On the Use of the AngellâWalden Equation To Determine the âIonicity

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On the Use of the Angell-Walden Equation to Determine the “Ionicity” of Molten Salts and Ionic Liquids Kenneth Robert Harris J. Phys. Chem. B, Just Accepted Manuscript • DOI: 10.1021/acs.jpcb.9b04443 • Publication Date (Web): 18 Jul 2019 Downloaded from pubs.acs.org on July 20, 2019

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The Journal of Physical Chemistry

On the Use of the Angell-Walden Equation to Determine the “Ionicity” of Molten Salts and Ionic Liquids Kenneth R. Harris* School of Science, The University of New South Wales, PO Box 7916, Canberra BC, ACT 2610, Australia.

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ABSTRACT In this work the Angell analysis of Walden plots of the conductivity of ionic liquids and other electrolytes against viscosity is used to examine simple molten salts at high temperatures, a test that does not appear to have been made previously. It is found that many simple salts such as alkali metal fluorides and chlorides are predicted to be “superionic” as their Walden plots fall above the arbitrary reference line introduced by Angell which passes through the datum point for 1M aqueous KCl at 25 oC. This contradicts long-standing molecular dynamics evidence in the literature showing that these salts conduct simply by ion motion in an electric field. Zinc chloride is also predicted to be “ideal” whereas one would expect it to be “sub-ionic” in Angell’s terminology given that it is an associated salt. Results for certain protic ionic liquids are also contradictory. Therefore Angell-Walden analyses of this type do not convey any useful information other than a qualitative ranking of the conductivity of similar ionic liquids at a given viscosity and their use for estimating “ionicity” is best discontinued. It can not and should not be used for classifying the interactions in ionic liquids. Instead, it is argued that an examination of Laity resistance coefficients is more useful in any discussion of true association in molten salts and ionic liquids where known examples show negative like-ion resistance coefficients with NE deviation parameters close to unity. Such an approach could be more fruitful in understanding the transport properties of molten salts and ionic liquids rather than simple comparisons of viscosity and conductivity.


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Introduction In a series of papers, Angell and co-workers have employed Walden plots of molar conductivity against fluidity, or reciprocal viscosity, in the form of log10-log10 plots, to categorize the behavior of polymer electrolytes,1 concentrated electrolyte solutions2 and ionic liquids.2,3,4 In doing so they have employed a reference line, of unit slope when units of S·cm2·mol-1 and 100 mPa·s (1 P in pre-SI cgs units) are employed, set by the conductivity5 and viscosity6 of 1 mol·L-1 aqueous KCl at 25 oC, which they termed “ideal”. Salts with Walden plots above the Angell reference line are termed “super-ionic”, by analogy with highly conductive solid-state materials,7,8 where special mechanisms govern the conductivity. Salts with Walden plots well below the Angell line are labelled “poor” ionic liquids or “sub-ionic” liquids. Such low conductivity is then usually attributed to ion association or aggregation.2 Walden himself first suggested this possibility in molten salts.9,10 It is well known that Walden’s rule – the inverse proportionality of molar conductivity and viscosity for ions at infinite dilution in a continuum solvent – can be derived from Stokes’ law, relating the velocity of a sphere to its radius and the viscosity of the surrounding fluid.11 In its empirical application to pure ionic liquids with the use of aqueous KCl as a reference for comparison, differences in (mean) ionic radius are generally neglected, though some authors have attempted to take this into account.12 Angell et al. assumed an “ideal” quasi-lattice liquid structure of ions of alternating charges where “ionic motions are not correlated in ways that lead to diminished conductivity”, with the caveat that allowance had to be made for friction between oppositely charged ions (see below).3,4 Nevertheless, the position of a Walden plot for a given substance relative to the reference line (W) is commonly used as a quantitative measure of the “ionicity” of the salt13,14 or degree


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of dissociation (Figure 1). This assumption that molten salts and ionic liquids contain ion-pairs or aggregates, based on Angell-Walden analyses, without independent experimental evidence, is widespread and persistent in the ionic liquid literature. This author has criticized “ionicity” arguments in previous work. 15,16,17 The primary method for determining “ionicity” 18,19,20,21,22,23 is to apply the NernstEinstein equation which relies on measurements of ion self-diffusion coefficients as well as the conductivity. As self-diffusion coefficients are not always available, and accurate determinations are not necessarily straightforward,24,25,26 a secondary method based on the more readily measurable conductivity and viscosity employing the Walden plot is attractive, particularly in screening large numbers of compounds for potential electrochemical applications of ionic liquids. The Nernst-Einstein (NE) equation estimates the molar conductivity () of a binary (twocomponent) electrolyte solution and the self-diffusion coefficients (Dsi) of the salt ions,

 NE 

F

2

/ RT   z 2 DS   z 2 DS 

(2)

F and R are the Faraday and gas constants in this equation, T is the absolute temperature, i are the stoichiometric numbers for the dissociated salt and zi are the ion charges, including the signs. It has long been known that this is exact only in the case of infinite dilution of the salt, where the ions do not interact. Nevertheless, it has been widely applied to other situations,27 especially neat ionic liquids, despite the unsuitable nature of the model in such cases. The “ionicity” (Y) is given by the ratio of the measured conductivity to that estimated from eq 1, with the assumption that any reduction in conductivity is due to some form of ion association forming long-lived neutral molecular pairs or more lowly charged aggregates,


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Y

  (1  )  NE

(3)

Y is given this name on the basis that when values are less than unity, it is proportional to the fraction of ions in the IL, again with the usually unstated assumption of no other kind of interaction.2,12,28  is known as the NE deviation parameter and is also commonly used in discussions of the difference between experimental molar conductivities and those given by eq 1. It was shown many years ago in the early molecular dynamics simulations of molten salts by Hansen and MacDonald 29,30 and by Trullàs and Padró31 that  is non-zero in simple molten salts such as NaCl where the ions are clearly not associated. Padró, Trullàs and Sesé32 subsequently showed that  is given by the difference between the velocity cross-correlation coefficient (VCC or fij) of the cation and anion, (an ensemble and time average of the GreenKubo velocity cross-correlation function) and the mean of those for cation-cation and anionanion velocity cross-correlation coefficients:



c(2   z z f    2 z2 f    2 z2 f   ) (  z2 DS   z2 DS )

c(2 f    f    f   )  ( DS /    DS /   )

(4)

where

fij 

N AV  vi (0)gv j  (t ) dt , i, j  ,  3 0

(5)

with  and  denoting different cations or anions.33 It is easily seen from eq (3) that the NE equation holds formally for a 1:1 salt when f+- for the oppositely charged ions is the arithmetic mean of f++ and f— for the like-charged ions. These same equations (3 and 4) also hold for binary electrolyte solutions.34 (Though there are three additional VCC in such cases due to the presence of a solvent, these do not appear in expressions for the conductivity and ion self-diffusion


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coefficients. They do appear in those for ion transport numbers, solvent-salt inter-diffusion coefficient and the solvent self-diffusion coefficient.) The ion-ion VCC tend to zero at infinite dilution,34,35 consistent with the original Nernst-Einstein relation. The VCC are negative in a neat molten salt or ionic liquid due to the principle of conservation of momentum in the fluid, showing that ion velocities are anti-correlated.33 This applies to the interactions of like-charged ions as well as ions of opposite charge, though the latter are generally much smaller than the former. Put simply, if an ion moves, the rest of the system moves in the opposite direction in proportion to the ratio of the ion mass to that of the rest of the system so that the total momentum is conserved. The concept of such an anticorrelation in a molten salt is counter-intuitive, with many, including Angell et al.3 assuming that positive correlations of the velocities of opposite charged ions must occur in molten salts and ionic liquids as in dilute electrolytes. But an additional component, a solvent, or another salt is required before this can be possible: a third species is needed to absorb momentum to allow positive correlations between the velocities of the other two species (called “momentum buffering”35). Interestingly, VCC calculations from experimental transport property data show that cation-anion velocity anti-correlations also occur in moderately concentrated electrolyte solutions.34,36 It is important to realise that the VCC are (non-equilibrium) thermodynamic quantities directly calculable from the experimentally measurable transport properties.33 As such they can be estimated from molecular dynamics simulations using appropriate force fields (interionic potentials), though this has yet to be satisfactorily achieved, other than as a demonstration of anti-correlated ion motions.35 In principle, they are calculable from theory. Most of the discussion around the Nernst-Einstein relation (eq 1) in the current literature centres on assumed


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ion association as the explanation for positive  However it has been shown that negative likeion Laity resistance coefficients are a better indicator and that common aprotic ionic liquids do not demonstrate such behaviour.17 Charge transfer from anion to cation has also been suggested, but in actual practice this still assumes the validity of eq 1, albeit with reduced zi values.37,38 Both approaches (ion association and charge transfer) neglect the assumptions inherent in the derivation of eq 1, that is that the charged species do not interact. There are no such assumptions in the VCC description. In a pure IL, the ion-ion VCC are negative and hence the anticorrelations they represent are not due to the characteristics of any model but are a real consequence of the Law of Conservation of Momentum. Deviations of measured conductivities from those predicted by eq 1 are the result of the difference between the cation-anion VCC and the mean of those for the cation-cation and anion-anion VCC (eq 3). At this stage, it remains for the necessary theory to be developed to examine empirical trends found for  and for the VCC as functions of temperature, pressure and viscosity.17,39,40,41,42,43,44,45,46,47 The choice of the word “ideal” to describe Angell’s reference line has been unfortunate as some have interpreted this to mean a thermodynamically ideal electrolyte solution, that is, one with activity coefficients of unity on the molarity concentration scale, which aqueous KCl is certainly not.48 In ref. 3 it was made clear that the choice of a reference solution was influenced by the observation that the electrical mobilities of the potassium and chloride ions, deduced by combining conductivities and transport numbers, are equal at a given temperature and concentration, though of course both are composition dependent. Aqueous KCl solutions appear unusual at first sight in that transport numbers are almost independent of concentration to approximately 3 mol·L-1, as is the viscosity, to around 2.5 mol·L-1,49 so at one time this was interpreted to mean that these ions have little effect on the structure of water.50 However, ion and


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water self-diffusion measurements show more complex composition dependences,51,52 and the mutual or inter-diffusion coefficient actually has a minimum at 0.25 mol·L-1, increasing strongly at higher compositions.53.More sophisticated comparisons of the transport properties of ions in solution, including mutual and self-diffusion coefficients, along with the conductivity and transport numbers, employing velocity correlation coefficients derived from these quantities, show that water-ion and ion-ion interactions in this system form part of a logical progression from H+ to Cs+ and from Cl- to I-.34 In other words, aqueous KCl solution transport properties should not be examined in isolation of one another. Aqueous KCl solutions at 25 oC do not comply with Walden’s rule due to this constancy of the viscosity to moderate concentrations. This is very clearly shown in Fig. 4 of Schreiner et al.54, where the Walden plot is given as a vertical line between infinite dilution and 1.5 mol·L-1! As these latter authors point out, one can obtain a linear Walden plot by using infinite dilution conductivities over a range of temperatures, but with a slope of about 0.87, not unity. Such behaviour has been known in aqueous electrolytes for well over a century,17,55 but ref. 3 incorrectly asserts that “the system obeys the Walden rule very well”. Schreiner et al. further criticize the arbitrary nature of the choice of a line of unit slope passing through the 1 mol·L-1 aqueous KCl reference point, saying a) “there is no theoretical foundation justifying the comparison of pure ILs’ (sic) conductivity and viscosity to that of any (semi)diluted electrolyte solution”56; b) that the very similar mobilities of K+ and Cl- ions in aqueous solution is of no relevance in examining the behaviour of neat (single component) ionic liquids; and c) that an “infinitely diluted KCl solution would be more reasonable as electrolyte theories are far more developed for that than for any concentrated solutions.” The present author strongly agrees with criticisms (a) and (b), but not the third point, as water is very different in structure and properties


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to a molten ionized salt. Despite the qualitative convenience of the Angell-Walden reference line for ranking liquids based on their conductivity at a given viscosity, as demonstrated in many studies, it seems merely coincidence that many aprotic ionic liquids yield Walden plots that lie close to it, and have slopes close to unity. In this work the Angell-Walden analysis is used to examine simple molten salts at high temperatures, a test that does not appear to have been made previously. It is found that though these salts consist of unassociated ions, many appear “super-ionic” in the Angell sense. Another salt where association and ion-complex formation does occur, ZnCl2, also appears nearly “ideal” on the plot. These contradictory examples show that comparisons of salt and ionic liquid Walden plots with Angell’s reference line can lead to mistaken conclusions and do not give rise to useful information. It can not and should not be used for classifying the interactions in ionic liquids.

Walden Plots for Molten Salts Table 1 lists the data sources for the conductivity and viscosity of the high and moderate temperature molten salts examined,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78 including the recent critically examined reference correlations for the viscosity of some halides made by Tasidou et al.79 (The Supporting Information lists the values employed.) Data sources for an intermediate-temperature molten salt (tetrabutylammonium tetrabutylborate, [NBu4][BBu4],80,81,82 the eutectic mixed nitrate Rb3Na2(NO3)583,84,85and three room-temperature ionic liquids , the aprotic 1-butyl-3-methylimidazolium bis(trifluoromethylsulfonyl)imide, [BMIM][Tf2N],86,87,88 and the protic salts, 1-methyl-2-oxopyrrolidinium tetrafluoroborate, [PyrOMe][BF4],89 and 1,8-diazabicyclo-[5,4,0]-undec-7-eneium methanesulfonate, [DBUH][CH3SO3]90 are also listed. Table 2 lists Walden slopes, the classification obtained from


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the Angell analysis and Nernst-Einstein deviation parameters, , derived from conductivity and self-diffusion data.43,60-70,91,92,93,94,95,96

Discussion Figure 2 is a Walden plot for the high-temperature molten salts, with melting points (mp) from 334 oC (KNO3) to 993 oC (NaF). The Angell KCl solution reference line is the main diagonal. Figure 3 is an extension to higher viscosities and lower conductivities to show plots for the associated salt ZnCl2, mp 290 oC, and three ionic liquids, the aprotic archetype [BMIM][Tf2N] and two protic examples, [PyrOMe][BF4] and [DBUH][CH3SO3], together with the intermediatetemperature example of [NBu4][BBu4], mp 109 0C. The black circle is the Angell reference point for 1M aqueous KCl at 25 oC. Figure 4 is a deviation plot showing the (logarithmic) difference between the conductivity ( log10 ) of each of the salts considered and the reference at the same viscosity: this allows the reader to distinguish individual plots more easily. The negative slopes in this plot are due to the Walden plot slopes being less than unity.  log10  values below the x-axis are equivalent to W, used to quantify ionicity.11 A W value of 1 indicates the salt has only 10% of the ionic conductivity that it would exhibit if it followed the 1M KCl Angell reference line. Values of 0.1 and 0.5, correspond to 79 % and 32 % respectively, as such plots are of course nonlinear. A value of -0.5 (above the x-axis) corresponds to more than three times the reference value (316 %). The Walden plots of the six “basic” alkali metal (Li, Na and K) fluorides and chlorides (except KCl) that are known to have free halide ions with no cation-anion association97 lie all or in part well above the aqueous KCl reference line, and therefore should be classified as “superionic” by Angell’s scheme. But these salts “can be depicted by a simple picture of charged


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hard spheres, and are thus easily described by an alternation of cationic and anionic shells around a central ion”,98 and have low, and very similar, NE deviation parameters (Table 2). CaCl2 also falls into this group. Clearly, the Angell analysis fails for these salts, and the conduction mechanism is simply that due to ion migration in an electric field. The plots for KCl, CsCl and KNO3 fall very close the reference line, but again there is nothing known from their liquid structures98,99,100 to differentiate them from the alkali halides in terms of their conductivities. It is interesting that the CuCl melt is also wrongly predicted to be superionic: the Nernst-Einstein deviation parameter is similar to that of the alkali and alkaline earth metal halides (Table 2), though it is known to have a super-ionic high-temperature phase in the solid state. There is said to be some intermediate range ordering101 in this case with the cations being more mobile, and occupying voids in a lattice of anions,102,103 and the salt is regarded as fully ionic. ZnCl2 (see Figures 3 and 4) is predicted to be “ideal”. Yet this is well known to be an associated salt with a complex liquid structure,70,104,105,106,107 and has a very large NE deviation parameter between 0.8 and 0.9 (Table 2). Again, the Angell -Walden analysis fails. Given all these examples have positive NE deviation parameters, , that is conductivities less than those given by the NE approximation, the assertion made by some that “ionicity” calculations based on the Nernst-Einstein equation and Angell-Walden plots are equivalent13,108 and that hypothetical salts for which  is zero fall on the aqueous KCl Angell reference line, is, in fact, without substance. A referee questioned the examination of the molten salts listed above as they are much more fluid than the KCl reference solution and ionic liquids. Figures 2 to 4 now also include lower melting salts, lithium and sodium chlorates (solid triangles) and potassium thiocyanate (mp: 128,


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248 and 173 oC respectively): the Li salt has a higher viscosity (lower fluidity) than the reference point and the Na salt a lower one, so these examples help to bridge the temperature and fluidity gaps. Lithium chlorate appears “superionic” whereas sodium chlorate is borderline “ideal”. Potassium thiocyanate straddles the reference line. Also included is the mixed nitrate, Rb3Na2(NO3), which is a eutectic mixture84,85 (m.p ~177 oC) where the conductivity and viscosity data extend over a very broad temperature range: it also straddles the KCl reference point. In this case the Angell-Walden analysis predicts no association. No ion-self-diffusion data are available for the calculation of the NE deviation parameter for these examples, so it is not possible to compare the two approaches for these salts. Of the ionic liquids listed, the moderate temperature example [NBu4][BBu4] and the typical aprotic [BMIM][Tf2N] lie close to the Angell line and are classified as “good” ionic liquids. However; the two protic examples, [DBUH][CH3SO3] and[PyrOMe][BF4], which like ZnCl2 have negative Laity like-ion resistance coefficients, which can be regarded as an indicator of cationanion association,17,104 also lie close by on the Walden plot, but well below the seemingly “ideal” ZnCl2 plot. Clearly, the Angell analysis of Walden plots gives contradictory results for the examples given above. Even if “superionicity” is assumed for the salts with plots above the Angell-Walden reference line, this is contradicted by the positive Nernst-Einstein deviation parameters derived from experimental salt conductivity and ion self-diffusion data for the examples given in Table 2. So the arguments employed by MacFarlane et al.12,13 and others in attempting to quantitatively estimate degrees of association based on the Angell-Walden analysis are flawed.

Ionicity


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Given that the Angell-Walden analysis fails for simple molten salts and that interpretation of the Nernst-Einstein deviation parameter really requires molecular dynamics simulations of the VCC or new theoretical developments for its interpretation it is perhaps worth reiterating the usefulness of Laity resistance coefficients for dealing with true association in molten salts and ionic liquids.17,104 The Laity resistance coefficients (rik) are non-equilibrium thermodynamic quantities. Like the statistical mechanical VCC, they are derived from the measurable transport properties and are independent of any model, other than the presumption of the existence of charged ions in a salt melt. They are defined by N

X i  (grad i )T   rik xk (vi  vk )

(6)

k 1

where Xi is the thermodynamic or generalised frictional force on ion species i in an electrochemical potential gradient and xk is the mole fraction of k. (vi - vk) is the velocity of species i relative to that of species k, and, consequently, the rik are independent of any frame of reference (unlike VCC, except in single component systems). The Onsager reciprocity relation, rik = rki holds. For a one component system with two ionic species,109,110,111 r  z  ( z  z ) F 2 / 

(7)

and

rii 

 1  ( z  z ) RT  zi r  , i = +,-; j  i  DSi zj  

(8)

r+- is necessarily positive, but the two rii may be positive or negative. The Nernst-Einstein  can be written in terms of the Laity resistance coefficients.45


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

2 z z r 1  r r / r 

2z



z r  z2 r  z2 r 

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(9)

Formally, if

r2  r r

(10)

that is, the cation-anion resistance coefficient is the geometric mean of the cation and anion resistance coefficients, then  is zero.45,112 If either r   z r / z or

r   z r / z

(11)

then  = 1, a new result. (The two solutions are the real roots of a binomial expression: either one is sufficient.)

 approaches 0.5 as r++ and r— approach zero (from positive values). The self-diffusion coefficients are given by

DSi 

( z  z ) RT ( z j rii  zi r )

,

ji

(12)

Under these circumstances, the cation-anion resistance term dominates, and for a 1:1 salt, the cation and anion DSi are equal. Figure 5 illustrates the trend for 1-alkyl-3-methylimidazolium salts where the r-- become smaller (and in the case of [OMIM][BF4], negative) with increasing cation size.45 (The r++ are all positive). The cation and anion DSi ratios are > 1 for the [BMIM]+ salts, approximately 1 for the [HMIM]+ salts, and < 1 for the [OMIM]+ salts, so the Figure is only suggestive of a limit. In an examination of molten salt resistance coefficients negative like-ion rii were found only for ZnCl2 and TlCl (Figure 6).17 As discussed above, the former is well-known to contain complex chlorozincates. Laity suggested that weakly dissociating molecular liquids, such as water, could be formally treated as highly associated molecular liquids. This does in fact yield negative


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like-ion resistance coefficients for water, formic and sulfuric acids,17 and their  values are very close to unity, in conformity with eq. 10 (see Table 3 of ref. 17). (For ZnCl2 at the lowest experimental temperature (600 K), [2r+-/(-r++)] and [r+-/(-2r--)] are 1.10 and 1.20, for comparison, giving a measure of the association in this case). This result is relevant to the study of protic ionic liquids, which are the result of proton transfer from one molecular liquid to another. When the degree of transfer is incomplete, the liquid may in fact contain both ions and the molecular liquids, just as in Laity’s suggested examples. As mentioned above, [DBUH][CH3SO3] and [PyrOMe][BF4] have been found to have negative like-ion rii and therefore behave like weakly dissociating molecular liquids (Figures 5 and 6). The NE  is greater than 0.5. [r+-/(-r++)] and [r+/(-r--)] are 1.2 and 1.2 respectively at 303 K (and strongly temperature dependent) for [DBUH][CH3SO3] and 2.3 and 1.8 respectively, independent of temperature between 298 and 363 K, for [PyrOMe][BF4]. A more extreme case is that of pseudo-protic ionic liquids, conducting liquids that are 1:1 mixtures of molecular liquids.113 These exhibit a conductivity higher than expected for such a mixture, though still low relative to true ionic liquids (Figure 3). This is argued to be due to a Grotthuss-like conduction mechanism involving proton transfer.114 The example of Nmethylimidazolium acetate where this mechanism has been confirmed experimentally also has negative like-ion Laity resistance coefficients,115 and again the NE  is quite large and close to unity (Figure 5). In this case, [r+-/(-r++)] and [r+-/(-r--)] are both about 1.1 between 298 and 323 K.

Conclusions The Angell analysis of Walden plots has been applied to a number of inorganic hightemperature molten salts. It wrongly predicts “super-ionicity” for examples such as alkali metal


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fluorides, chlorides and chlorates, as well as for the associated salt zinc chloride. It is shown as a consequence to be inconsistent with negative deviations from the Nernst-Einstein equation relating conductivities with ion self-diffusion coefficients (positive NE). Results for certain protic ionic liquids are also found to show inconsistencies. It is concluded that Angell-Walden analyses of this type do not convey any useful information other for qualitative ranking of the conductivity of ionic liquids at a given viscosity and their use for estimating “ionicity” either qualitatively or semiquantitatively is best discontinued. It can not and should not be used for classifying the interactions in ionic liquids. It is argued that an examination of Laity resistance coefficients is more useful in any discussion of true association in molten salts and ionic liquids. Known examples show negative like-ion resistance coefficients with NE deviation parameters close to unity. This does not prove that such behaviour can be used to predict association – this requires a tractable link between molten salt theory and non-equilbrium thermodynamics. However it seems likely that such an approach could be more fruitful in understanding the transport properties of molten salts and ionic liquids rather than simple comparisons of viscosity and conductivity.

ASSOCIATED CONTENT Supporting Information The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jpcb.xx. Reference table for the transport properties of molten salts. See also the Supporting Information for ref. 17.


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AUTHOR INFORMATION Corresponding Authors E-mail: [email protected] Notes The author declares that he has no competing financial interests.

ACKNOWLEDGMENTS The author is grateful to Dr Mitsuhiro Kanakubo (AIST-Tohoku, Sendai, Japan) and the referees for helpful comments on the manuscript. Portions of this work were presented at the Joint Conference of EMLG/JMLG Meeting 2018 and 41st Symposium on Solution Chemistry of Japan, Nagoya University, Nagoya, Japan, on 6 November 2018.


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111 Laity, R. W. An Application of Irreversible Thermodynamics to the Study of Diffusion. J. Phys. Chem. 1959, 30, 80-83. 112 Takagi R.; Kawamura, K. An Explanation of the Nernst-Einstein Relation in Terms of Friction Coefficients. Bull. Tokyo Inst. Tech. 1975, 127, 57-62. 113 Doi, H.; Song, X.; Minofar, B.; Kanzaki, R.; Takamuku, T.; Umebayashi, Y. A New Proton Conductive Liquid with No Ions: Pseudo-Protic Ionic Liquids. Chem. Eur. J. 2013, 19, 1152211526. 114 Ingenmey, J.; Gerhke, S.; Kirchner, B. How to Harvest Grotthuss Diffusion in Protic Ionic Electrolyte Systems. ChemSusChem 2018, 11, 1900-1910. 115 Watanabe, H.; Umecky, T.; Nozaki, E.; Arai, N.; Nazet, A.; Takamuku, T.; Harris, K. R.; Kameda, Y.; Buchner, R.; Umebayashi, Y. Possible Proton Conduction Mechanism in PseudoProtic Ionic Liquids: The Concept of Specific Proton Conduction. J. Phys. Chem. B 2019, DOI: 10.1021/acs.jpcb.9b03185, in press.


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FIGURES

Figure 1. Angell’s classification of ionic liquids relative to the “ideal” aqueous KCl reference line, of unit slope, that passes through the conductivity and viscosity coordinates of 1M KCl at 25 oC (black circle). The blue line represents a Walden plot for [BMIM][Tf2N], an archetypal “good” ionic liquid (data86,87). W is the vertical shift used by some authors to estimate “ionicity”.12,13


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Figure 2. Walden plots for inorganic salts. The solid line is the Angell reference line. Symbols: black solid circle, Angell reference point for 1 M KCl(aq) at 25 oC; circles, fluorides – Li, black, Na, blue, K, red; squares, chlorides - Li, black, Na, blue, K, red, Cs green; inverted blue triangles, CaCl2; triangles, CuCl; diamonds, nitrates, K, - black, Rb3Na2(NO3)5, green; solid triangles, LiClO3, Note that the equivalent conductivity [/(+z+)] is plotted to normalize the conductivity of CaCl2 and Rb3Na2(NO3)5 relative to the 1:1 salts.

Figure 3. Figure 2 extended to higher viscosities and lower conductivities. Symbols: black solid circle, Angell reference point for 1 M KCl(aq) at 25 oC; solid triangles, chlorates - Li, black, Na,


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Page 36 of 41

blue; inverted red triangles, ZnCl2 and green open diamonds, Rb3Na2(NO3)5; coloured circles, ionic liquids: [NBu4][BBu4], red; [BMIM][Tf2N], blue; [DBUH][CH3SO3], magenta; [PyrOMe][BF4], green; the pseudo-protic N-methylimidazolium acetate, magenta triangles, (see text). Note that the equivalent conductivity [/(+z+)] is plotted to normalize the conductivity of ZnCl2 and Rb3Na2(NO3)5 relative to the 1:1 salts.

Figure

4.

Plot

of

the

difference

between

the

experimental

conductivities

[as

log10(equiv/S·cm2·mol-1)] and the Angell KCl reference line as a function of fluidity. The “ideal” region has been arbitrarily chosen to be between the limits of 0.1 and -0.1 (126 to 79% of the reference conductivity). It is not known why the Walden plots for [DBUH][CH3SO3] and [PyrOMe][BF4] appear to be convex: Walden plots can become concave at very low temperatures due to the decoupling of mass(charge) and momentum transfer. Symbols: as Figures 2 and 3.


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Figure 5. Nernst-Einstein deviation parameters for a) typical aprotic ionic liquids (data: ref. 45) and b) the protic ionic liquid [PyrOMe][BF4] (data: ref. 89) and the pseudo-protic ionic liquid, Nmethylimidazolium acetate (data: ref. 115). Note the apparent convergence for the former on  = 0.5 as the cations become larger. Symbols: 1-alkyl-3-methylimidazolium salts - [BMIM][PF6], black circles; [HMIM][PF6], red circles; [OMIM][PF6], blue circles; [BMIM][BF4], black open circles; [HMIM][BF4], red open circles; [OMIM][BF4], blue open circles; [PyrOMe][BF4], black squares; N-methylimidazolium acetate, black triangles.


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Figure 6. (a) Analogue Stokes-Einstein-Sutherland plot of Laity resistance coefficients rij for the molten salts NaCl (circles), KCl (squares), and KNO3 as a function of fluidity, ϕ , (reciprocal viscosity). Symbols: black, r+−; blue, r++; red, r−−; (b) and (c) Laity resistance coefficients rij for the associated molten salt ZnCl2 and the protic ionic liquid [PyrOMe][BF4]. The like-ion resistance coefficients are positive for the unassociated salts and negative where there is ion association or aggregation. Reproduced from ref. 17 [Figures 8, 10(a) and 10 (b)] with permission from the American Chemical Society.


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TABLES. Table 1. Sources of Transport Property and Density Data for Molten Salts and Ionic Liquids salt







LiF

ref. 57

ref. 58,59

ref. 59

NaF

ref. 60

ref. 59

ref. 60

KF

ref. 60

ref. 59

ref. 61

LiCl

ref. 62

ref. 79

ref. 61

NaCl

ref. 62

ref. 79

ref. 60

KCl

ref. 63

ref. 79

ref. 63

CsCl

ref. 63,64

ref. 79

ref. 63

CuCl

ref. 66

ref. 69

ref. 69

KNO3

ref. 65

ref. 65

ref. 65

CaCl2

ref. 67

ref. 68

ref. 67

ZnCl2

ref. 70

ref. 71

ref. 72

LiClO3

ref. 74

ref. 73

ref. 73

NaClO3

ref. 76

ref. 75

ref. 75

KSCN

ref. 77

ref. 78

ref. 78

Rb3Na2(NO3)2

ref. 85

ref.84

ref. 83

[NBu4][BBu4]

ref. 80

ref. 81

ref. 82

[BMIM][Tf2N]

ref. 86

ref. 87

ref. 88

[PyrOMe][BF4]

ref. 89

ref. 89

ref. 89

[DBUH][CH3SO3]

ref. 90

ref. 90

ref. 90


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Page 40 of 41

Table 2. Walden Data t



(Walden slope)

Angell-Walden analysis

(from ref. 36)

LiF

0.57 (1123-1340 K)

“super-ionic”

0.23 (1123-1173 K)

NaF

0.81 (1280-1400 K)

“super-ionic”

0.27 (1275 K)

KF

0.62 (1130-1300 K)

“super-ionic”

0.27 (1123-1173 K)

LiCl

0.45 (880-1130 K)

“super-ionic”

0.21 (880-1040)

NaCl

0.47 (1080-1200 K)

“super-ionic”

0.17-0.29 (1030-1300 K) a

KCl

0.58 (1060-1200 K)

“ideal”

0.22-0.26 (1080-1200 K)

CsCl

0.88 (950-1170 K)

“ideal”

n/a

CuCl

0.23 (720-1020 K)

“super-ionic”

0.21-0.36 (720-1020 K)

KNO3

0.83 (610-720 K)

“ideal”

0.24 (620-660 K)

CaCl2

0.66 (800-1000 K)

“super-ionic”

0.27 (1070-1270 K)

ZnCl2

0.88 (600-800 K)

““ideal”

0.94-0.81 (600-800 K)

LiClO3

0.79 (403-433 K)

“super-ionic”

n/a

NaClO3

0.92 (533-563 K)

“ideal”

n/a

KSCN

0.86 (450-520 K)

“ideal”

n/a

Rb3Na2(NO3)2

0.91 (450-745 K)

“ideal”

n/a

[NBu4][BBu4]

0.79 (385-454 K)

“good IL”

n/a

[BMIM][Tf2N]

0.94 (243-363 K)

“good IL”

0.37 (288-363 K, to 250 MPa)

[PyrOMe][BF4]

0.90 (298-363 K)

“poor IL”

0.76 (298-363 K)

[DBUH][CH3SO3]

1.04 (303-383 K)

“good IL”

0.93 – 0.51 (303-383 K)

salt

a

The available self-diffusion data for NaCl have a different temperature dependence to the

conductivity. The slope of the consistency plot of ln (T) versus ln(DS+ + DS-) is 0.60 for the data of Bockris et al.93 and 0.72 for that of Ejima et al.,94 rather than the expected value of unity. The  values given are derived from Ejima’s diffusion results as these give Stokes-Einstein-Sutherland plots with slopes < 1, consistent with other salts and ionic liquids.15-17,39-4


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On the Use of the AngellâWalden Equation To Determine the âIonicity

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