚10-6) and Tokuda et al.,26 but our results are

Figure 1. Pressure-dependence of D+ (open), D- (dotted), and κ (filled) at 25 (squares), 50 (triangles), and 70 (circles) °C. The solid line represents the least-squares result of the MVFT1 fit (eq 4) for each quantity.

systematically lower at higher temperatures. The atmospheric pressure values of D+, D-, and κ at 25, 50, and 70 °C are shown in Table 1 together with the literature results. The self-diffusion coefficients D+ and D- and the conductivity κ are plotted against pressure, p, at different temperatures, T, in Figure 1. All three quantities strongly decrease with increasing pressure down to about one-fifth of the atmospheric pressure value at 200 MPa. In our earlier studies,2,3 we found that both the temperature and the pressure dependence of the viscosity of ILs can be reproduced very well over a wide range of conditions by modified Litovitz (ML) and modified VogelFulcher-Tammann equations (MVFT1 and MVFT2).

ML: f(T, p) ) exp(a + bp + (c + dp + ep2)/T3)

(3)

MVFT1: f(T, p) ) exp(a′ + b′p + δ(p)T0/(T - T0)) with δ(p) ) (c′ + d′p + e′p2)/T0

(4)

MVFT2: f(T, p) ) exp(a′′ + b′′p + δT0(p)/(T - T0(p)) with T0(p) ) x + yp + zp2

(5)

The fitted coefficients are obtained by nonlinear regression. δ is the so-called Angell strength parameter, usually represented by D.28 For MVFT1, δ is allowed to be p-dependent, with T0 constant; for MVFT2, T0 is allowed to be p-dependent, with δ constant. These equations are applicable to D+ and D-, as well as κ and Λ. The coefficients are listed in Table 2, and those for the viscosity, η, are also given for comparison. The qualities of the fits for the three equations are similar, and they reproduce the experimental data within (3% for the diffusion coefficients and less than (2% for the conductivities, consistent with the experimental uncertainties. The modified Litovitz equation (eq 3), may be preferable for moderate temperature extrapolations, given the simplicity of the exp(1/T3) dependence and experience with the viscosity. The values for T0 and the strength parameter δ for the modified VFT equations do differ somewhat from those

Transport Properties of [BMIM]PF6

J. Phys. Chem. B, Vol. 111, No. 8, 2007 2065

TABLE 2: Coefficients of Best Fit for Equations 3, 4, and 5 for D+, D-, and K, Together with η in [BMIM]PF6a D+/10-12 m s-1

D-/10-12 m2 s-1

a b/10-3 MPa-1 c/106 K3 d/106 K3 MPa-1 e/K3 MPa-2 standard uncertainty of fit, %

7.682 (0.021) -2.04 (0.8) -152.31 (0.8) -0.258 (0.03) 35.52 (36) 2.5

ML, eq 3 7.732 (0.024) -6.79 (2.3) -161.45 (0.8) -0.150 (0.09) 522 (121) 2.4

a′ b′/10-3 MPa-1 c′/K d′/K MPa-1 e′/10-5 K MPa-2 T0/K standard uncertainty of fit, %

9.86 (0.38)

MVFT1, eq 4 10.97 (0.51)

-1067 (124) -1.539(0.09) 28.9 (13) 163.6 (9) 2.5

-1467 (19) -2.059 (0.14) 211.0 (48) 140.5 (12) 2.3

10.161 (0.36)

MVFT2, eq 5 10.671 (0.49)

-7.602 (1.2) 154.8 (9) 11.05 (0.66) -5.22 (1.0) 2.5

-9.207 (1.9) 147.4 (12) 11.84 (1.1) -14.45 (3.1) 2.4

a′′ b′′/10-3 MPa-1 δc x/K y/10-2 K MPa-1 z/10-5 K MPa-2 standard uncertainty of fit, % a

κ/S m-1

Λ/µS m2 mol-1

η/mPa sb

3.285 (0.018) -0.0514 (0.28) -138.47 (0.58) -0.328 (0.009) 64 (21) 1.5

8.698 (0.017) -0.5038 (0.26) -140.50 (0.55) -0.328 (0.009 88.49 (21) 1.4

-0.406 (0.014) 2.354 (0.21) 159.49 (0.43) 0.317 (0.007 -191.3 (17) 1.8

4.585 (0.11) 3.336 (0.33) -776.2 (31) -1.875 (0.08) 18.74 (6.6) 179.2 (2.7) 1.1

9.979 (0.11) 2.790 (0.32) -777.2 (30) -1.848 (0.078) 28.68 (6.6) 180.1 (2.6) 1.0

-2.654 (0.13) -2.323 (0.32) 1121 (37) 2.269 (0.07) -94.45 (8.3) 162.2 (2.4) 1.8

4.904 (0.11) -1.222 (0.25) -5.059 (0.28) 171.4 (2.8) 11.67 (0.24) -8.404 (0.52) 1.1

10.428 (0.11) -2.094 (0.22) -4.456 (0.23) 178.1 (2.5) 11.71 (0.24) -8.960 (0.81) 1.1

-2.975 (0.14) 2.521 (0.25) 7.802 (0.41) 156.2 (2.7) 9.971 (0.24) -9.74 (0.89) 1.8

The standard uncertainties are given in parentheses. b Taken from ref 3. c δ has opposite signs for η and the group (Di, κ, Λ).

obtained for the viscosity, but this may be due to the shorter temperature ranges to which the diffusion and conductivity results are constrained. Discussion Diffusivity-Conductivity Relations. The electric conductivity is frequently related to the ionic self-diffusion coefficients through the modified Nernst-Einstein equation:

κ)

ne2 2 (z D + z-2D-)(1 - ∆) kT + +

(6)

where n is the number density; e, the elementary charge; k, the Boltzmann constant; and z+ and z-, the charge numbers of the cation and anion, respectively. Positive values of the deviation parameter, ∆, are often interpreted as implying ion association or aggregation,29 but summaries of older experimental work show ∆ to be a positive fraction for unassociated molten salts containing monatomic ions.30,31 This result is supported by computer simulations using models of varying sophistication,32-36 although theory can sometimes yield negative values.37 We note that negative values have also been reported for LiBr, for which the ions differ markedly in size,8d and for AgI.36 Clearly, the situation warrants further work on simple systems. Watanabe et al.26,38-40 have investigated transport properties for a variety of ILs with different pairs of cations and anions as a function of temperature, in which the molar conductivity ratio Λimp/ΛNMR (usually termed the Haven ratio by solid-state physicists but called “ionicity” by Watanabe et al., with Λimp being the measured conductivity and ΛNMR, that calculated from the ionic self-diffusion coefficients using the unmodified Nernst-Einstein equation) corresponds to the factor (1 - ∆) in eq 6. For all the ILs studied, they found (1 - ∆) to be much smaller than unity (i.e., ∆ is not negligible) and insensitive to changes in temperature between -10 and 80 oC. Consistent with their results for [BMIM]PF6,26 we obtain ∆ ) 0.36-0.39 at atmospheric pressure between 40 and 80 oC. The mean value of 0.38 ( 0.02 for [BMIM]PF6 is larger than those for molten alkali halides30,32-36 and comparable to those for alkali nitrates30,41 and tetraalkylammonium salts.42 At 50 and 70 °C,

the value of ∆, 0.37 ( 0.02, is essentially independent of pressure to 50 and 150 MPa, respectively (see Figure 2). Velocity correlation coefficients (VCCs), corresponding to the integrals of velocity correlation functions, provide more detailed information about collective motions in solutions. There have been several expressions proposed for VCCs. In the present study, we use the following, derived by Scho¨nert43 for salt melts with two ionic constituents in the form of Rυ+βυ- h ν+Rz+ + ν-βz-:

( ) ( )

f++ ≡

N AV 3

∫∞0 〈V+R(0)V+R(t)〉 dt ) RTκ z- FcM

f- - ≡

N AV 3

+ ∫∞0 〈V-β(0)V- β(t)〉 dt ) RTκ z+FcM

M

2

M

2

-

-

D+ ν+c (7) Dν-c (8)

and

f+- ≡

N AV 3

∫∞0 〈V+R(0)V-β(t)〉 dt ) RTκ z

M + M-

2 +z- (FcM)

(9)

where NA is the Avogadro constant; V, the volume of the ensemble; R, the gas constant; F, the Faraday constant; c, the amount concentration (molarity) of salt; and ν+ and ν- the stoichiometric numbers of cation and anion, with M, M+, and M- the molar masses of salt, cation, and anion, respectively. Equations 7-9 have been obtained for the mass-fixed (or barycentric) frame of reference. The Scho¨nert VCCs are closely related to the distinct diffusion coefficients of Friedman44 in that they are expressions for velocity cross-correlations for different particles. Padro´, Trulla`s, and Sese´45 have provided an expression for the deviation parameter that in our notation for the VCCs and for a 1:1 salt is

∆)

0.5(f++ + f- - - 2f+- ) f++ + f- -

(10)

2066 J. Phys. Chem. B, Vol. 111, No. 8, 2007

Figure 2. Pressure-dependence of the deviation ∆ from the NernstEinstein equation at 50 (triangles) and 70 (circles) °C.

Figure 3. Pressure-dependence of the velocity correlation coefficients f++ (open), f- - (dotted), and f+- (filled) at 50 (triangles) and 70 (circles) °C.

(This is analogous to a similar expression derived by McCall and Douglass46 for deviations from the Hartley-Crank relationship between mutual diffusion and self-diffusion coefficients in binary systems.) Equation 10 shows clearly that although it might seem reasonable to argue that ion-association mechanisms should reduce the conductivity relative to that obtained from the simple Nernst-Einstein value based on the ionic diffusion currents, such interpretations may naively neglect the effects of the ion velocity cross-correlations between different ions of the same species. Hence, the positive values obtained for ∆ for simple ionic melts referred to above. Nevertheless, the interpretation of VCCs is not straightforward because they must be compared with the predictions of a specific theoretical model47 or the results of computer simulation. Here, we restrict the discussion to the pressure-dependence. Future work will report comparisons between different ionic liquids. From the experimental results of D+, D-, and κ with the known molarity, c,48 we can calculate the auto- and cross-VCCs for the cation-cation, anion-anion, and cation-anion pairs. The values of f++, f- - and f+- are plotted against pressure at 50 and 70 °C in Figure 3. Numerical values are given in the Supporting Information document. All the VCCs increase asymptotically and monotonically with increasing pressure. The magnitude follows f+- < f- - < f++ at each temperature, whereas fij’s have larger magnitudes (are more negative) at the higher temperature. It should be noted that in f++ and f- -, the last term (-Di/νic) on the right-hand sides of eqs 7 and 8 is the predominant factor in the pressure- and temperature-dependence because κ is very small with respect to a large c in the first term; f+- has a negative value by definition and a negative temperature derivative due to the positive (∂κ/∂T)p. The calculation would be improved if more pVT data were to become available, the c values being derived from two isotherms (200 MPa maximum pressure at (25 and 50 °C) measured by Gu and Brennecke4 and our own densities at atmospheric pressure (0-90 °C).2 It is instructive to plot the pressure dependence of the VCCs relative to their atmospheric pressure values. The curves for

Kanakubo et al.

Figure 4. Pressure-dependence of the normalized velocity correlation coefficients [fij/fij (0.1 MPa)] at 50 and 70 °C. The solid lines represent the least-squares results of eqs 11a and 11b at 50 and 70 °C, respectively. Symbols as in Figure 3.

Figure 5. Pressure-dependence of D+η/kT (open) and D-η/kT (dotted) at 50 (triangles) and 70 (circles)°C.

Figure 6. Plot of ln(ΛT) against ln(T/η) for high-pressure isotherms at 25 (squares), 50 (triangles), and 70 (circles) °C and for atmospheric pressure isobar (diamonds). The solid line represents the linear leastsquares fit: ln[(ΛT)/(K µS m2 mol-1)] ) 0.913 ln[(T/η)/(K/mPa s)] + 8.993.

each isotherm are essentially coincident (Figure 4) and can be fitted to the functions

fij (50 °C, p)/fij (50 °C, 0.1 MPa) ) exp(-1.0445‚10-2 p + 2.3924‚10-6 p2) (11a) fij (70 °C, p)/fij (70 °C, 0.1 MPa) ) exp(-9.5713‚10-3 p + 5.4388‚10-6 p2) (i, j ) +, -) (11b) with standard uncertainties of 1.6 and 2.6%, respectively. (The p2 term is a reflection of the higher-order terms needed in the ML and MVFT fits for the transport properties, eq 3 to 5. A deviation plot is also included in the Supporting Information document. The fit is worst for f- -(70 °C, p), but still within the expanded experimental error, (6%.) This result might have been anticipated because the pressure should affect the interionic velocity correlations in the same way for each type of interaction. Nevertheless, we believe this is the first time such a regular behavior has been observed for VCCs. This is perhaps the explanation for the fact that despite the strong pressure

Transport Properties of [BMIM]PF6

J. Phys. Chem. B, Vol. 111, No. 8, 2007 2067 fitting and interpreting results.52,53 Zwanzig and Harrison54 have argued that such behavior is due to the effect of intermolecular forces on the effective hydrodynamic radius of the diffusing species, and in general, constancy of the Stokes-Einstein product is more closely approached the larger the diffusing species, the lower the viscosity, and the weaker the intermolecular forces.52 Voronel et al.55 have applied the fractional Stokes-Einstein (or Walden17) relation to the conductivity and viscosity of ionic melts in the form

Figure 7. Plots of ln(Di) vs ln(T/η) for the cation (open) and anion (dotted) for high pressure isotherms at 50 (triangles) and 70 (circles) °C and for atmospheric pressure isobar (diamonds). The solid lines represent the linear least-squares fits for the cation and anion, respectively: ln[D+/(10-12 m2 s-1)] ) 0.896 ln[(T/η)/(K/mPa s)] + 1.84 and ln[D-/(10-12 m2 s-1)] ) 0.923 ln[(T/η)/(K/mPa s)] + 1.60.

Figure 8. Plots of ln(ΛT) vs ln(Di) for the cation (open) and anion (dotted) and ln(D+ + D-) (filled) for high-pressure isotherms at 50 (triangles) and 70 (circles) °C and for the atmospheric pressure isobar (diamonds). The solid lines represent the linear least-squares fits for the cation, anion, and the sum (D+ + D-), respectively: ln[(ΛT)/(K µS m2 mol-1)] ) 1.01 ln[D+/(10-12 m2 s-1)] + 7.15; ln[(ΛT)/(K µS m2 mol-1)] ) 0.959 ln[D-/(10-12 m2 s-1)] + 7.50; and ln[(ΛT)/(K µS m2 mol-1)] ) 0.986 ln[(D+ + D-)/(10-12 m2 s-1)] + 6.60.

dependence of the VCCs, the Nernst-Einstein deviation parameter, ∆, is essentially constant under the conditions studied. The cross-correlations of the ionic velocities are all reduced to ∼20% of their atmospheric pressure values at 200 MPa. Thus, the effect of pressure appears to be to reduce the effect the movement of one ion has on the velocity of another, both like and unlike, with velocities being more quickly randomized as collisions occur at the higher pressures. Of course, the VCCs are ensemble averages and time integrals, so this interpretation may well be simplistic; therefore, it would be interesting to test the time evolution of the velocity cross-correlation functions by computer simulation as a function of pressure. The interactions between cations and anions in this system are complex; they include H-bonding between the anion and the C2 ring H atom49,50 and interestingly, given D([BMIM]+) > D(PF6-), the channel formation postulated in the liquid on the basis of an X-ray scattering study of the solid, crystallized by shock induction from the supercooled liquid.51 Diffusivity-Viscosity Relations. The product Diη/kT derived from the Stokes-Einstein relation,

D)

kT kT ) (4 e n e 6) ζ [nπη(σ/2)]

κT ∝ D ∝

(ηT)

t

(13)

obtaining t ∼ 0.8 for a number of pure and mixed nitrates and a single halide mixture. Voronel et al. claim that this value is a “universal” for ionic melts. However, this seems merely to be a special case given the general diffusion-viscosity relationships for liquids, and experiment and simulation show t can take a range of values.56 Strictly speaking, it is the molar conductivity, Λ, that should be used in such an analysis because it is the sum of the ionic molar conductivities, λi, each in turn proportional to the ionic mobilities, ui, and consequently, a more appropriate property than k for the investigation of ionic motion.

F2 κ (ν z 2D + ν-z-2D-) ) Λ ≡ ) ν+λ+ + ν-λ- ) c RT + + + F(ν+z+u+ + ν-z-u-) (14) It is also Λ that appears in Walden’s law, the conductivityviscosity relation derivable from Stokes’s law for electrolytes. That Λ is the more fundamental quantity, particularly for high pressure work, is also shown in the analysis of high-pressure conductivities of the simple molten salt NaCl in which maxima are apparent in higher temperature κ(p) isotherms, but Λ isotherms decrease monotonically with increasing pressure.8b The maxima in κ, inversely proportional to molar volume (see eq 6 and 14), are due to the competing effects of pressure on the compressibility, which is higher at higher temperatures, and the ionic mobilities, which are reduced by higher pressure. Figures 6 and 7, respectively, show plots of ln ΛT and ln(Di) against ln T/η for [BMIM]PF6 using the smoothed results (modified Litovitz parameters) for atmospheric pressure and the 25, 50, and 70 °C high-pressure isotherms. Remarkably, the isotherms are coincident in each case, and the atmospheric isobar results fall on the same lines. In the three cases, t is 0.91, 0.92, and 0.92 ( 0.05, so a consistent value is obtained, independent of both temperature and pressure within experimental error (conservatively taken as 3% for each property) for the three independently determined quantities. Strictly speaking, ΛT ∝ (D+ + D-) (eq 6), not Di, and one might expect slightly different exponents for the ΛT - η and Di - η relationships, but this is not distinguishable within experimental error. Figure 8 is a plot of ln ΛT against ln(D+ + D-). This acts as a check on our results because the slope, m, should be unity: here m ) (0.99 ( 0.07). The slopes for the cation and anion plots are (1.01 ( 0.05) and (0.96 ( 0.05), respectively.

(12)

where ζ is the frictional coefficient and σ is the diameter of the diffusing species52, has a positive pressure-dependence for both the cation and anion diffusion, as shown in Figure 5. An increase in this quantity with increasing viscosity is typical of many liquid systems, and the quantity Dηt, with 0 e t e 1, is often used in

Conclusions The self-diffusion coefficients of the cation and anion in the ionic liquid 1-butyl-3-methylimidazolium hexafluorophosphate have been determined together with the electrical conductivity under high pressure at different temperatures. All three quantities strongly decrease with increasing pressure to ∼20% of their

2068 J. Phys. Chem. B, Vol. 111, No. 8, 2007 atmospheric pressure values at 200 MPa. D(PF6-) is always less than D([BMIM]+), despite the larger van der Waals volume of the cation. It was found that all the velocity cross-correlation coefficients for the cation-cation, anion-anion, and cation-anion pairs are negative and strongly pressure-dependent, increasing (becoming less negative) with increasing pressure. The values of the VCCs, normalized relative to the corresponding atmospheric pressure values, are essentially coincident for a given isotherm and can be fitted to the form exp(Rp + βp2). This might be expected on the basis that the pressure should affect the ionic velocity cross-correlations in the same way for each type of interaction: cation-cation, anion-anion, and cation-anion. The NernstEinstein deviation parameter, ∆, which depends on the differences between the like-like ion and unlike ion VCCs, (f++ + f- - - 2f+-), is very nearly constant under the conditions examined. The diffusion and molar conductivity data are found to fit fractional forms of the Stokes-Einstein relationship with the viscosity, ΛT ∝ (T/η)t and Di ∝ (T/η)t , with t ) 0.92 ( 0.05, independent of both temperature and pressure within the ranges studied and common to the three independently determined properties. At present, it is difficult to explain the origin of the pressure effect on the transport properties in [BMIM]PF6 in terms of the detailed interactions between the ions. Better pVT data than are presently available would help in determining the effect of density because transport properties generally show regularities when expressed as functions of (molar) volume.52,56c Molecular dynamics experiments with pressure as a variable should also help. Fortunately, however, we can choose a variety of cation (with different alkyl chain lengths in the imidazolium unit) and anion (BF4-, PF6-, (CF3SO2)2N-, and so on) pairs in ILs. Such an investigation is now in progress, and we hope to report comparisons of these ILs with [BMIM]PF6 in due course. Acknowledgment. M.K. thanks the Japan Society for the Promotion of Science and the University of New South Wales for their financial support and the Australian Academy of Science for its administration of the JSPS Fellowship through the International Science Linkages Programme. We thank Dr. Lawrie Woolf (UNSW@ADFA) for many helpful discussions. Supporting Information Available: Numerical data for D+ (Table S1), D- (Table S2), and κ (Table S3), VCCs (Table S4), and residuals for fits of the normalized VCCs to eq 11 (Figure S1). This material is available free of charge via the Internet at http://pubs.acs.org. References and Notes (1) Ionic Liquids in Synthesis; Wasserscheid, P., Welton, T., Eds.; Wiley-VCH: Weinheim, 2003. (2) Harris, K. R.; Woolf, L. A.; Kanakubo, M. J. Chem. Eng. Data 2005, 50, 1777. (3) Harris, K. R.; Kanakubo, M.; Woolf, L. A. J. Chem. Eng. Data 2006, 51, 1161. (4) Gu, Z.; Brennecke, J. F. J. Chem. Eng. Data 2002, 47, 339. (5) (a) Gomes de Azevedo, R.; Esperanc¸ a, J. M. S. S.; NajdanovicVisak, V.; Visak, Z. P.; Guedes, H. J. R.; Nunes da Ponte, M.; Rebelo, L. P. N. J. Chem. Eng. Data 2005, 50, 997. (b) Gomes de Azevedo, R.; Esperanc¸ a, J. M. S. S.; Szydlowski, J.; Visak, Z. P.; Pires, P. F.; Guedes, H. J. R.; Rebelo, L. P. N. J. Chem. Thermodyn. 2005, 37, 888. (c) Esperanc¸ a, J. M. S. S.; Guedes, H. J. R.; Blesic, M.; Rebelo, L. P. N. J. Chem. Eng. Data 2006, 51, 237. (6) Shah, J. K.; Brennecke, J. F.; Maginn, E. J. Green Chem. 2002, 4, 112. (7) Palmer, G.; Richter, J.; Zeidler, M. D. Z. Naturforsch. 2004, 59a, 59.

Kanakubo et al. (8) (a) Kushiro, I. J. Geophys. Res. 1976, 81, 6347. (b) To¨dheide, K. Angew. Chem., Int. Ed. 1980, 19, 606. (c) Angell, C. A.; Cheeseman, P. A.; Kadiyala, R. R. Chem. Geol. 1987, 62, 83. (d) Yamaguchi, T.; Nagao, A.; Matsuoka, T.; Koda, S. J. Chem. Phys. 2003, 119, 11306. (e) Palmer, G.; Richter, J.; Zeidler, M. D. Z. Naturforsch. 2004, 59a, 51. (9) Mills, I.; Cvitasˇ, T.; Homann, K.; Kallay, N.; Kuchitsu, N. Quantities, Units and Symbols in Physical Chemistry, 2nd ed.; IUPAC, Blackwell Science: Oxford, 1993. (10) Umecky, T.; Kanakubo, M.; Ikushima, Y. Fluid Phase Equilib. 2005, 228-229, 329. (11) (a) Sørland, G. H.; Aksnes, D. Magn. Reson. Chem. 2002, 40, S139. (b) Hayamizu, K.; Price, W. S. J. Magn. Reson. 2004, 167, 328. (12) Harris, K. R.; Mills, R.; Back, P. J.; Webster, D. S. J. Magn. Reson. 1978, 29, 473. (13) Harris, K. R.; Alexander, J. A.; Goscinska, T.; Malhotra, R.; Woolf, L. A.; Dymond, J. H. Mol. Phys. 1993, 78, 235. (14) Harris, K. R.; Newitt, P. J. J. Phys. Chem. B 1998, 102, 8874. (15) Hoshina, T.; Tsuchihashi, N.; Ibuki, K.; Ueno, M. J. Chem. Phys. 2004, 120, 4355. (16) Ueno, M.; Tsuchihashi, N.; Shimizu, K. Bull. Chem. Soc. Jpn. 1985, 58, 2929. (17) Robinson, R. A.; Stokes, R. H. Electrolyte Solutions, 2nd ed.; Butterworths: London, 1959. (18) (a) Robbins, G. D. J. Electrochem. Soc. 1969, 116, 813. (b) Braunstein, J.; Robbins, G. D. J. Chem. Educ. 1971, 48, 52. (19) Hoover, T. B. J. Phys. Chem. 1970, 74, 2667. (20) Benson, G. C.; Gordon, A. R. J. Chem. Phys. 1945, 13, 473. (21) Litovitz, T. A. J. Chem. Phys. 1952, 20, 1088. (22) Widegren, J. A.; Saurer, E. M.; Marsh, K. N.; Magee, J. W. J. Chem. Thermodyn. 2005, 37, 569. (23) Seddon, K. R.; Stark, A.; Torres, M. J. Pure Appl. Chem. 2000, 72, 2275. (24) Kanakubo, M.; Umecky, T.; Aizawa, T.; Kurata, Y. Chem. Lett. 2005, 34, 324. (25) Widegren, J. A.; Laesecke, A.; Magee, J. W. Chem. Comm. 2005, 1610. (26) Tokuda, H.; Hayamizu, K.; Ishii, K.; Susan, M. A. B. H.; Watanabe, M. J. Phys. Chem. B 2004, 108, 16593. (27) Nama, D.; Anil Kumar, P. G.; Pregosin, P.; Geldbach, T. J.; Dyson, P. J. Inorg. Chim. Acta 2006, 359, 1907. (28) Angell, C. A. Phys. Chem. Solid 1998, 49, 863-871. (29) Margulis, C. A.; Stern, H. A.; Berne, B. J. J. Phys. Chem. B 2002, 106, 12017. (30) Angell, C. A. J. Phys. Chem. 1965, 69, 399. (31) Young, R. E.; O’Connell, J. P. Ind. Eng. Chem. Fundam. 1971, 10, 418. (32) Hansen, J.-P.; McDonald, I. R. J. Phys. C: Solid State Phys. 1974, 7, L384. (33) Hansen, J.-P.; McDonald, I. R. Phys. ReV. A: At., Mol., Opt. Phys. 1975, 11, 2111. (34) Trulla`s, J.; Padro´, J. A. Phys. ReV. B: Condens. Matter Mater. Phys. 1997, 55, 12210. (35) Sharma, R.; Tankeshwar, K. J. Chem. Phys. 1998, 108, 2601. (36) Koishi, T.; Tamaki, S. J. Non-Cryst. Solids 1999, 250-252, 501. (37) Koishi, T.; Tamaki, S. J. Chem. Phys. 2005, 123, 194501. (38) Tokuda, H.; Hayamizu, K.; Ishii, K.; Susan, M. A. B. H.; Watanabe, M. J. Phys. Chem. B 2005, 109, 6103. (39) Tokuda, H.; Ishii, K.; Susan, M. A. B. H.; Tsuzuki, S.; Hayamizu, K.; Watanabe, M. J. Phys. Chem. B 2006, 108, 2833. (40) Tokuda, H.; Tsuzuki, S.; Susan, M. A. B. H.; Hayamizu, K.; Watanabe, M. J. Phys. Chem. B 2006, 110, 19593. (41) Cleaver, B.; Herdlicka, C. J. Chem. Soc. Faraday Trans. 1 1976, 72, 1861. (42) Weiden, N.; Wittekopf, B.; Weil, K. G. Ber. Bunsen-Ges. Phys. Chem. 1990, 94, 353. (43) Scho¨nert, H. J. Phys. Chem. 1984, 88, 3359. (44) (a) Friedman, H. L.; Mills, R. J. Solution Chem. 1981, 10, 395. (b) Friedman, H. L.; Mills, R. J. Solution Chem. 1986, 15, 69. (c) Rainieri, F. O. Friedman, H. L. J. Chem. Phys. 1989, 91, 5642. (45) Padro´, J. A.; Trulla`s, J.; Sese´, G. Mol. Phys. 1991, 72, 1035. (46) McCall, D. W.; Douglass, D. C. J. Phys. Chem. 1967, 71, 987. (47) Miller, D. G. J. Phys. Chem. 1981, 85, 1137. (48) The value of c was calculated from the mass density, F. The necessary equation of state was obtained by refitting the original data from ref 4 as previously described in ref 2. (49) (a) Antony, J. H.; Mertens, D.; Breitenstein, T.; Do¨lle, A.; Wasserscheid, P.; Carper, W. R. Pure Appl. Chem. 2004, 76, 255. (b) Antony, J. H.; Mertens, D.; Do¨lle, A.; Wasserscheid, P. Anal. Bioanal. Chem. 2004, 378, 1548. (c) Talaty, E. R.; Raja, S.; Storhaug, V. J.; Do¨lle, A.; Carper, W. R. J. Phys. Chem. B 2004, 108, 13177. (50) Heimer, N. E.; Del Sesto, R. E.; Meng, Z.; Wilkes, J. S.; Carper, W. R. J. Mol. Liq. 2006, 124, 84. (51) Dibrov, S. M.; Kochi, J. K. Acta Crystallogr. 2006, C62, o19.

Transport Properties of [BMIM]PF6 (52) Tyrrell, H. J. V.; Harris, K. R. Diffusion in Liquids; Butterworths: London, 1984; p 322, ff. (53) Pollock, G. L.; Kennan, R. P.; Himm, J. F.; Stump, D. R. J. Chem. Phys. 1990, 92, 625. (54) Zwanzig, R.; Harrison, A. K. J. Chem. Phys. 1985, 83, 5861. (55) (a) Voronel, A.; Veliyulin, E.; Machavariani, V. Sh.; Kisliuk, A.; Quitmann, D. Phys. ReV. Lett. 1998, 80, 2630. (b) Veliyulin, E.; Shasha,

J. Phys. Chem. B, Vol. 111, No. 8, 2007 2069 E.; Voronel, A.; Machavariani, V. Sh.; Seifer, Sh.; Rosenberg, Yu.; Shumsky, M. G. J. Phys. Condens. Matter 1999, 11, 8773. (56) (a) Saiki, H.; Takami, K.; Tominaga, T. Phys. Chem. Chem. Phys. 1999, 1, 303. (b) Schweizer, K. S.; Saltzman, E. J. J. Phys. Chem. B 2004, 108, 19729. (c) Heyes, D. M.; Bran´ka, A. C. Mol. Simulation 2005, 13, 945. (d) Ito, N.; Richert, R. J. Phys. Chem. B DOI: 10.1021/jp0640023.