J. Phys. Chem. B 2008, 112, 8285–8290
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Concentration Dependence of Ionic Transport in Dilute Organic Electrolyte Solutions Matt Petrowsky and Roger Frech* Department of Chemistry and Biochemistry, UniVersity of Oklahoma, Norman, Oklahoma 73019 ReceiVed: February 8, 2008; ReVised Manuscript ReceiVed: April 3, 2008
Ion transport is studied in dilute organic liquid electrolyte solutions in which close cation-anion interactions are minimized either through steric hindrance imposed by the bulky tetrabutylammonium cation or by strong solvation of alkali metal cations by DMSO or 1-propanol. In these solutions, the molar conductivity does not appear to depend on either the solvent viscosity or the size of the solvated charge carrier in a manner consistent with Walden’s rule. The molar conductivities plotted as a function of the solvent dielectric constant from ε ) 5.48 to 63.5 appear to lie on a smooth curve for a set of 0.0055 M solutions of tetrabutylammonium trifluoromethanesulfonate in a variety of aprotic solvents. The molar conductivity smoothly increases with increasing dielectric constant to a maximum at roughly ε ) 33 and then decreases with further increase of the dielectric constant. The conductivity appears to depend only on the dielectric constant and not the specific functional group in this broad family of solvents. A similar plot for a series of linear alcohols as solvents also led to a smooth curve, although the values of the molar conductivity were lower than values in the aprotic solvents by almost an order of magnitude at corresponding values of the solvent dielectric constant. 1. Introduction Ion transport in liquid electrolytes has been studied for over a century, highlighted by concerted fundamental studies including the pioneering work of Kohlrausch,1 Walden,2 Debye and Huckel,3 Fuoss,4 and Kraus.5 Despite extensive work, there is no clear molecular-level picture of how ion transport is governed by the salt concentration, dielectric constant, temperature, and the choice of salt and solvent. Part of the complexity arises because the dielectric constant itself depends on the other four variables.6–9 Early work focusing on the concentration dependence of the conductivity resulted in Kohlrausch’s law, an empirical description of the dependence of the molar conductivity Λ on the salt concentration c
Λ ) Λo - Ac1/2
(1)
where Λo is the molar conductivity at infinite dilution and the constant A is determined from the slope of a plot of Λ against c1/2. This equation was subsequently placed on a theoretical basis by considering the ionic cloud about a central ion; the constant A could then be calculated from the solvent dielectric constant and viscosity, the temperature, and the ionic charges. In that form, eq 1 is usually termed the Debye-Hu¨ckel-Onsager equation.3,10,11 Later extensions of this model by Fuoss and Onsager included effects of electrophoresis, the relaxation field, and finite ion sizes;12–14 these effects have also been treated by linear response theory.15,16 Cation-anion interactions play an important role in ion transport and complicate the analysis of conductivity data. The concentration dependence of conductivities in liquid and polymeric organic electrolytes have been interpreted in terms of the formation of ionically associated species such as ion pairs and more highly associated ionic aggregates.17,18 However, it has been argued that the association constants obtained from these analyses do not necessarily correspond to formation of contact ion pairs or complexes; rather, these constants should * To whom correspondence should be addressed.
be viewed as a way of representing the effects of strong Coulombic forces at short interionic distances.19,20 There is a range of cation-anion interactions that result in what is somewhat loosely termed “ionic association”. One example is the coordinative interaction between a lithium ion and the oxygen atom(s) of a trifluoromethanesulfonate or triflate ion, CF3SO3- (abbreviated Tf). These interactions result in longlived, discrete ionic species whose presence can be spectroscopically detected.21–23 The strength of these interactions leads to electronic redistribution within bonds that result in normalmode frequency shifts and spectral intensity changes that allow an experimental distinction between a “free” triflate ion, a contact ion pair (LiTf) or a triple anion (LiTf2-), a triple cation (Li2Tf+) or a dimer (Li2Tf2), and a quadruple cation (Li3Tf2+).23 In contrast, the interaction between a tetrabutylammonium (Tba) cation and a triflate anion is primarily a monopole-monopole interaction and is relatively weak because a close approach of the anion to the nitrogen atom of the cation is frustrated by the bulky butyl groups. Between these two extremes lie other degrees of cation-anion interaction strengths; however, it is important to note that Coulombic (monopole-monopole) and non-Coulombic interactions between cations and anions are always present, leading to some degree of ion association. On the time scale of IR spectroscopy, solutions of tetrabutylammonium triflate, TbaTf, contain only “free” cations and anions in the sense that the triflate anion undergoes no electronic redistribution. Therefore, from a vibrational spectroscopic point of view, discrete species do not exist in TbaTf solutions. Of course, ions (particularly cations) that are spectroscopically “free” are solvated by solvent molecules. The electrolyte solutions considered in this paper are chosen so that cation-anion interactions do not result in spectroscopically distinct, ionically associated species. In most cases, this occurs because the cation is the charge-protected Tba cation, as discussed above. Although alkylammonium electrolyte data have been interpreted in terms of the presence of discrete species and lead to calculations of equilibrium constants for the association process,9,24 these constants may only be a convenient
10.1021/jp801146k CCC: $40.75 2008 American Chemical Society Published on Web 06/21/2008
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representation of the effect of Coulombic interactions as noted above. In a few dilute solutions considered here, the solvent molecules strongly solvate the cation such that spectroscopically distinct, ionically associated species do not form. In this study, the solutions in Figures 2 and 3 have concentrations < 0.01 M. Most of the solutions studied here utilize the triflate anion because of its spectroscopic sensitivity to the formation of discrete cation-anion species. The transport of ions in an electrolyte solution is usually described by either the molar conductivity Λ or the conductivity σ, where Λ ) σ/c and c is the salt concentration. The conductivity of an electrolyte solution is given by the following wellknown expression
σ )
∑ ni|zie|µi
(2)
to propose a modification of Walden’s rule by incorporating an explicit salt concentration dependence c
µ ∝ c/∆η
(6)
where µ is the ionic mobility and ∆η is the change in solvent viscosity with addition of salt to pure solvent to bring the salt concentration to the value c.26 However, Prego et al. found that Walden’s rule successfully explained the conductivity resulting from self-ionization in a series of alcohols.27 Most of the extensive literature on the subject of Walden’s rule concludes that this relationship is, at best, a rough approximation, but a more accurate alternative is not known. In general, the concentration dependence of Λ results from factors contributing to the nonideality of the solution including ion-ion and ion-solvent interactions.
i
where i designates a charge-carrying species, ni is the number density, zi is the charge number, µi is the ionic mobility, e is the proton charge, and the sum is over all ions in the solution. When the salt MmAa is completely dissociated in a solution with concentration c yielding only ions Mz+ and Az-, eq 2 may be written
σ ) mcz+Fµ+ + ac|z-|Fµ-
(3)
where F is Faraday’s constant and z+ and z- are the charge numbers of the cation and anion, respectively. The requirement of overall charge neutrality quickly leads to the well-known relationship between the sum of the ionic mobilities and the molar conductivity Λ
(µ+ + µ-) ) Λ/Fmz+
(4)
The sum of the cation and anion mobilities can be calculated for TbaTf electrolyte solutions from eq 4 with mz+ ) 1 by measuring the conductivity of the system since these solutions contain only “free” ions. Therefore, studying TbaTf-containing electrolytes in a wide variety of solvents at varying salt concentrations will help elucidate the concentration dependence of the ionic mobility, which is the primary goal of this work. In the initial part of this study, the application of Walden’s rule to these electrolyte solutions will be briefly examined since it is expected that Walden’s rule would apply to dilute solutions that do not form discrete ionically associated species if the rule has any semiquantitative validity. Early studies of ion transport in electrolyte solutions have often invoked Walden’s rule, which can be written as
constant Λ) ηr
(5)
where η is the solution viscosity and r is the radius of the mobile charged species, the solvated ion. There is no explicit concentration dependence of Λ in Walden’s rule, and in principle, it should only be used for very dilute solutions. The applicability of Walden’s rule to electrolyte solutions has been widely tested with mixed results. For example, Longsworth and MacInnes studied solutions of sodium chloride and lithium chloride in water-methanol mixtures, reporting that the product Λη “deviated greatly from Walden’s rule” and that the apparent size of the ion-solvent entity increased with increasing methanol concentration.25 Moreira et al. examined solutions of 2,6lutidinium chloride in a variety of solvents and at different temperatures, reporting a breakdown of Waldens rule.20 The failure of Walden’s rule in solutions of CoBr2 in liquid oligomeric poly(ethylene glycol) led Mendolia and Farrington
2. Experimental Section All solvents (g99% pure) were obtained from Aldrich and used as received. Salts were obtained from Aldrich (purity g 99%) and dried under vacuum at 85 °C for 24 h. The samples were prepared by dissolving weighed amounts of salt into the appropriate amount of solvent and then stirring for 24 h. The sample preparations (excluding aqueous KCl to determine the cell constant) and all measurements were carried out in a glovebox ( 10, that is, ethanol and 1-hexanol, the molar conductivity decreases with increasing salt concentration in a nonlinear manner. The molar conductivity values also decrease from ethanol to 1-hexanol at comparable concentrations by almost an order of magnitude. As the length of the alkyl chain increases and ε < 10, that is, 1-octanol and 1-decanol, the molar conductivities continue to decrease. However, the salt concentration dependence of Λ exhibits a somewhat unusual form; the molar conductivity initially decreases with increasing salt concentration to a minimum (region I), then increases to a maximum (region II), and again decreases at high salt concentrations (region III). The designation of three distinctive regions of behavior follows Albinsson et al.30 The behavior of Λ shown in Figure 2 is typical for low dielectric electrolytes and was first described by Kraus and Fuoss.5 There are numerous other examples of lithium salts in low dielectric solvents that exhibit this behavior.31–35 In the very dilute region of low dielectric constant (ε < 10) solutions, the decrease in Λ with increasing salt concentration (Figure 2, region I) is generally attributed to the increasing formation of ion pairs that can be either contact ion pairs as in LiClO4 solutions36 or solvent-separated ion pairs as in LiAsF6 solutions.35 The decrease in Λ at high salt concentrations (Figure 2, region III) is usually attributed to the increase of bulk viscosity
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Figure 3. Molar conductivity versus the square root of the salt concentration at 25 °C for solutions of TbaTf-solvent (solvent ) methyl acetate, tetrahydrofuran, diglyme).
Figure 2. Molar conductivity versus square root of salt concentration for a series of LiTf-alcohol electrolytes at 25 °C.
in the liquids.31 However, various authors have offered very different explanations for the increase in molar conductivity from the minimum to the maximum value (Figure 2, region II). This increase has been attributed to the formation of charged triple ion aggregates37 in solutions of LiClO4 dissolved in tetraglyme31 and tetrahydrofuran.32 Alternatively, Cavell and Knight suggested that the change of solution permittivity with salt concentration played a significant role in the dependence of the molar conductivity on salt concentration, particularly in region II.6 In general, the dissociation of salt would decrease with increasing salt concentration according to the Le Chatelier principle. However, according to Cavell and Knight, the solution permittivity and therefore the equilibrium constant governing the salt dissociation increase with salt concentrations. These two trends lead to a minimum in the molar conductivity. At higher salt concentrations, the increase in the equilibrium constant leads to an increasing degree of dissociation, which in turn results in increasing values of Λ. Later authors have referred to this as “redissociation”. For example, in a study of thiocyanate salts dissolved in low molecular weight liquid copolymers of ethylene oxide and propylene oxide, Cameron et al. attributed the increase in Λ to redissociation of aggregated ionic species into simple cations and thiocyanate anions.38 Bruce has described redissociation as occurring through stabilization of “free” ions relative to ion pairs by the increasing number of long-range ion interactions.39 As noted above, ionic association has been assumed to play a significant role in the behavior in region I through the formation of neutral ion pairs and in region II through the formation of charged aggregates and/or redissociation. Discrete ionic species in a variety of LiTf-based electrolytes have been widely studied using vibrational spectroscopic techniques.21–23 However, a comparison of LiTf-based and TbaTf-based organic electrolytes yields very similar Λ(c1/2) curves with the same general shape as those of the 1-octanol and 1-decanol solutions shown in Figure 2.40 The very different degrees of ionic
Figure 4. Molar conductivity versus TbaTf concentration at 25 °C in dilute solutions of alcohols: ethanol (ε ) 23.7), 1-propanol (ε ) 19.8), 1-butanol (ε )17.1), 1-hexanol (ε ) 12.75).
association in LiTF and TbaTf solutions suggest that although ionic association in some form plays a role in determining the shape of the Λ(c1/2) curves, other factors are important. The data summarized in Figures 2 and 3 point to the major role of the dielectric constant in determining the salt concentration dependence of the molar conductivity, Λ(c1/2). Specifically, not only do the values of Λ decrease with decreasing dielectric constant, but the qualitative shape of the Λ(c1/2) curve depends on the dielectric constant. The complexity of Λ(c1/2) is dramatized by Figure 3, which shows Λ as a function of c1/2 for solutions of TbaTf in three organic solvents. The values of Λ in the different solutions are similar to each other in region I at comparable salt concentrations because the dielectric constants of the solvents are similar (methyl acetate ) 6.69, tetrahydrofuran ) 7.33, diglyme ) 7.19). However, the values of Λ deviate significantly from each other at concentrations > 0.01 M. The concentration dependence of the mobility at higher salt concentrations is briefly discussed in the Conclusions section. The remainder of this paper will focus on TbaTf solutions at concentrations < 0.01 M, that is, region I. As noted earlier, measurement of the molar conductivity in solutions of TbaTf is equivalent to measuring the sum of the cation and anion mobilities because ionic association is minimized by the bulky, charge-protected tetrabutylammonium ion. Figure 4 illustrates the change in the ionic mobilities with concentration in dilute TbaTf-alcohol solutions. These alcohols are structurally similar in that the OH group occurs at the terminus of an alkyl chain. Therefore, the interaction of the cation with the coordinating heteroatom should be very similar for each member of this particular solvent family. The molar conductivity decreases with increasing salt concentration and decreasing dielectric constant over this limited
Ionic Transport in Dilute Organic Electrolyte Solutions
Figure 5. Molar conductivity versus dielectric constant for solutions of 0.0055 M TbaTf in various aprotic solvents at 25 °C.
concentration range, as expected. This behavior is qualitatively similar to the trends seen in Figure 2 for solvents with ε > 10; however, the strong ionic association present in those LiTf solutions is not present in these TbaTf solutions. Therefore, a simple identification of Λ with the ionic mobilities can be made for the data in Figure 4. Experiments have shown that in organic liquid electrolytes at concentrations less than 0.01 M, the ionic mobility is a function of the concentration of “free” ions.40 This paper now extends that work to consider the role of the dielectric constant and the solvent functional group on ionic mobility. It is important to examine the dependence of Λ on the solvent dielectric constant over the widest possible range of solvent dielectric constant values. Linear alcohols are excluded from the chosen set of solvents because significant differences between protic and aprotic solvents are expected. Figure 5 summarizes the dependence of Λ on ε for 0.0055 M solutions of TbaTf in a variety of aprotic solvents. The molar conductivity is measured for each TbaTf solution, while the dielectric constant of the pure solvent is measured. Studies have shown that the dielectric constant for a solvent does not significantly differ from the dielectric constant of a dilute solution at concentrations < 0.01M.24 In Figure 5, the molar conductivity increases with increasing dielectric constant for small dielectric constants, reaches a maximum at a dielectric constant of approximately 33, and then decreases with increasing dielectric constant. For solvents with a dielectric constant below a threshold value (ε approximately < 5), the addition of salt has very little effect on the ion transport (i.e., the resistance of the solvent is comparable to that of the electrolyte). For example, the conductivity of 1,4-dioxane (ε ) 2.42) is σ ) 2.5 × 10-7 S cm-1 and does not change appreciably with added TbaTf in the dilute region (salt concentration < 0.01 M). Solutions of TbaTf-hexyl acetate (ε ) 4.62) also demonstrate conductivity values that are independent of concentration in the dilute region. For the particular solvent families represented in Figure 5, the data lie on a relatively smooth curve. This trend results primarily from the dependence of Λ on the solvent dielectric constant for these several solvent families and not on the specific identity of the solvent. For example, a 75% propyl acetate/25% 2-pentanone mixture by volume has a dielectric constant of 7.24, which is very similar to the dielectric constant of diglyme (ε ) 7.19). The molar conductivity of this mixture at a TbaTf concentration of 0.0055 M is 2.38 S cm2 mol-1, while the molar conductivity of diglyme at the same TbaTf concentration is 2.32 S cm2 mol-1 (data from Table 2). It is important to note that the nature of the solvent functional group can play a major role in ion transport, as shown by the
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Figure 6. Molar conductivity versus dielectric constant at 25 °C for 0.0055 M TbaTf-solvent, where the aprotic solvents are solvents 1-7 in Figure 5 and the alcohols are 3-octanol, 1-decanol, 1-octanol, 1-hexanol, 1-butanol, and 1-propanol, in order of increasing ε.
TABLE 3: Molar Conductivity of 0.0055 M TbaTf-Alcohol, Solvent Dielectric Constant, And Viscosity at 25 °C for Alcohols of Varying Chain Length alcohol
molar conductivity (S cm2 mol-1)
dielectric constant
viscosity (cp)
ethanol 1-propanol 1-butanol 1-hexanol 1-octanol 1-decanol
26.67 13.5 6.18 1.75 0.432 0.175
23.7 19.8 17.1 12.75 9.93 7.96
1.02 1.87 2.46 4.56 7.20 10.53
data in Figure 6 where Λ is plotted as a function of ε for a series of linear alcohols and the first seven aprotic solvents of Figure 5. Although the alcohol curve has the same general form as that of the aprotic solvents, the values of Λ for the alcohols are substantially less than those for the aprotic solvents at comparable values of the dielectric constant. 4. Conclusions In many dilute organic liquid electrolytes, the ionic mobility does not appear to depend on either the solvent viscosity or the size of the solvated charge carrier in a manner consistent with Walden’s rule (Tables 1 and 2). However, it is possible to find systems in which Walden’s rule is qualitatively correct, for example, the study of 1-alcohols from methanol to 1-pentanol by Prego et al.27 Table 3 summarizes data for a number of other 1-alcohols; these data indeed show that the molar conductivity decreases as the viscosity increases, as predicted by Walden’s rule. It should be noted that as the length of the alcohol chain increases, the solvent dielectric constant decreases (third column). Studies of dilute solutions of TbaTf described in section 3.2 show that the molar conductivity and hence the sum of the ionic mobilities strongly depend on the dielectric constant at concentrations < 0.01 M. Therefore, the decrease in molar conductivity observed by Prego et al. may be actually due to the decrease in the dielectric constant with increasing alkyl chain length. The most significant finding of this study is that values of the molar conductivity plotted as a function of solvent dielectric constant appear to lie on a smooth curve for a set of 0.0055 M solutions of TbaTf in a variety of aprotic solvents including an acetate, a cyclic ether, four ketones, a nitrile, DMSO, and propylene carbonate (Figure 5). As the dielectric constant increases from an initial value of 5.48 (propyl acetate), the molar
8290 J. Phys. Chem. B, Vol. 112, No. 28, 2008 conductivity smoothly increases to a maximum value and then decreases with further increase of the dielectric constant until a low value of Λ is asymptotically reached. A similar plot for linear alcohols (Figure 6) also leads to a smooth curve, although the values of Λ for the alcohols are lower than values of selected aprotic solvents in Figure 6 by almost an order of magnitude at corresponding values of the solvent dielectric constant. This suggests that the form of Λ versus ε may be generalized to solvent families with some degree of commonality in their functional groups. What is the commonality? It is relatively easy to recognize that the linear alcohols can be grouped into a family in which the molar conductivity might fall on a smooth curve as the dielectric constant changes. These compounds are structurally very similar, with the change in the dielectric constant primarily governed by the alkyl chain length. It is not as clear why the particular group of solvents in Figure 5 can be grouped as a family whose Λ(ε) data pairs fall on a smooth curve. One factor common to all members of this group is that in each molecule there is only one heteroatom that is primarily involved in the cation-solvent interaction. The two exceptions to this generalization might appear to be propylene carbonate and propyl acetate. However, in a study of lithium perchlorate dissolved in binary solutions comprised of ethylene carbonate mixed with propylene carbonate, diethyl carbonate, or dimethyl carbonate, Klassen et al. established that it was the carbonyl oxygen atom that interacted with the cation.41 Similarly, a Raman spectroscopic study of the lithium cation in methyl acetate solutions suggested that solvation occurred through the carbonyl group.42 It is noteworthy that acetonitrile can be included in the particular solvent family despite the fact that the heteroatom is nitrogen rather than an oxygen atom that is common to other members of this family. Although each solvent in Figure 5 has one heteroatom that interacts with cations, not all solvents with one such heteroatom lie on this curve. The family of linear alcohols has already been discussed and summarized in Figure 6. There are other examples of solvents with one heteroatom whose Λ and ε values are significantly divergent from the curve in Figure 5. For example, acetophenone has a dielectric constant of 17, which would translate into a molar conductivity of about 75 S cm2 mol-1 according to Figure 5. However, the experimental value is 21.2 S cm2 mol-1. Several other aromatic electrolytes solutions studied had molar conductivity values lower than that predicted from Figure 5. Plots of Λ versus c1/2 for methyl acetate, THF, and diglyme were similar in the dilute region (salt concentrations < 0.01 M) because the dielectric constants of the solvents are similar (methyl acetate ) 6.69, tetrahydrofuran ) 7.33, diglyme ) 7.19), and in dilute solutions, the value of the dielectric constant is essentially the solvent dielectric constant. However, the values of Λ deviate significantly from each other at larger salt concentrations, which may be due to a different dependence of ε on salt concentration for the three solvents at higher salt concentrations. It is well-known that the solution dielectric constant begins to increase from that of the pure solvent for salt concentrations > approximately 0.01 M,7,8 which is roughly the same concentration where Λ begins to increase with increasing salt concentration. Because Λ is proportional to the sum of ionic mobilities in these TbaTf solutions, the increase in solution dielectric constant may be responsible for the increase in the ionic mobilities as observed in Figures 5 and 6. It is also
Petrowsky and Frech possible that a different mechanism of ion transport may begin to operate in concentrated regimes in addition to the mechanism governing ion transport in the dilute region. This additional mechanism may depend on the local viscosity as suggested by Albinsson et al.30 Methyl acetate, the solvent with the lowest viscosity, has the largest molar conductivity maximum, while the solvent with the highest viscosity (diglyme) has the smallest molar conductivity maximum (see Figure 3). Although the focus of this work is on the dilute region, studies are currently underway to understand ion transport in concentrated regimes. References and Notes (1) Kohlrausch, F. Z. Phys. Chem. 1907, 58, 630. (2) Walden, P. Z. Elektrochem. Angew. Phys. Chem. 1906, 12, 77. (3) Debye, P.; Huckel, E. Z. Phys. 1923, 24, 305. (4) Fuoss, R. M. Z. Phys. 1934, 35, 59. (5) Kraus, C. A.; Fuoss, R. M. J. Am. Chem. Soc. 1933, 55, 21. (6) Cavell, E. A. S.; Knight, P. C. Z. Phys. Chem. 1968, 57, 331. (7) Gestblom, B.; Svorstol, I.; Songstad, J. J. Phys. Chem. 1986, 90, 4684. (8) Sigvartsen, T.; Gestblom, B.; Noreland, E.; Songstad, J. Acta Chem. Scand. 1989, 43, 103. (9) Barthel, J.; Wachter, R.; Schmeer, G.; Hilbinger, H. J. Solution Chem. 1986, 15, 531. (10) Onsager, L.; Fuoss, R. M. J. Phys. Chem. 1932, 36, 2689. (11) Onsager, L.; Kim, S. K. J. Phys. Chem. 1957, 61, 215. (12) Fuoss, R. M.; Onsager, L. J. Phys. Chem. 1962, 66, 1722. (13) Fuoss, R. M.; Onsager, L. J. Phys. Chem. 1963, 67, 621. (14) Fuoss, R. M.; Onsager, L. J. Phys. Chem. 1964, 68, 1. (15) Bernard, O.; Kunz, W.; Turq, P.; Blum, L. J. Phys. Chem. 1992, 96, 3833. (16) Durand-Vidal, S.; Turq, P.; Bernard, O. J. Phys. Chem. 1996, 100, 17345. (17) Fuoss, R. M.; Hsia, K.-L. Proc. Natl. Acad. Sci. U.S.A. 1967, 57, 1550. (18) Delsignore, M.; Maaser, H. E.; Petrucci, S. J. Phys. Chem. 1984, 88, 2405. (19) Cote, J.-F.; Perron, G.; Desnoyers, J. E. J. Solution Chem. 1998, 27, 707. (20) Moreira, L.; Leitao, R. E.; Martins, F. Mol. Phys. 2006, 104, 1905. (21) Papke, B. L.; Ratner, M. A.; Shriver, D. F. J. Phys. Chem. Solids 1981, 42, 493. (22) Schantz, S.; Sandahl, J.; Bo¨rjesson, L.; Torell, L. M.; Stevens, J. R. Solid State Ionics 1988, 28-30, 1047. (23) Huang, W.; Frech, R.; Wheeler, R. A. J. Phys. Chem. 1994, 98, 100. (24) Cachet, H.; Cyrot, A.; Fekir, M.; Lestrade, J.-C. J. Phys. Chem. 1979, 83, 2419. (25) Longsworth, L. G.; MacInnes, D. A. J. Phys. Chem. 1939, 43, 239. (26) Mendolia, M. S.; Farrington, G. C. Chem. Mater. 1993, 5, 174. (27) Prego, M.; Cabeza, O.; Carballo, E.; Franjo, C. F.; Jimenez, E. J. Mol. Liq. 2000, 89, 233. (28) Berns, D. S.; Fuoss, R. M. J. Am. Chem. Soc. 1961, 83, 1321. (29) Barthel, J.; Buchner, R. Pure Appl. Chem. 1986, 58, 1077. (30) Albinsson, I.; Mellander, B.-E.; Stevens, J. R. J. Chem. Phys. 1992, 96, 681. (31) Gray, F. M. Solid State Ionics 1990, 40/41, 637. (32) Jagodzinski, P.; Petrucci, S. J. Phys. Chem. 1974, 78, 917. (33) Onishi, S.; Farber, H.; Petrucci, S. J. Phys. Chem. 1980, 84, 2922. (34) Matsuda, Y.; Morita, M.; Yamashita, T. J. Electrochem. Soc. 1984, 131, 2821. (35) Farber, H.; Irish, D. E.; Petrucci, S. J. Phys. Chem. 1983, 87, 3515. (36) Watanabe, M.; Sanui, K.; Ogata, N.; Kobayashi, T.; Ohtaki, Z. J. Appl. Phys. 1985, 57, 123. (37) Fuoss, R. M.; Kraus, C. A. J. Am. Chem. Soc. 1933, 55, 2387. (38) Cameron, G. G.; Ingram, M. D.; Sorrie, G. A. J. Electroanal. Chem. 1986, 198, 205. (39) Bruce, P. G.; Gray, F. M. Polymer Electrolytes II: Physical Principles. In Solid State Electrochemistry; Bruce, P. G., Ed.; Cambridge University Press: Cambridge, U.K., 1995. (40) Petrowsky, M.; Frech, R.; Suarez, S. N.; Jayakody, J. R. P.; Greenbaum, S. J. Phys. Chem. B 2006, 110, 23012. (41) Klassen, B.; Aroca, R.; Nazri, M.; Nazri, G. A. J. Phys. Chem. B 1998, 102, 4795. (42) Deng, Z.; Irish, D. E. Can. J. Chem. 1991, 69, 1766.
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