Solute Diffusion in Ionic Liquids, NMR Measurements and

Article pubs.acs.org/JPCB

Solute Diffusion in Ionic Liquids, NMR Measurements and Comparisons to Conventional Solvents Anne Kaintz,† Gary Baker,‡ Alan Benesi,† and Mark Maroncelli*,† †

Department of Chemistry, The Pennsylvania State University, University Park, Pennsylvania 16802, United States Department of Chemistry, University of Missouri, Columbia, Missouri 65211, United States

‡

S Supporting Information *

ABSTRACT: Diffusion coefficients of a variety of dilute solutes in the series of 1-alkyl-1methylpyrrolidinium bis(trifluoromethanesulfonyl)imides ([Prn1][Tf2N], n = 3, 4, 6, 8, and 10), trihexyltetracedecylphosphonium bis(trifluoromethanesulfonyl)imide [P14,666][Tf2N], and assorted imidazolium ionic liquids are measured using pulsed field gradient 1H NMR. These data, combined with available literature data, are used to try to uncover the solute and solvent characteristics most important in determining tracer diffusion rates. Discussion is framed in terms of departures from simple hydrodynamic predictions for translational friction using the ratio ζobs/ζSE, where ζobs is the observed friction, determined from the measured diffusion coefficient D via ζobs = kBT/D, and ζSE = 6πηR is the Stokes friction on a sphere of radius R (determined from the solute van der Waals volume) in a solvent with viscosity η. In the case of neutral solutes, the primary determinant of whether hydrodynamic predictions are accurate is the relative size of solute versus solvent molecules. A single correlation, albeit with considerable scatter, is found between ζobs/ζSE and the ratio of solute-to-solvent van der Waals volumes, ζobs/ζSE = {1 + a(VU/VV)−p}, with constants a = 1.93 and p = 1.88. In the case of small solutes, the observed friction is over 100-fold smaller than predictions of hydrodynamic models. The dipole moment of the solute has little effect on the friction, whereas solute charge has a marked effect. For monovalent solutes of size comparable to or smaller than the solvent ions, the observed friction is comparable to or even greater than what is predicted by hydrodynamics. These general trends are shown to be quite similar to what is observed for tracer diffusion in conventional solvents.

1. INTRODUCTION Ionic liquids (ILs) have recently been a focus of much research because their many unusual properties and their tunability make them attractive options for a wide variety of prospective applications. Low vapor pressure and high chemical and thermal stability make them ideal solvents for many organic and inorganic reactions1 and separations.2 Their wide electrochemical windows and high conductivities are clear assets as electrolytes in solar cells and batteries.3−5 In addition, the ease with which cations or anions can be interchanged allows ionic liquids to be designed for specific purposes. For example, changing the length of an alkane chain by one methylene unit enables fine-tuning of many properties. To further research into these and other applications, a thorough understanding of the properties of ionic liquids is necessary. The present work seeks to gain a more global understanding of solute diffusion in ionic liquids. The high viscosities and correspondingly slow diffusion in ionic liquids may limit both reaction rates and charge transport, making these dynamical properties of particular concern in synthetic and electrochemical applications. Gaining a predictive understanding of solute diffusion in ionic liquids will therefore help guide selection and design of solvents better suited to an intended purpose. Self-diffusion in ionic liquids has already received considerable attention. For example, the benchmark pulsed-field-gradient © 2013 American Chemical Society

NMR (PFG-NMR) measurements of ion self-diffusion and its relationship to electrical conductivity by Watanabe and coworkers6−10 have been influential in the debate concerning the “ionicity” of ionic liquids.11−13 A number of other groups have also applied PFG-NMR for measuring self-diffusion coefficients of ionic liquids.14−25 It has generally been observed that diffusion coefficients of the cation and anion components of a given ionic liquid are similar, despite sometimes substantial differences in size. Moreover, they are typically within a factor of 2 of hydrodynamic predictions. In contrast, there has been less focus on tracer diffusion, the diffusion of dilute solutes in ionic liquids. With few exceptions,26,27 the results available to date are either from electrochemical measurements28−45 or one of a variety of methods that measure diffusion of gases into the bulk liquid.46−51 These techniques carry the drawback of applying only to redox active solutes or to gaseous solutes, respectively. Surprisingly, very little has been done in the way of measuring solute diffusion in ionic liquid solutions using PFG-NMR. Most studies to date concern mainly Li+ and a few other solutes, often at significantly higher concentration than may be considered infinitely Received: May 31, 2013 Revised: August 22, 2013 Published: August 22, 2013 11697

dx.doi.org/10.1021/jp405393d | J. Phys. Chem. B 2013, 117, 11697−11708

The Journal of Physical Chemistry B

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dilute.52−55 Infinite dilution is preferable in these measurements, as it simplifies interpretation by eliminating solute−solute interactions. One general observation available from these various studies is that the diffusion coefficient of a given solute will exhibit an approximate proportionality both to the hydrodynamic factor of temperature/viscosity within a single solvent (as a function of temperature) and to 1/viscosity across a range of similar ionic liquid solvents at fixed temperature. The degree of proportionality for different solutes and the variation demonstrated in ionic liquids with substantially different properties, however, is not so well understood. Further investigation is required before any other such general statements can be made about solute diffusion in ionic liquids. In this study, we report PFG-NMR measurements of the diffusion coefficients of a variety of solutes at near-infinite dilution, primarily in the 1-alkyl-1-methylpyrrolidinium bis(trifluoromethylsulfonyl)imides ([Prn1][Tf2N], n = 3, 4, 6, 8, and 10) and trihexyltetradecyl-phosphonium bis(trifluoromethylsulfonyl)imide ([P14,666][Tf2N]) shown in Figure 1. Through systematic variation of the cation alkyl

DSE =

kBT 6πηR

(1)

This equation combines the Einstein relation, D = kBT/ζ, where ζ is the translational friction coefficient, with Stokes’ law for the frictional drag acting on a sphere of radius R in a continuous fluid of viscosity η, ζsphere = 6πηR. When R is chosen to be the radius of a sphere of volume equal to the van der Waals volume of the diffusing molecule, eq 1 usually provides the correct order of magnitude for the diffusion coefficient. Remarkably, the predicted value of D is often within a factor of 2 of the measured value. Stokes−Einstein predictions tend to be most accurate when the diffusing solute is much larger than the solvent (for example, proteins diffusing in water56), or even when the solute and solvent are comparable in size, as long as solute−solvent and solvent−solvent interactions are also similar. Figure 2 illustrates what is often observed in the case of selfdiffusion, in which the diffusing species is identical to the solvent,

Figure 2. Illustration of the accuracy of Stokes−Einstein predictions for self-diffusion coefficients in dipolar and ionic liquids at 25 °C. Dipolar liquid data (64 common solvents) are from Marcus57 and ionic liquid data from the literature (68 ionic liquids, Table SI-3, Supporting Information). The solid and dashed lines in the bottom panels indicate the average and twice the standard deviation in the data.

Figure 1. Ionic liquid components and solutes examined in this study.

content, we hope to better understand the effects of ion size, shape, charge density, and viscosity on tracer diffusion. Similarly, series of solutes have been chosen to focus on the effects of specific solute properties. One series, comprised of unsubstituted aromatics of increasing size and gradually varying aspect ratio (benzene, naphthalene, biphenyl, anthracene, and pyrene), addresses solute size and shape. The effect of solute−solvent interactions is examined through a series of benzene derivatives capable of nonpolar, dipolar, and ionic interactions. Other assorted ionic liquids and solutes are also considered for purposes of comparison to literature data obtained using other techniques.

or to one component of it in the case of ionic liquids. The left panels of Figure 2 contain data on a wide variety of conventional solvents compiled by Marcus;57 the right panels show data on 68 ionic liquids collected from the literature. (Data and sources are listed in Table SI-3 of the Supporting Information.) The top panels of Figure 2 show observed versus SE-predicted diffusion coefficients. The solid lines indicate agreement, and the dashed lines show the correlations: Dobs = aDpSE with (a, p) = (1.68, 0.93) and (1.92, 0.98) for dipolar and ionic liquids, respectively. The bottom panels show the ratio of observed to SE-predicted friction coefficients: ζobs/ζSE = 0.60 ± 0.24 (2σ) for dipolar solvents and 0.54 ± 0.24 for ionic liquids. As indicated by these data, self-diffusion in ionic liquids is quite similar to self-diffusion in conventional solvents. The SE equation provides a good first estimate of self-diffusion coefficients but systematically over-

2. STOKES−EINSTEIN PREDICTIONS The simplest model for predicting diffusion coefficients (D) is embodied in the venerable Stokes−Einstein (SE; more properly the Sutherland−Einstein56) equation 11698



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prepared from silver benzoate (Sigma-Aldrich, 99% purity) and tetraphenylphosphonium chloride (TCI, 98% purity). The tetraphenylphosphonium chloride was dissolved in methanol at 0.65 M. After complete dissolution, excess silver benzoate was added to the solution, which was stirred for several hours and the insoluble AgCl removed by filtration. The filtrate was evaporated to form a brownish resin which tested negative for chloride with application of silver nitrate. All sample preparation was performed within a nitrogenpurged glovebox. NMR samples were prepared by weight to be ∼50 mM in solute and were vacuum sealed in 5 mm sample tubes prior to measurement. Unless otherwise specified, all measurements were made at 25 °C. Viscosities were measured on a Brookfield Programmable DVIII+ rheometer fitted with a CPE 40 spindle and calibrated with Cannon calibration standard RT100. The rheometer was enclosed in a nitrogen-purged bag, with low temperatures measured last, to eliminate water uptake during measurements. A recirculating water bath was used to record temperaturedependent data (±0.2 °C) in increments of 5 °C between 5 and 65 °C. Comparisons of current data to earlier measurements of the same ionic liquids, made using a less rigorous technique, resulted in an overall assignment of 10% uncertainty for these viscosities. 1 H NMR measurements employed a DRX-400 Bruker NMR spectrometer with a 5 mm inverse broadband probe (BBI) as well as a Bruker AV-III-850 MHz NMR spectrometer with a Diff30 probe, both with triple axis gradients. Temperature calibrations were made using a sample of neat methanol and the reference data of Raiford et al.64 Diffusion coefficients were measured using the longitudinal-eddy-current delay (LED) stimulated echo pulse sequence with bipolar gradient pulse pairs as developed by Wu et al.65 Gradient strengths were calibrated using known self-diffusion coefficients of several neat liquids (water, dimethylsulfoxide, n-hexadecane). Diffusion coefficients were obtained by fitting peak area (DRX-400) or intensity (AV-III-850) data to the equation65

estimates the friction by roughly a factor of 2. (If slip rather than stick boundary conditions are used in calculating the Stokes slip friction ζslip sphere = 4πηR, one finds ζobs/ζSE ∼ 1.) Added to these results for self-diffusion is the fact that observed tracer diffusion coefficients agree with SE(stick) predictions in the limit of large solute/solvent size ratio.58,59 In light of the above, we attempt to understand diffusion in ionic liquids by examining deviations from Stokes−Einstein predictions, specifically by focusing on the ratios ζobs/ζSE = DSE/Dobs. In so doing, we assume that the frictional dissipation of solute kinetic energy should be proportional to solvent viscosity (as η1)60 and interpret departures of ζobs/ζSE from unity as reflecting the effects of solute−solvent packing and attractive interactions. It should be noted that eq 1 was derived for a spherical solute. Given that most molecules are nonspherical, it is relevant to ask how hydrodynamic predictions for ζ depend upon solute shape. In general, there should be distinct friction coefficients for motion parallel to different molecular axes, but in an isotropic environment, measured diffusion coefficients report only orientationally averaged values, ζav = kBT/Dav. In the case of ellipsoidal objects having semiaxis lengths a, b, and c Hubbard and Douglas61 showed that for stick boundary conditions the angularly averaged friction coefficient is given by ζav = Cζsphere with C=

2 ∞

∫0 [(a + x)(b + x)(c 2 + x)]−1/2 dx 2

2

≥1 (2)

Use of ellipsoid approximations for the least spherical solutes illustrated in Figure 1 (the planar aromatics) in eq 2 predicts values of C in the range 1.04 ≤ C ≤ 1.10. For symmetric ellipsoids, even with aspect ratios (a/c or c/a) as large as 5, C differs from unity by less than 25%. Slip boundary conditions might be supposed to produce different results. Analytical solutions to the hydrodynamic equations do not exist for slip boundary conditions. However, the calculations of Tang and Evans for prolate ellipsoids again show that, even for aspect ratios of 5, average friction coefficients are only ∼25% larger than slip predictions for a sphere. On the basis of these observations, we conclude that the effect of solute shape on hydrodynamic predictions is such as to increase ζobs/ζSE only by a modest amount. Given that the observed friction ratios vary by much more than 25%, eq 1 should be adequate for the present purposes.

⎧ ⎛ 1 1 ⎞⎫ I(g ) = I0 exp⎨−D(γδg )2 ⎜Δ − δ − τ ⎟⎬ ⎝ ⎩ 3 2 ⎠⎭

(3)

where I(g) and I0 represent integrated peak intensities (or areas) in the presence and absence of gradient pulses of amplitude g. δ is the gradient duration, γ the gyromagnetic ratio of the nucleus observed, Δ the separation between gradient pulse pairs, and τ the time allowed for gradient recovery before the next pulse. In the experiments on the DRX-400 spectrometer, τ was generally set to 2 × 10−4 s, the shortest time allowed by the electronics. Values of Δ and δ were manually optimized with respect to the diffusion coefficient of interest. A larger diffusion coefficient requires a larger Δ or δ, or both. These values were automatically optimized by the Bruker AV-III-850 software, based on the expected diffusion coefficient of the sample. Measurements resulting in diffusion coefficients that differed significantly from their expected value were repeated with a more accurate estimation, to ensure correct optimization. For solution measurements, parameters were optimized with respect to the anticipated solute diffusion coefficients, but such runs also provided accurate diffusion coefficients of the ionic liquid constituents. Evaluation of NMR peak areas was carried out in SpinWorks 3.1.7,66 which was far more time-efficient than the same analysis using the software available on the DRX-400. In contrast, calculation of diffusion coefficients for a given peak was

3. EXPERIMENTAL SECTION The pyrrolidinium ionic liquids [Prn1][Tf2N] with n = 3, 4, 6, 8, and 10 used in this work were synthesized and characterized as described in ref 62. Trihexyltetradecylphosphonium bis(trifluoromethanesulfonyl)imide ([P14,666][Tf2N]) was prepared from the chloride salt as described in ref 63. 1-Ethyl-3methylimidazolium bis(trifluoromethanesulfonyl)imide ([Im 2 1 ][Tf 2 N]), 1-butyl-3-methylimidazolium bis(trifluoromethanesulfonyl)imide ([Im41][Tf2N]), 1-butyl-3methylimidazolium tetrafluoroborate ([Im41][BF4]), and 1butyl-3-methylimidazolium hexafluorophosphate ([Im41][PF6]) were purchased from Iolitec (>99%). All ionic liquids were dried under a vacuum (10−2 Torr) for several hours until the water content was found to be significantly less than 100 ppm by Karl Fischer coulometry. Most of the solutes studied (Figure 1) were obtained from Sigma-Aldrich and were the highest grade available. In the case of the tetraphenylphosphonium and benzoate ions, the salt was 11699



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Prior to initiating the present study, very few measurements of tracer diffusion coefficients had been reported using NMR methods. Because we correlate data collected here with data measured using a variety of other techniques, several of our samples were chosen solely for comparison to literature values using some of these other techniques. Table SI-2 (Supporting Information) shows the available comparisons, which involve the solutes ferrocene, cobaltocenium, benzophenone, and anthracene in five ionic liquids. Of the 18 comparisons (excluding one clearly erroneous case), the unsigned average deviation between our values and literature data is 34%. Except for two instances, the literature diffusion measurements were made using electrochemical methods, primarily cyclic voltammetry. Among the different electrochemical methods used, our values agree best with chronoamperometric measurements (9% average deviation in five comparisons), mostly reported in the paper by Rogers et al.34 We note that, although electrochemical measurements of ionic liquids have become commonplace, there nevertheless remains considerable uncertainty associated with measurements of even the prototypical system ferrocene/ferrocinium.69,45 For example, Hussey and co-workers45 recently compared electrochemically measured diffusion coefficients of ferrocene in three imidazolium and pyrrollidinium ionic liquids. Their compilations showed standard deviations of 25−55% among the results reported by four to five different groups, with pairs of values often differing by a factor of 2 or more. Even different electrochemical approaches by the same group on the same samples yield diffusion coefficients that differ systematically by ∼50%.43 Thus, while the comparisons presented in Table SI-2 (Supporting Information) cannot be used to validate the accuracy of the present NMR measurements, they do provide an appreciation for the likely uncertainties in the electrochemically based diffusion data discussed later. B. Temperature Dependence. Although we are not primarily interested in the temperature dependence of diffusion rates, we first examine some temperature dependent data in order to examine the validity of a basic prediction of hydrodynamic theories, D ∝ T/η, or equivalently that Dη/T ∝ (ζobs/ζSE)−1 is a temperature (and pressure) independent quantity. If we are to use the friction ratios ζobs/ζSE and their departure from unity to help correlate diffusion coefficients in ionic liquids, these ratios should be at least approximately constant with temperature. Temperature-dependent results for a series of aromatic hydrocarbons in two pyrrolidinium ionic liquids, [Pr41][Tf2N] and [Pr10,1][Tf2N], are shown in Figure 4. The observed diffusion coefficients (top panels) are linear in T/η; however, the best fit lines do not pass through the origin. The ratios of the observed to SE-predicted friction coefficients, ζobs/ζSE ∝ (Dη/ T)−1 (bottom panels), therefore are not strictly temperature independent. In general, relative variations in ζobs/ζSE are small (mostly 1). Harris71 has shown that, in the case of self-diffusion coefficients, p typically falls between 0.9 and 1.0, whereas slightly smaller values are usually observed for tracer diffusion in a wide variety of conventional liquids (including both temperature and pressure variations). In the case of several imidazolium ionic liquids, Harris also showed that self-diffusion coefficients exhibit values of p = 0.90 ± 0.05. For comparison, we find that the data in Figure 4 can be fit to fractional Stokes−Einstein relations with most values of p in the range 0.85−0.95. C. Diffusion of Uncharged Solutes. We now consider how the friction ratios ζobs/ζSE depend upon solute and solvent properties, focusing on data near 298 K. As already clear from the lower panels of Figure 4, ζobs/ζSE varies systematically with solute, tending to increase as the size of the solute increases. In Figure 5, data on the same aromatic hydrocarbons in the

Figure 4. Diffusion data on five aromatic solutes in two pyrrolidinium ionic liquids measured at several temperatures in the range 275−316 K. The top panels show the diffusion coefficients and the bottom panels the ratios of observed to Stokes−Einstein predicted friction coefficients ζ/ ζSE. Solutes are Bz = benzene, Na = naphthalene, Bp = biphenyl, An = anthracene, and Py = pyrene.

Table 1. Characteristics of the Temperature Dependence of the Data in Figure 4a [Pr41][Tf2N] ζobs/ζSE benzene naphthalene biphenyl anthracene pyrene

0.19 0.29 0.26 0.33 0.44

δ (10

−3

6 2 3 2 −3

[Pr10,1][Tf2N] −1

K )

ζobs/ζSE

δ (10−3 K−1)

0.09 0.14 0.14 0.20 0.27

15 9 7 7 8

ζobs/ζSE is the average value of ζobs/ζSE over the temperature range 275−316 K, and δ is the logarithmic temperature derivative defined by eq 5. Uncertainties in δ are (1−2) × 10−3 K−1. a

correlation between the derivatives δ and the average values of ζobs/ζSE (R2 = 0.82) such that the smaller ζobs/ζSE (i.e., the larger the departure from SE predictions) the greater the variation of ζobs/ζSE with temperature. Similar behavior is found in several prior studies where the temperature dependence of diffusion was reported. For example, Watanabe’s group6,7,9,10 measured self-diffusion coefficients of the ions of 14 ionic liquids over the temperature range 263−353 K using PFG-NMR. The values of δ derived from their data range from 3 to 14 × 10−3 K−1 (average 6.7 × 10−3 K−1) for cations and 2 to 12 × 10−3 K−1 for anions (average 5.3 × 10−3 K−1). Evans et al. used chronoamperometry to measure the diffusion of N,N,N′,N′-tetramethyl-para-phenylenediamine (TMPD) and its radical cation in four Tf2N− ionic liquids ([Im21], [Im10,1], [Py4], and [P14,666] + [Tf2N]) between 298 and 348 K.70 Values of δ were in the range (2−5) × 10−3 K−1 except in [P14,666][Tf2N] where TMPD+• showed a small negative temperature derivative (−1.2 × 10−3 K−1). Finally, Taylor et al. recently measured diffusion coefficients of a ferrocene derivative having a 1,3dimethyl-imidazolium substituent on one cyclopentadienyl ring.43 Electrochemical data in five imidazolium ionic liquids

Figure 5. Ratio of observed to SE friction coefficients of aromatic hydrocarbons plotted versus the ratio of solute-to-solvent molecular volumes. (Solvent volumes are the average of cation and anion values.) Solutes are Bz = benzene, Na = naphthalene, Bp = biphenyl, An = anthracene, and Py = pyrene. Solid symbols with error bars are data in the pyrrolidinium ionic liquids [Prn1][Tf2N] (n = 3, 4, 6, 8, 10). For n = 4 and 10, temperature-dependent data are included, whereas the remaining data are at 298 K. Larger open symbols are data in the phosphonium ionic liquid [P14,666][Tf2N] and in several imidazolium ionic liquids: [Im41][Tf2N], [PF6], [BF4], and [Im21][Tf2N]. 11701



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[P14,666][Tf2N] partly to determine whether the large nonpolar domains engendered by the unusually large P14,666+ cation75,78 might show a clear effect. The fact that the values of ζobs/ζSE of the aromatic solutes in [P14,666][Tf2N] correlate with those of the [Prn1][Tf2N] solvents indicates otherwise. Thus, at least for nonpolar solutes as large or larger than benzene, if domain formation in ionic liquids has an effect on solute diffusion, it is much more subtle than the simple effect of the solute−solvent size ratio. A more global perspective on solute diffusion can be gained by considering the larger range of solute + solvent systems plotted in Figure 6. In this figure, we have included all of our own neutral

[Prn1][Tf2N] series, as well as a few other ionic liquids, are plotted versus the solute-to-solvent volume ratio VU/VV. Volumes here and below are derived from atomic van der Waals increments,72,73 and the ionic liquid solvent volumes used are the averages of cation and anion values. Focusing first on the [Prn1][Tf2N] solvents, a strong dependence of ζobs/ζSE upon VU/VV is evident in Figure 5. For each of the solutes, there is a good linear correlation between ζobs/ζSE and VU/VV within this homologous series of solvents (average correlation coefficient R̅ = 0.97 at 298 K). Different solutes in a given solvent do not, however, follow a single correlation with VU/VV. Extending the individual regression lines to the midpoint of the VU/VV range, we find values of ζobs/ζSE vary in the order Bp < An < Py < Na < Bz and differ by about a factor of 2 between Bp and Bz. Given the similar interactions expected for all of these solutes and the fact that this order does not follow that of solute size, it seems reasonable to ascribe this solute dependence to some effect of solute shape. We examined several properties that could be used to characterize the shapes of these solutes, including the inertial asymmetry factor74 κ=

(2IB−1 − I A−1 − IC−1) (I A−1 − IC−1)

(6)

where IA ≤ IB ≤ IC are the principal moments of inertia. This parameter is defined such that for a sphere κ = 0, for an oblate molecule like benzene κ = +1, and for a prolate molecule κ = −1. On the basis of geometries derived from B3LYP/6-31G(d,p) calculations, we find that the values of κ for these solutes follow the same ordering as the values of ζobs/ζSE: κ(Bp, An, Py, Na, Bz) = (−0.94, −0.91, −0.69. −0.40, +1). There is also a reasonable linear correlation (R = 0.88) between ζobs/ζSE and these values of κ. More study is required to determine whether these observations concerning solute shape will hold true for a wider range of solutes. For now, we note that differences in ζobs/ζSE at fixed VU/VV = 0.8 are much larger than would be predicted from the hydrodynamic calculations discussed in section 2 and probably reflect details of solute packing within the chargestructured environment provided by ionic liquids. Also shown in Figure 5 are friction ratios of these aromatic hydrocarbons in other ionic liquids. In the case of the phosphonium liquid [P14,666][Tf2N], the point for each solute falls close to what would be predicted from the trends set by the [Prn1][Tf2N] liquids. However, data on anthracene in assorted imidazolium-based ionic liquids (open squares) depart substantially from these trends. The largest departures are found for ionic liquids having anions different from Tf2N−, the anion of most of the ionic liquids measured here. These deviations, as well as the trends shown by the [Prn1][Tf2N] ionic liquids themselves, indicate that there are factors other than the relative size of the solute and solvent ions dictating the observed departures from Stokes−Einstein predictions. It is now widely recognized that ionic liquids containing ions with long (n ≥ 6) alkyl chains contain nanoscopic domains comprised of polar and nonpolar fragments,75 and one might expect this heterogeneous structure to influence solute diffusion. For example, X-ray scattering measurements and simulations76,77 suggest that the degree of domain formation should vary considerably with n in the [Prn1][Tf2N] series of solvents examined here. However, there is little in the data illustrated in Figure 5 that would suggest that this domain formation has an effect on the diffusion of the aromatic solutes studied here. We measured diffusion coefficients of these same solutes in

Figure 6. Ratio of observed to SE friction plotted versus the ratio of solute-to-solvent volumes for neutral solutes in ionic liquids. Filled symbols are data from this work (Table SI-4, Supporting Information), open symbols are literature data (Table SI-5, Supporting Information), and the curve is a least-squares fit to eq 7.

solute data (filled symbols) as well as data from literature measurements (open symbols). The literature values include gaseous solutes (squares) primarily from volumetric measurements46−51 and data on larger solutes (triangles), primarily from electrochemical measurements.28−45 Included within this expanded data set (323 solute/solvent combinations, 37 solutes, and 56 ionic liquids) are only data on simple ionic liquids without functional groups that might form specific associations with neutral solute molecules. Tables SI-4 and SI-5 of the Supporting Information provide listings of all data used. Considering first data from the present study, Figure 6 shows that there is no distinction between dipolar (red) and nondipolar (blue) solutes, even for relatively small solutes having dipole moments as large as 5.6 D (2-fluorobenzonitrile).79 Addition of data on nonpolar gaseous solutes (open squares), which vary in size between O2 and n-butane, greatly extends the range of size ratios available. There is much more scatter in the gaseous solute data than in our data set. For the smallest size ratios (VU/VV < 0.3), variations of ζobs/ζSE as large as a factor of 10 for a given value of VU/VV are found. Some of this scatter is undoubtedly the result of experimental uncertainties. For example, among the eight direct comparisons available, we find an average difference of 47% among gaseous solute data reported by different workers. However, such uncertainties are far smaller than the scatter. As noted above, there must be other factors beyond the size ratio contributing to the variation in values of ζobs/ζSE observed for these gaseous solutes. Literature data on larger solutes (open triangles) such as decamethylferrocene and N,N,N′,N′-tetrabutyl-para-phenylenediamine extend the range to larger values of VU/VV than studied here and suggest an approach to the limit ζobs/ζSE = 1 as VU/VV ≫ 1. Among these data, the scatter is also 11702



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quite large, and is only partially explained by the uncertainties in electrochemical measurements discussed in section 4A. Despite the scatter, it is still clear from Figure 6 that the dominant factor giving rise to deviations from Stokes−Einstein predictions is the relative size of the solute and solvent molecules. A simple function that describes this dependence is ζobs/ζSE = {1 + a(VU /VV )−p }

ln(D/T ) = −23.14 − 1.77 ln(VU) + 0.84 ln(VV ) + −0.52 ln(η)

where the diffusion coefficient D is expressed in units of m s , the temperature T in K, VU and VV are van der Waals volumes72,73 of the solute and solvent in Å3, and η the viscosity in (m Pa s). The fit using eq 8 is illustrated in Figure 7. Equivalently, the fit

(7)

The heavy curve shown in Figure 6 is a least-squares fit of ln(ζobs/ ζSE) vs ln(VU/VV) with parameters a = 1.93 and p = 1.88. The standard error in ln(ζobs/ζSE) for the 323 mostly distinct solute/ solvent pairs is ±0.65, meaning that eq 7 predicts ζobs/ζSE, or equivalently tracer diffusion coefficients, to an average accuracy of about a factor of 2. In an attempt to find more predictive correlations, we compared friction ratios, or the equivalent Dobs/T data, to a number of additional solute and solvent properties. Such attempts met with little success. We first correlated ln(ζobs/ ζSE) and ln(Dobs/T) to logarithmic values of the volumes and masses of the solute and the cations (C) and anions (A) of the solvent, various combinations of these quantities such as VU/VV and VC/VA (recall VV = (VC + VA)/2), as well as the solution viscosity. Attempts to include solute shape as we did with the aromatic/[Prn1][Tf2N] series did not appear fruitful based on the fact that nearly all of the gaseous solutes, which exhibited the largest scatter in Figure 6, fell close to the prolate limit and thus showed little variation in the asymmetry parameter κ. Table 2

Figure 7. Observed versus predicted values of D/T determined using eq 8. The data and symbols are the same as in Figure 6.

using ln(ζobs/ζSE) as the independent variable and ln(VU/VV), ln(VU), and ln(η) as dependent variables is ln(ζ /ζSE) = −2.05 + 0.84 ln(VU /VV ) − 0.60 ln(VU)

Table 2. Correlation Coefficients between Dobs/T and ζobs/ζSE and Solute or Solvent Propertiesa ζobs/ζSE Dobs/T VU VV VC VA VU/VV |VC/VA| η

range

ζobs/ζSE

Dobs/T

930 880 17 5.7 5.0 17 49 29 131

1 −0.79 0.81 −0.52 −0.58 0.08 0.89 −0.43 −0.55

−0.79 1 −0.88 0.17 0.11 0.05 −0.81 0.09 −0.07

(8) 2 −1

− 0.48 ln(η)

(9)

In both cases, the standard error in the logarithmic fit is 0.54 (∼55% average error), little better than the single-variable fit using eq 7. Data collected only in the present study (larger filled points in Figure 7) provide a tighter correlation, having ∼25% average error, due to the greater homogeneity of the much more limited set of solute/solvent systems investigated. The same is true of the prior correlations of CO2 in assorted ionic liquids,51 or a standard set of solutes in different classes of ionic liquids47,48,51 which are all subsets of the data examined here. Given that the improvement beyond the correlation to the single variable VU/ VV as in Figure 6 is only modest, we do not attempt to interpret these correlations further. The importance of the relative solute and solvent sizes in tracer diffusion revealed here in ionic liquid solvents has long been known in conventional solvents.80−83 Figure 8 shows the analogue of Figure 6 using data on uncharged solutes in a variety of conventional organic solvents. (Data are provided in Tables SI-6 and SI-7, Supporting Information.) These data are divided into two sets: aprotic solutes diffusing in either nonpolar (blue circles) or alcohol solvents (red diamonds). For comparison, the ionic liquid correlation from Figure 6 is reproduced as the heavy dashed curve. Two features of Figure 8 are noteworthy. First, the dependence on the solute/solvent size ratio is substantially greater in ionic liquids than it is in conventional solvents. At the smallest values of VU/VV shown here, the ratios ζobs/ζSE are roughly 10-fold smaller in ionic liquids, ∼0.005 versus ∼0.05. The second noteworthy feature of Figure 8 is the fact that solutes diffusing in the two different classes of conventional solvents, nonpolar vs hydrogen bonding, show distinct correlations with respect to departure from SE predictions. For VU/VV < 1, aprotic solutes diverge approximately 2-fold more in hydroxylic solvents compared to nonpolar solvents. The divergence of hydrogen bonding solutes in

a

Range is the ratio of the maximal to minimal value of a given property. The remaining values are linear correlation coefficients (R) in fits of the logarithms of both variables. Volumes (V) of the solute (U), solvent (V), and its cation (C) and anion (A) components are all van der Waals volumes and η is the solution viscosity.

summarizes these correlations. Because molecular masses and volumes are strongly correlated (R ∼ 0.96), the masses add no additional information and are omitted here. The results in Table 2 indicate that departure from SE predictions, quantified by ζobs/ ζSE, is most closely tied to the solute-to-solvent size ratio VU/VV (R = 0.89). In addition to volumes, the viscosity also shows a large correlation coefficient (R = −0.55) with this ratio. With respect to the solute diffusion coefficient itself (Dobs/T but with T ∼ 298 K), the solute volume is the most predictive (R = −0.89). Viscosity alone is a remarkably poor indicator of D/T. We also examined multilinear regressions of ln(ζobs/ζSE) and ln(Dobs/T) with the logarithms of the most highly correlated of these variables. Similar to what was observed with several subsets of the gas-phase data analyzed in this manner previously,47,48,51 the best representations of the data required the use of three solute/solvent properties, for example, 11703



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Figure 8. Ratio of observed to SE friction plotted versus the ratio of solute-to-solvent volumes for (non-hydrogen bonding) aprotic solutes diffusing in nonpolar (blue circles, N = 180) and alcohol (red diamonds, N = 76) solvents near room temperature. The dashed curve labeled “IL” is the correlation in ionic liquid solvents from Figure 6. Data used here are primarily from the literature, supplemented by some of our own measurements (Tables SI-6 and SI-7, Supporting Information). The curves shown here are fits to eq 7 with a = 0.65, p = 0.79 for nonpolar solvents and a = 1.95, p = 0.86 for alcohol solvents.

Figure 9. Ratio of observed to SE friction plotted versus the ratio of solute-to-solvent volumes for ionic and neutral solutes in ionic liquids. The neutral solute data (blue) are from the present work, and the blue curve is the fit from Figure 6. Circles are tracer diffusion data, and triangles are self-diffusion data. Smaller “+” symbols are assorted solution data from electrochemical measurements (Tables SI-3−5, Supporting Information). The Li+ datum overlaid with an arrow indicates a deviant point at coordinates (0.23, 9.3).55

between the diffusion of dilute ionic solutes (circles) and the selfdiffusion of solvent ions (triangles). Whereas the ionic and neutral solute data are nearly the same for VU/VV > 1, at smaller volume ratios, the two data sets diverge markedly. When VU/VV ∼ 0.2, the friction ratio of ionic solutes is roughly 100-fold greater than that of neutral solutes. At such small volume ratios, the friction on monovalent ions often exceeds SE predictions. An analogous increase in the friction on small ions is found in aqueous solutions, as illustrated by the data in Figure 10. Similar

hydrogen bonding solvents (Table SI-6, Supporting Information) is similar to the aforementioned dipolar aprotic solutes diffusing in nonpolar solvents. One can interpret the latter variations as reflecting differences in the solute−solvent versus solvent−solvent interactions acting in these different solute + solvent pairings. The viscosity (“solvent−solvent friction”) of alcohol solvents is greatly enhanced by the presence of hydrogen bonding among solvent molecules. When a small non-hydrogen-bonding solute diffuses in an alcohol solvent, it feels less friction than would a molecule that can participate in hydrogen bonding, and it is thereby immune to some portion of the full friction. When solute− solvent and solvent−solvent interactions are qualitatively the same, as they are in the case of aprotic solutes diffusing in nonpolar liquids, more of the friction expressed by the solvent viscosity is operative. The even greater reduction in ζobs/ζSE when nonionic solutes diffuse in ionic liquids can likewise be seen as resulting from the absence of solute−solvent interactions comparable to the strong ion−ion interactions present in ionic liquids. It is these ion−ion interactions that are largely responsible for the high viscosities of ionic liquids. Thus, at least for sufficiently small solutes, the full friction implied by the high solvent viscosities of ionic liquids is not felt by neutral solutes. D. Diffusion of Charged Solutes. The situation for charged solutes differs considerably from that of neutral solutes. The difference is illustrated in Figure 9, where we consider data for both neutral and monovalent solutes diffusing in ionic liquids. Large circles indicate results for tracer diffusion and triangles selfdiffusion in neat ionic liquids, all from PFG-NMR measurements. Smaller symbols (+) denote literature data from electrochemical measurements. (Note that the relatively greater scatter of these electrochemical measurements is consistent with their larger uncertainties, as discussed in section 4A.) Dilute solutes from the present work (circles) include the cobaltocenium and tetraphenylphosphonium cations (red) and the BF4− and benzoate anions (green). Magenta circles denote literature values for Li+ at concentrations below 0.5 M. (Data are collected in Tables SI-4 and SI-5, Supporting Information.) Although there is considerable scatter even among the NMR data, the general pattern is clear. There is no significant distinction

Figure 10. Ratio of observed to SE friction plotted versus the ratio of solute-to-solvent volumes for dilute ionic and neutral solutes in water. Neutral solute data are those compiled by Edwards72 and ionic data from Marcus.101

non-monotonic trends with solute ion size are also found in other dipolar solvents like acetonitrile and methanol. Although the small size of water compared to ionic liquid ions shifts the horizontal scale in Figure 10, there is no significant difference between the behavior of monovalent ions in water and in ionic liquids. The scatter about the nominal trends suggested by the dashed curves is larger in the ionic liquid case, but this greater scatter can be readily ascribed to larger uncertainties in the data and to the fact that many different ionic solvents are represented in Figure 9 as compared to the single solvent, water, in Figure 10. In the case of aqueous solutions, the non-monotonic dependence of friction on solute size has been known for many years.84 Sophisticated theories85−88 and simulations89−91 describing the molecular basis of this behavior are available. An 11704



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It should also be noted that in the present study we have not included systems (i.e., water as a solute) for which solute− solvent hydrogen bonding or other specific interactions are likely to be important. Such specific interactions will further hamper attempts to find simple empirical descriptions, except within suitably constrained data sets. Solute charge has a clear influence on diffusion rates in ionic liquids. The diffusion coefficients of large (VU/VV ≥ 1) monovalent solutes are comparable to those of neutral solutes and generally close to SE predictions. There is considerable scatter in the literature data on such solutes, perhaps due to uncertainties in the electrochemical techniques used to generate them but also potentially because details of the charge distribution become even more important in the case of charged solutes. When the size of an ionic solute is less than the solvent size, diffusion coefficients are dramatically different from those of their neutral counterparts. In the case of Li+, well studied by virtue of its relevance to battery technology,99 simulations suggest that two or more anions are bound to the Li+ cation and diffuse as a “solventberg” unit.96−98 The data compiled here suggest that Li+ provides a limiting case of a continuous range of frictional interactions as a function of solute charge density. Remarkable similarity is found between the non-monotonic variation of ζobs/ζSE with VU/VV in ionic liquid solvents and in water. Overall, it appears that solute diffusion in ionic liquids bears much in common with diffusion in conventional solvents. The conventional solvent case is more familiar and has been studied for a long time. But beyond hydrodynamic approaches and empirical characterization of departures from them, in neither conventional nor ionic liquids do tractable models exist that can accurately predict diffusion coefficients in the variety of systems examined here. We hope to contribute to further progress in building such predictive capabilities with future experimental and computational studies.

intuitive and qualitatively correct description is provided by the “solventberg” picture. When the solute charge density becomes large enough, lengthy residence times of water about the solute cause the solute + first solvation shell to move as a unit, which results in a larger hydrodynamic radius and greater friction compared to a like-sized neutral solute. Given the nearly quantitative similarity in the behavior shown in Figures 9 and 10, it is reasonable to expect that a similar description also applies to ions in ionic liquids. In the case of Li+, Raman measurements92−95 support the idea of distinct solvates, and molecular dynamics simulations suggest that the lifetimes of these solvates are many nanoseconds,96−98 supporting such a solventberg interpretation.

5. SUMMARY AND CONCLUSIONS In this work, we have reported new PFG-NMR measurements of the tracer diffusion coefficients of a variety of solutes in a homologous series of pyrrolidinium and other assorted ionic liquids. The majority of the new data involve neutral aromatic solutes of moderate size. When combined with available data on self-diffusion in ionic liquids, tracer diffusion of small gaseous solutes, and other literature data on charged and uncharged species, the data summarized here provide a framework for a coarse empirical description of diffusion rates in ionic liquids. The Stokes−Einstein hydrodynamic prediction DSE = kBT/6πηR provides a convenient reference point, in the form of the friction ratios ζobs/ζSE = ζobs/6πηR, for discerning how diffusion depends upon solute and solvent properties. In the case of neutral solutes, the foremost contributor to deviations from hydrodynamic predictions is the relative size of the solute and solvent. For small gaseous solutes, the ratios of solute-to-solvent van der Waals volumes, VU/VV, can be as small as 0.1 in ionic liquids and in such cases values of ζobs/ζSE as small as 0.01 are observed. An empirical equation of the form ζobs/ζSE = {1 + a(VU/VV)−p} captures this size dependence. The dependence of ζobs/ζSE upon VU/VV is not unique to ionic liquids. Similar behavior in the case of neutral solutes diffusing in conventional dipolar liquids has been known for a long time.81,82 In the latter case, however, values of ζobs/ζSE < 0.1 are quite rare, and the size dependence is appreciably weaker than in ionic liquids. The dependence on relative size alone is insufficient to provide useful quantitative predictions of diffusion coefficients in ionic liquids, except perhaps in cases of series of similar solute−solvent pairs. It is clear that other factors are involved, but attempts to develop more predictive empirical models based on simple solute and solvent properties have thus far been unsuccessful, at least for the diverse data set examined here. Within the selected set of aromatic solutes and pyrrolidinium ionic liquids we have measured, no dependence on solute dipole moment was found. Neither was there any clear evidence that the polar/ nonpolar domain formation present in the higher homologues of the [Prn1][Tf2N] series or in [P14,666][Tf2N] influences diffusion of solutes as small as benzene. There did appear to be some systematic variation in ζobs/ζSE with solute shape among the aromatic solutes, but further study is needed to determine the correctness of this observation. It seems likely that the substantial departures observed from the single-variable correlation between ζobs/ζSE and VU/VV result from subtle packing effects and to a lesser extent intermolecular interactions which will require more detailed descriptions of the shapes and charge distributions of both the solute and solvent than attempted here. We suspect that these effects may prove difficult to quantify in any simple manner.

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ASSOCIATED CONTENT

S Supporting Information *

A table containing parameters of fits to temperature-dependent viscosities, a comparison of selected diffusion coefficients measured here to literature values, and tabulations of diffusion coefficients collected here and from a variety of literature sources. This material is available free of charge via the Internet at http:// pubs.acs.org.

■

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS A.K. thanks Kirk Marat for rewriting the Spinworks code to enable efficient analysis of stacked NMR spectra. A.K. and M.M. were supported by the Division of Chemical Sciences, Geosciences, and Biosciences, Office of Basic Energy Sciences of the U.S. Department of Energy through grant DE-FG0212ER16363.

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REFERENCES

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