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J. Phys. Chem. B 2010, 114, 9572–9577
Relations between the Fractional Stokes-Einstein and Nernst-Einstein Equations and Velocity Correlation Coefficients in Ionic Liquids and Molten Salts Kenneth R. Harris* School of Physical, EnVironmental and Mathematical Sciences, UniVersity College, UniVersity of New South Wales, Australian Defence Force Academy, Canberra, ACT 2600, Australia ReceiVed: March 25, 2010; ReVised Manuscript ReceiVed: June 9, 2010
It is often asserted that deviation from the Nernst-Einstein relation (NE) between the molar conductivity (Λ) and ion self-diffusion coefficients (Di) in ionic liquids (ILs) and molten salts is evidence for ion pairing. The NE was originally derived for noninteracting ions, as in an infinitely dilute electrolyte solution. In reality, mass, charge, momentum, and energy transport processes in ILs and molten salts involve correlated interionic collisions, caging, and vortex motions, as in any other dense liquid. Phenomenological theory using nonequilibrium thermodynamics and literature molecular dynamics simulations shows that deviations from the simple NE expression occur due to differences in cross-correlations of ionic velocities. ILs have also been shown, like molecular liquids generally, and model fluids such as the Lennard-Jones, to fit the fractional form of the Stokes-Einstein relation, Di/T ∝ (1/η)t and Λ ∝ (1/η)t, where η is the shear viscosity. Here, it is shown that when this is the case, the NE deviation parameter ∆ is then a constant, independent of temperature and pressure (consistent with experiment) and the value of the parameter t; it is a function of the ionic charges and volumes, but not the masses. Therefore, ∆ is not a measure of “ionicity”: it is necessary to seek other independent evidence to determine whether ion pairing is present in a given ionic liquid or molten salt. The use of “apparent” transport numbers derived from self-diffusion coefficients to describe charge transport in pure salts is argued to be unnecessary. Introduction Ionic liquids are salts that have low melting points due to the low (generally single) charge on their constituent ions, coupled with a large surface area and an asymmetric structure of either the cation, the anion, or both. Attempts have been made to correlate thermo-physical properties using the concept of “ionicity”, that is, the degree to which the substances can be considered to be composed of discrete ions as opposed to neutral ion-pairs or larger aggregates. Ionicity as such is a somewhat ill-defined concept as any estimate of association or aggregation depends on the time-scale of the experiments used for its determination. It is not a well-defined quantity such as the degree of dissociation of a weak electrolyte, for example, which is determined by a thermodynamic quantity, the pKa. There are two methods in use for the estimation of ionicity.1 The first is based on the Walden Rule, an empirical but useful observation that the molar conductivity of electrolyte solutions is often approximately proportional to the fluidity (reciprocal viscosity).2 Plots are made of the logarithm of the molar conductivity against that of the fluidity, generally yielding linear plots. These are then compared to a similar plot for a reference material, 0.01 M1 or even 1 M3-5 aqueous KCl, which is assumed to be fully dissociated. The distance below the reference line is taken to be a measure of ionicity, and the smaller this latter quantity is, the more the salt is believed to contain associated or aggregated ions.1,6-8 Schreiner et al.5 have criticized the use of KCl solutions of arbitrary composition as a reference because the ion-solvent interactions are different from the ion-ion interactions of a molten salt and the K+-Cl- interactions are certainly not zero * Corresponding author. E-mail:
[email protected].
at other than infinite dilution, as has been long evidenced in transport property studies.9,10 Schreiner et al.5 have also observed that the slopes of Walden plots for KCl solutions, like those for many molten salts and electrolyte solutions, are actually fractional, not unity (Λ ∝ (1/η)t, t ≈ 0.87), in common with other electrolytes and molten salts. It is important to note that the original paper3 employing this method recognized “Such (an ionic) liquid must have a high degree of correlation in the motion of its cations and anions to explain the poor conductivity at a given fluidity” (although it then confuses the issue by discussing ion association rather than such correlations). It is one of the main points of the present Article that such correlations of ion velocities are the primary contribution to the “ionicity”. The second method11 defines the ionicity, here given the symbol Υ, as the ratio of the measured molar conductivity (Λ) of the salt to that calculated from ionic self-diffusion coefficients (Di) using the simple form of the Nernst-Einstein equation (NE). This is then a quantitative measure. The ionicity defined in this way can be shown12 to be the reciprocal of the Haven ratio, Hr,13 used to interpret conductivities in condensed matter physics. The NE can be written as
Λ)
F2 2 D-)(1 - ∆) (ν z2 D + ν-zRT + + +
(1)
where νi and zi are stoichiometric and charge numbers, respectively, for a salt formally dissociating as
10.1021/jp102687r 2010 American Chemical Society Published on Web 07/01/2010
Relations between the SE and NE Equations in ILs
J. Phys. Chem. B, Vol. 114, No. 29, 2010 9573
Aν+Bν- h ν+Az+ + ν-Bz-
(2)
R and F are the gas and Faraday constants, T is the temperature, and ∆ is the NE deviation parameter.
∆ ) 1 - Y ) 1 - (1/Hr)
(3)
The NE was derived for noninteracting ions, as in an infinitely dilute electrolyte solution,14 where ∆ is then zero. In reality, mass, charge, momentum, and energy transport processes in ILs and molten salts involve correlated collisions, caging, and vortex motions, as in any other dense liquid.15 In fragile ionic liquids where the structure is dominated by packing effects, the structural relaxation, which determines the transport properties, is that of the cage around each particle.16 It was shown by Hansen and McDonald17 in early molecular dynamics simulations for a fluid of equally sized, charged, hard spheres that deviations from the simple NE expression occur due to differences in cross-correlation functions of ionic velocities. This has been reinforced in many subsequent molecular dynamics studies for molten salts,18-21 and ionic liquids,22-26 but these results have been neglected or overlooked by the advocates of ionicity. We have shown elsewhere that the NE deviation parameter can be expressed directly in terms of the velocity correlation coefficients for like-ion and unlike-ion interactions for 1:1 salts.27,28 Thus
∆)
c(2f+- - f++ - f--) (D+ + D-)
(4)
2 2 2 2 c(2ν+ν-z+z-f+- + ν+ z+f++ + νz-f--)
+ c(f++ + f-- - 2f+-) ) ((D+/ν+) + (D-/ν-)) 2 (ν+z+ D+
2 ν-zD-)
(5)
using the electroneutrality condition ν+z+ ) -ν-z-. The fij’s in these expressions are the velocity cross-correlation coefficients (VCC) defined by Scho¨nert for the mass-fixed frame of reference:29
(
f++ ≡
NAV 3
∫0∞ 〈V+R(0)V+β(t)〉 dt ) RTκ z-FcM
f-- ≡
NAV 3
+ ∫0∞ 〈V-R(0)V-β(t)〉 dt ) RTκ z+FcM
(
M
M
)
-
)
-
2
2
D+ ν+c (6) Dν-c (7)
and
f+- ≡
NAV 3
Di ≡
∫0∞ 〈ViR(0)ViR(t)〉 dt
1 3
M+M-
∫0∞ 〈V+R(0)V-β(t)〉 dt ) RTκ z
2 +z-(FcM)
(8)
(9)
Equation 5 is obtained by substituting expressions for D+, D-, and Λ (≡κ/c) derived from eqs 6-8 into the Nernst-Einstein relation, eq 1. These expressions for ∆ are also inherent in the equations of Hertz and others in the early demonstrations of the use of velocity correlation coefficients in the phenomenological description of the transport properties of electrolyte solutions.10,30 (Indeed, the forms of eqs 4 and 5 are unchanged for a binary electrolyte.) These works derived from that of McCall and Douglass,31 who showed that in a binary nonelectrolyte solution (components 1 and 2), the mutual diffusion coefficient in the volume-fixed frame of reference (that normally measurable by experiment), D12V, and the two self-diffusion coefficients are related by an expression also containing a difference between velocity cross-correlation coefficients:
(
DV12 ) 1 +
∂ ln γ1 ∂ ln x1
)[
x2D1 + x1D2 +
T,p
x1x2
More generally, for any stoichiometry,
∆)-
where NA is the Avogadro constant, V is the volume of the ensemble, κ is the conductivity, c is the amount concentration (molarity) of salt, and M, M+, and M- are the molar masses of salt, cation, and anion, respectively. The VCCs are time integrals of the ensemble average of the velocity correlation functions of different ions as indicated by the subscripts R and β, whereas the self-diffusion coefficients are time integrals of the ensemble average of the velocity correlation functions of single ions of a given species.
(
f11 x21
+
f22 x22
-2
f12 x1x2
)]
(10)
where these fij values are defined in an analogous way to eqs 6-8: γ1 is the activity coefficient of component 1 in the mixture. If the VCC term is set to zero, one obtains the well-known Hartley-Crank or Darken relation.30,32,33 Other relations used in the study of silicate melts can be deduced from alternative conditions.33 VCCs in binary nonelectrolyte systems have been thoroughly analyzed in the literature.34 Harris, Kanakubo, and co-workers27,28 have calculated ∆ values as a function of T and p from high pressure measurements of the conductivities and ionic self-diffusion coefficients for a number of 1-alkyl-3-methylimidazolium tetrafluoroborates and hexafluorophophates. For these salts, ∆ is found to be independent of temperature and pressure within the range (25-80) °C and pressures to 150 MPa (see Figure 1 for the example of the tetrafluoroborates or Figure 7 of ref 28), lying in the range 0.37-0.52. For a given anion, ∆ depends on the size of the cation, and vice versa, consistent with the Watanabe group’s observations for Υ1,11 where temperature was the experimental variable. It is of importance that the molecular dynamics studies of molten salts referred to above17,18 also show nonzero values of ∆, even for very simple inorganic salts where association is not contemplated. The authors of these papers clearly recognized the contribution of cross-correlations between ion velocities to the ∆ parameter, but this has been overlooked in later work, perhaps because of the complex nature of the mathematical expressions in some of these papers.19,20 Instead, simpler hypotheses of ion association and aggregation producing neutral
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Harris concluding that “there is no indication of a mode arising from the reorientation of stable ion pairs.” FSE Equivalent to Viscosity-Dependent Slip-Stick Parameter It has been shown previously35 that the FSE is formally equivalent to a viscosity-dependent slip-stick parameter. Here, the argument is repeated. For tracer diffusion in solvents of viscosity η, it is found from experiment that ∞ t DT2 η ) constant
Figure 1. Nernst-Einstein deviation function, ∆, as a function of pressure for 1-alkyl-3-methylimidazolium tetrafluoroborates (data from ref 28). Symbols: [BMIM]BF4, 50 °C, b; 75 °C, O; [OMIM]BF4, 50 °C, 9; 75 °C, 0.
species that contribute to diffusion but not to the movement of charge have found favor.11 Regrettably, as noted elsewhere,28 experimental conductivity and self-diffusion coefficient data for high temperature molten salts show inconsistencies (different viscosity dependency), perhaps due to the difficult nature of the techniques employed, and are not reliable enough for the calculation of VCCs or the temperature and pressure dependence of ∆. Clearly, it would be most useful if MD studies for the transport properties of a fluid of charged hard spheres or charged Lennard-Jones particles could be made over a wide range of liquid states to determine the dependence of the Nernst-Einstein parameter ∆ on ion size and charge. This would then give a reference fluid against which ionic liquids could be gauged. In the interim, one can use the simple hydrodynamic Stokes-Einstein model to examine the effect of ion size and charge. Although the Stokes-Einstein equation was derived for a tracer particle moving in a solvent continuum, and is commonly applied to colloidal and macromolecular systems, it has often been used to rationalize both tracer and self-diffusion in pure liquids, solutions of electrolytes, and solutions of molecular liquids.35 An examination of diffusion and viscosity data for the Lennard-Jones fluid36 has shown them to fit the fractional form of the Stokes-Einstein relation (FSE), Di/T ∝ (1/η)t, with t ) (0.921 ( 0.003).37 Isotherms and isobars condense onto single curves over a very wide range of dense fluid states at densities above twice critical and temperatures ranging from the triple point to supercritical. This seems to be a general relationship, as a range of non-H-bonded molecular liquids for which high pressure data are available behave similarly, with 0.79 < t < 1.37 Ionic liquids also fit the FSE, with the molar conductivity and the self-diffusion coefficients having the same exponent for a given salt, so Λ ∝ (1/η)t (i.e., the fractional form of the Walden Rule) and Di/T ∝ (1/η)t.27,28,37 Here, it is shown that if this is the case, the NE deviation parameter ∆ is then a constant, independent of temperature and pressure (as is consistent with experiment) and the value of the parameter t; it is instead a function of the ionic charges and volumes. Therefore, ∆ < 1 is not in itself a measure of “ionicity”: it is necessary to seek alternative evidence to determine whether ion pairing might be present in a given ionic liquid or molten salt. Evidence of pairing, or at least clustering,38 on a short time scale, has been suggested in the IR spectroscopic and density functional theory studies of Ludwig and co-workers.39 On the other hand, Sangoro et al.40 have ruled this out in the case of [BMIM]BF4, on the basis of broadband dielectric and terahertz spectroscopy,
(11)
where DT2∞ is the tracer diffusion coefficient of a given solute (2) at infinite dilution. The Stokes-Einstein number is defined in terms of the standard Stokes-Einstein equation
n≡
kT ∞ DT2 ηπr
(12)
where r is the effective hydrodynamic radius of the tracer species. Taking logarithms,
[ ]
ln n ) ln
kT - ln η ∞ DT2 πr
(13)
Let n0 be n at unit viscosity, and with (kT/noπr) equal to the right-hand side of eq 11, then
ln n ) ln no - (t - 1) ln η
(14)
That is, n is viscosity-dependent. In classical hydrodynamic theory, it varies from 4 to 6: in real molecular systems, this range is extended to both higher and lower values.35 FSE and Velocity Correlation Coefficients Various forms of the Stokes-Einstein equation have been applied empirically to pure liquids where the condition of infinite dilution for diffusion of a tracer species does not apply.37 Of these, there are two fractional versions in current use, one a natural extension of the ordinary Stokes-Einstein equation, here termed FSE:
D 1 ∝ T η
t
()
(15)
( ηT )
(16)
and the other
D∝
s
here referred to as FSE-T. It is the first of these that best fits the computer simulation results for the Lennard-Jones fluid in the liquid and dense supercritical states. As mentioned above, there are, unfortunately, as yet no comparable data available for a fluid salt of charged hard-spheres or similar models. For ionic liquids based on 1-methyl-3-alkylimidazolium cations, it has been found from experiment27,28 that
Relations between the SE and NE Equations in ILs
ΛT ∝
( ηT ) ) q′ ( ηT ) s
s
J. Phys. Chem. B, Vol. 114, No. 29, 2010 9575
(
(17)
1
and
∆) 1-
q1R F (q2+ + q2-) 2
)
(25)
If D+ ) D- (equally sized particles),
Di ∝
( ηT ) ) q′ ( ηT ) s
s
i ) +,-
2i
(18)
where q1′ and q2i′′ are fitted constants, with s ) (0.90 ( 0.05). These expressions fitted data for 1-alkyl-3-methyl-imidazolium salts from 25-80 °C at pressures to 300 MPa. However, reexamination of the ionic liquid data showed that fits to the simpler FSE form found to best fit the Lennard-Jones fluid and a broad range of molecular liquids are just as good.37 Thus
Λ∝
( η1 ) ) q ( η1 ) t
t
and
()
t
()
q1R 2q2F2
[ ( )[
1 ) q1RT η
t
Λ ) ν+λ+ + ν-λ) ν+z+u+ + ν-z-u-
z+z-(FM)2
] ]
( ) ( )( )
D+ M- 2 cf++ ) RTΛ z-FM ν+ 2 q M T 1t 2+ 1 ) q1RT η z-FM ν+ η
()
ui )
( ) ( )( )
) q1RT
1 η
t
Dνq2-T 1 ν- η
M+ z+FM
M+ z+FM
2
2
(21)
t
(22)
(
)
k R/L ) nπr nπr
(29)
(30)
(L is Avogadro’s number) and eq 25 then becomes
∆)0
(31)
as is expected for noninteracting charges, irrespective of the value of t. A positive value of ∆ means t
(23)
q1R 2 2 F2(ν+z+ q2+ + ν-zq2-)
2 2 2 2 (ν+ z+f++ + νz-f--) < ν+ν-z+z-f+2
(32)
or, for a 1:1 salt,
Substitution of these forms into eq 5 yields
∆)1-
(28)
nπηtri
2 2 ν-z2F2/L ν+z+ + q1 ) nπ r+ r-
q2 )
-
()
zie
so that
and
cf-- ) RTΛ
(27)
(20)
2
z+z-(FM) M+M-
(26)
where the ui are the mobilities. If it is assumed that the fractional Stokes-Einstein equation applies to pure liquid salt, and the charges are not interacting (cf., infinitely dilute electrolyte), then
i ) +,-
M+M-
< 1 (q1 > 1, q2 >
consistent with the molecular dynamics results of Hansen and McDonald.17 The molar conductivity is the sum of the ionic molar conductivities:
Using the expressions above for the VCCs, one obtains
cf+- ) RTΛ
)
1, by definition; assuming q1R < 2q2F2)
(19)
1
Di 1t 1 ) q2i ∝ T η η
(
∆) 1-
(24)
which is a constant, dependent on ionic volume (radius) through the qi and on the salt stoichiometry (ν+z+ ) -ν-z-), but independent of T and p. The ionic masses do not appear in this expression, consistent with the assumptions inherent in the Stokes-Einstein relation. For 1:1 salts, eq 24 becomes
(f++ + f--) < f+2
(33)
Therefore, the coupling between the velocities of like ions is greater (less negative) than that between the ions of opposite sign (see eq 8, where f+- is negative by definition). This is the behavior observed for the imidazolium ionic liquids, and that to be expected for the majority of salts with normal ion conduction mechanisms.27,28 For these salts, t ≈ 0.9, so the different ∆ values observed are due to the effects of ion size
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Harris s t+ ) c+λ+/cΛ ) ν+λ+/Λ
(35)
which at infinite dilution (i.e., in the absence of ion-ion interactions46) becomes s,∞ ∞ ∞ ∞ t+ ) ν+z+D+ /(ν+z+D+ + ν-z-D)
Figure 2. Plot of ln[(-cfij/T)/(10-15 m2 s-1)] versus ln[(mPa s)/φ] for [BMIM]BF4 (data from ref 28). The fij are velocity cross-correlation coefficients, and φ is the fluidity or reciprocal viscosity. This is analogous to a Stokes-Einstein plot of ln(Di/T) or a Walden plot of ln Λ against ln φ. Note the similarity of the slopes (t ≈ 0.9). Symbols: blue, f++; red, f--; black, f+-; solid, atmospheric pressure isobar; open, high pressure isotherms at (25, 50, and 75) °C.
and shape, and hence liquid structure, on the ionic motion. There is no need to postulate ion-pairing without other specific evidence that might be obtainable by spectroscopic means or from neutron or X-ray scattering determinations of the liquid structure. It is interesting that negative values of ∆ are also possible, reversing the inequality in eq 33, and this has been observed in molecular dynamics simulations of certain salts (LiBr, Ag, and CuI halides).20,41,42 This finding suggests that in these cases the motion of one species is more affected by the collisions with like ions than by the attractive electrostatic interactions with counterions. As yet, there seems to have been no direct experimental observation of such an example in molten salts or ionic liquids, although such “superionicity” has been observed in the solid state.43 Superionicity has been claimed for the protic ionic liquids pyrrolidinium nitrate and pyrrolidinium hydrogen sulfate on the basis of the Walden plot comparison with aqueous KCl,44 but the measurement of the diffusion coefficients, which is difficult for the anions in these two liquids, has not been attempted. In the high pressure work27,28 on ionic liquids, it was observed that the isotherms for the velocity correlation coefficients and the atmospheric pressure isobar collapsed onto a single curve when plotted against the fluidity (reciprocal viscosity). It is now apparent from the discussion above that this is simply a consequence of the (fractional) Stokes-Einstein relationship. From eq 8, f+- will depend on viscosity in the same manner as Λ/c; similarly, given that the diffusion coefficients and the molar conductivity have identical Stokes-Einstein parameters t, the same scaling applies to the other VCCs (Figure 2). Comment on Transport Numbers In much of the work on ionicity, “apparent” transport numbers are given, calculated from the experimental self-diffusion coefficients with the formula
t+ )
D+ D+ + D-
(34)
for a 1:1 salt. This is presumably written in an analogy with the expression for the transport number of an ion in a binary electrolyte in the solvent-fixed frame45
(36)
using the Nernst-Hartley relation between the ionic molar conductivities and the self-diffusion coefficients at infinite dilution. However, eq 34 is often incorrectly applied to a pure ionic liquid. It was shown nearly 50 years ago that in a one-component molten salt, transport numbers are defined simply by the choice of the frame of reference, irrespective of the relative values of the ionic self-diffusion coefficients.47,48 In the mass-fixed frame, for example,
m t+ )
ν-MM
(37)
Only in mixtures of a salt with one or more other components can transport numbers have any physical meaning. So the use of eq 34 is misleading in terms of attributing greater charge transport to one ion or the other and is, at best, an alternative way of saying one ionic species diffuses more rapidly than the other. Conclusions Deviations from the Nernst-Einstein relation linking the electrical conductivity and ion self-diffusion coefficients in ionic liquids and molten salts have been attributed wholly to ionpairing or aggregation in parts of the literature. Literature molecular dynamics simulations and phenomenological theory show instead that these deviations, expressed as the quantity ∆, are due to differences between the velocity cross-correlation coefficients for unlike ions and the mean of those for like ions (fij, i,j ) +,-). There is thus no need to postulate ion-pairing or aggregation in the absence of independently determined liquid structural or spectroscopic information. In the majority of salts with normal conduction mechanisms, ∆ lies between zero and unity, and the general relationship between the fij and ∆ is given for salts of any stoichiometry. Using the fractional StokesEinstein relation, which is consistent with data for ionic liquids, it is shown that ∆ is independent of temperature and pressure, ionic masses, and the SE exponent, t; experimentally, it appears to be a function of ion size and shape. The use of “apparent” transport numbers derived from self-diffusion coefficients to describe charge transport in pure salts is argued to be unnecessary. Acknowledgment. I thank Lawrie Woolf and Mitsuhiro Kanakubo for their comments on the manuscript. An earlier version of this work was presented in poster form at the third Congress on Ionic Liquids, 31 May-4 June 2009, Cairns, Queensland, Australia. References and Notes (1) Ueno, K.; Tokuda, H.; Watanabe, M. Phys. Chem. Chem. Phys. 2010, 12, 1649. (2) For example: Apelblat, A. J. Phys. Chem. B 2008, 112, 7032. (3) Xu, W.; Cooper, E. I.; Angell, C. A. J. Phys. Chem. B 2003, 107, 6170. (4) Xu, W.; Angell, C. A. Science 2003, 302, 422.
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