5040
J. Phys. Chem. B 2000, 104, 5040-5044
Conductivity and Solvation of Li+ Ions of LiPF6 in Propylene Carbonate Solutions Kazutaka Kondo,† Mitsuru Sano,*,† Akio Hiwara,‡ Takehiko Omi,‡ Miho Fujita,§ Akio Kuwae,§ Masayasu Iida,| Koichi Mogi,⊥ and Haruhiko Yokoyama# Graduate School of Human Informatics, Nagoya UniVersity, Nagoya 464-8601, Japan, Functional Materials Laboratory, Mitui Chemicals, Sodegaura 299-0265, Japan, Institute of Natural Sciences, Nagoya City UniVersity, Nagoya 467-8501, Japan, Department of Chemistry, Nara Women’s UniVersity, Nara 630-8506, Japan, Department of Molecular and Material Sciences, Kyushu UniVersity, Kasuga, Fukuoka 816-8580, Japan, and Department of Chemistry, Yokohama City UniVersity, Yokohama 236-0027, Japan ReceiVed: January 11, 2000
Solutions of lithium hexafluorophosphate, LiPF6, in propylene carbonate (4-methyl-1,3-dioxolan-2-one; denoted by PC hereafter) in the concentration range from 0.0 to 3.29 M (M ) mol dm-3) have been studied regarding their conductivities, viscosities, and self-diffusion coefficients of PC by the NMR field gradient technique, Raman spectra, and NMR spectra. Walden’s products are almost constant in the range up to and over 3.0 M. Therefore, Li ions are considered to be quite free from the firm interaction with anions even in such concentrated solutions. The appearance of the maximum conductivity at about 0.8 M is explained by associating with the concentration dependence of the solution viscosity. A remarkable increase in the solution viscosity was observed in a concentration beyond 2.0 M, and it can be ascribed to the cluster formation of lithium ions with PC molecules of the solvent. Such an idea of clusters can reasonably interpret some of the characteristic changes of the viscosities, the diffusion coefficients, the Raman spectra, and the NMR spectra at concentrations over 2 M.
Introduction Lithium ion secondary batteries are used in various kinds of portable electric devices because of their high performance with much capacity and high output voltage, and their importance has been lately increasing. The battery consists of three main components: cathode, anode, and electrolyte solution. The electrolyte solution requires high ionic conductivity, low melting point, high boiling point, and high chemical and electrochemical stabilities for the battery’s high performance as well as in practical usage.1 However, to date there have been no electrolyte solutions satisfying all such requirements. Solvents of electrolyte solutions for the lithium ion battery currently used are almost always organic carbonates in which proper lithium salts are dissolved. LiPF6 in an ethylene carbonate-dimethyl carbonate system is one of the frequently used ones.1 However, the conductivities of these electrolyte solutions are extremely low in comparison with corresponding aqueous solutions.1 Development of electrolyte solutions with high ionic conductivity is needed for highperformance Li ion batteries. For this, it is necessary to acquire more knowledge about the solvation of lithium salts and the mechanism of ionic conduction in the solvents. Many studies have already been performed on organic electrolyte solutions.2 There is a little information available for the LiPF6 solutions in organic solvents which is now in practical use for Li ion batteries.3,4 One of the main objects of this study is understanding the ionic conductivity of LiPF6 and the behavior of solvated Li+ †
Nagoya University. Mitsui Chemicals. Nagoya City University. | Nara Women’s University. ⊥ Kyushu University. # Yokohama City University. ‡ §
ion species in organic solvents from the macroscopic and microscopic points of view. We selected propylene carbonate as the solvent because it is commonly used in a primary Li battery and is also liquid at room temperature. We measured conductivities, viscosity coefficients, selfdiffusion coefficients, Raman spectra, and NMR spectra of LiPF6-PC solutions in the range of 0.0-3.29 M concentration. The high solubility of LiPF6 in PC and the high viscosity of the solution are discussed from the point of view of molecular level interactions in solutions, and the maximum conductivity appearing when the concentrations were changed is also discussed in association with the concentration dependence of solution viscosities. Experimental Section Materials. Commercially available propylene carbonate was used after refining by the fine distillation. The purity and the water content of the purified PC were greater than 99.99% by analysis with gas chromatography and less than 5 ppm with the Karl Fischer titration method. Lithium hexafluorophosphate used as the electrolyte salt was the one available from Tomiyama Chemical Co. It was of special grade for electrochemical use, which was certified to contain more than 99.9% of LiPF6 and less than 10 ppm of water and to include less than 50 ppm of HF as impurity. All the electrolyte solutions of LiPF6 in PC were prepared in a glovebox in an atmosphere of dry nitrogen gas (with a dry point below -50 °C). Instrumentation. The pulse gradient NMR measurements were made by a JEOL FX 90 NMR spectrometer operating at 90 MHz for protons. The spectrometer was equipped with an apparatus producing a field gradient in the range of 0.6-1 T m-1. The gradient strength was calibrated and cross-checked using the known self-diffusion coefficient of H2O at 25 °C (2.23
10.1021/jp000142f CCC: $19.00 © 2000 American Chemical Society Published on Web 04/27/2000
Conductivity and Solvation of Li+ Ions of LiPF6
J. Phys. Chem. B, Vol. 104, No. 20, 2000 5041
TABLE 1: Conductivities, Viscosities, and Diffusion Coefficients of PC in LiPF6-PC at 25 °C concentration c/mol dm-3
Li+: PC molar ratio
conductivity κ/mS cm-1
molar conductivity Λ/S cm2 mol-1
viscosity η/cPa
Walden’s product Λη/S cm2 mol-1 cP
0.0 0.118 0.565 0.992 1.10 1.60 2.07 2.50 2.91 3.29
1:100 1:20 1:11 1:10 1:6.7 1:5 1:4 1:3.3 1:2.85
2.1 5.8 6.0 5.8 4.1 2.3 1.2 0.61 0.43
(25.0)b 18 10 6.0 5.3 2.6 1.1 0.46 0.21 0.13
2.55 (2.53)c 2.77 4.69 8.03 9.93 18.6 46.9 120 223 389
63.3 50 48 48 53 48 52 56 47 51
a
diffusion coeff D/10-10 m2 s-1 4.9 4.2 2.7 1.4 0.74 0.37 0.17 0.090 0.061
viscosity ratio η/η0 1 1.09 1.84 3.15 3.89 7.29 18.4 47.1 87.5 153
1 cP ) 1 mPas. b Reference 4. c Janz, J.; Tomkins, R. B. T. Nonaqueous Electrolytes Handbook; Academic Press: New York, 1972; Vol. 1.
× 10-9 m2s-1).5 The simple spin echo pulse sequence was used for the diffusion measurements, and the free diffusion echo signal attenuation, E, is related to the experimental parameters by6
ln(E) ) ln(S/Sg)0) ) -γ2g2dδ2(d - δ/3) where S is the spin echo signal intensity, δ is the duration of the field gradient with magnitude, g is the gyromagnetic ratio, and d is the interval between the two gradient pulses. Typical acquisition parameters were d ) 40 ms, δ ) 0-250 ms, and g ) 0.44-0.90 Tm-1. The temperature was controlled at 25 ( 0.5 °C with a JEOL GVT2 temperature-control unit. The accuracy of the diffusion coefficients measured is estimated to be better than (5%. WILMAD Roto Tite Valve NMR Tubes were used. Proton and 7Li NMR spectra were recorded on a JEOL R-400 spectrometer with a pulse delay of 3 s and 64 acquisitions; tetramethylsilane in CD3Cl and 1 M LiCl aqueous solution were the external standard at 25 °C. FT-Raman spectra were recorded on a Perkin-Elmer 2000R spectrometer equipped with a quartz beam splitter and InGaAs detector. The 1064 nm line of a Spectron Laser System SL300 Nd:YAG laser was used as the exciting source with an output power of about 200 mW at the sample position. The scattered light was collected in 180° configuration. All spectra were accumulated for 60 scans at 4 cm-1 resolution. The samples were contained in the NMR tubes having a 5 mm outside diameter. The conductivity measurements were carried out at 25 ( 0.1 °C with a Toa Denpa Industries CM-40S conductivity meter at 3000 Hz. The viscosity measurements were made with a Tokyo Instrument VISCONIC-ELD viscosity meter at 25 ( 0.1 °C. Results and Discussion Propylene carbonate solutions of LiPF6 were studied in terms of measurements of conductivities (κ), viscosity coefficients (η), self-diffusion coefficients (D) of PC, Raman spectra, and NMR spectra in the concentrations between 0.0 and 3.29 M. Table 1 lists the values of κ, η, and D obtained as a function of the concentrations of LiPF6 at 25 °C, also the data of the molar ratio of PC/Li+, molar conductivity Λ, and Walden’s product Λη. The relationship between c and κ or η is also shown in Figure 1. The maximum ionic conductivity (κmax) is observed at about 0.8 M LiPF6, which is very close to the 0.787 M reported by Christie et al.4 The κmax for LiPF6 and LiAsF6 solutions are larger than those for the others according to Werblan et al.7 who studied the conductivities of Li salt electrolytes in PC solutions. The viscosity of the solutions increases with concentration and shows a remarkable jump of
Figure 1. Conductivity κ and viscosity η of PC-LiPF6 as a function of salt concentration.
Figure 2. Walden’s product Λη and a product of ηD of PC-LiPF6 as a function of salt concentration.
η in the concentrations above 2 M. As seen in Figure 2, where the values of Λη and ηD are plotted vs concentration, the Λη values are almost constant over the observed concentration range, while the values of ηD increase slightly in the range up to 1.6 M and then increase significantly beyond that concentration. Walden’s Product and Salt Dissociation. We discuss the degree of dissociation R of the salt in concentrated solutions on the basis of Λ and η. The molar conductivity Λ of a 1:1 electrolyte completely dissociated in dilute solution is theoretically expressed as follows:8
Λ ) Λ∞ - Sc1/2 + Ec log c - J1c + J2c3/2
(1)
5042 J. Phys. Chem. B, Vol. 104, No. 20, 2000
Kondo et al.
where Λ∞ is the limiting molar conductivity, c is the concentration of the salt, and the other parameters are conventional ones. Among them, S, E, J1, and J2 are functions of temperature, dielectric constant, and viscosity, and in addition, J1 and J2 are also functions of Λ∞ and the closest distance of approach of the ions. As we have no theoretically exact expressions of Λ in concentrated electrolyte solutions, a practical one for the solution in question would be expressed by the introduction of a higher order term, O(c2), to eq 1 as follows:
Λ ) Λ∞ - Sc1/2 + Ec log c - J1c + J2c3/2 + O(c2) (2) ) Λ∞Q(c)
(3)
where
association theory,14 assuming an arbitrary interionic distance. His equation deriving ion-association constants is said to include a serious problem giving unrealistically large values on increasing the interionic distance.15 Consequently, his conductivity equation leads to a value of KΛ larger than the real one. For example, Fuoss’ equation gives the values of KΛ for LiCl, NaCl, and KCl as 4.02, 2.86, and 2.11 M-1 in water at 25 °C, respectively, while other conventional conductivity equations give no appreciable values of KΛ. Thus, the value of KΛ reported for LiPF6 in the PC would be much smaller, probably close to zero. Maximum Ionic Conductivity. As seen from Figure 1, the ionic conductivity κ of LiPF6 has the maximum value at about 0.8 M LiPF6. We discuss the maximum ionic conductivity. The conductivity κ is given by
O(c2) ) O1c2 + O2c3 + O3c4 + ... Q(c) ) 1 - {Sc
1/2
+ Ec log c - J1c + J2c
3/2
κ ) Λc/103
+ O(c )}/Λ 2
When the viscosity of the solutions varies to a large extent with electrolyte concentrations, eq 3 is further modified by an additional factor of the ratio of solvent to solution viscosities (η0/η), as follows:9,10
Λ ) (η0/η)Λ∞Q(c)
(4)
Λη ) Λ∞η0Q(c)
(5)
The value of Q(c) is 1.0 at infinite dilution, and it decreases with increasing concentrations due to ionic interactions. When ion pairs form, eq 5 is deformed as follows:
Λη ) Rη0Λ∞Q(Rc)
(9)
∞
(6)
where R is the degree of dissociation. Figure 2 shows experimental values of Walden’s product Λη. The values remain almost the same, 50.3 ( 2.9 S cm2 mol-1cP (cP ) mPas), over the concentration range observed but are apparently smaller than 63 S cm2 mol-1 cP at infinite dilution.4 Equation 6 is then expressed as
50.3 ) R63Q(Rc)
(7)
0.8 ) RQ(Rc)
(8)
The R is at least larger than 0.8, because Q(Rc) is expected to be smaller than 1.0. Sodium chloride and potassium chloride completely dissociate in water (R ) 1), even in high concentrations. The ratio of Walden’s products Λη/η0Λ∞ for NaCl and KCl in water is between 0.70 and 0.75 at 25 °C in the concentration range 1.0-3.0 M.10,11 This means that the values of Q(c) are between 0.70 and 0.75. Because the relative dielectric constant () of PC is 64.4 and slightly smaller than that of water (78.5) and the ratio of Λη/η0Λ∞ for LiPF6 in the PC is similar to that of NaCl or KCl in water, the R value of LiPF6 in the PC is close to 1.0 even in higher concentration solutions. Christie and Vincent4 reported a value of 5.67 M at 25 °C for the overall formation constant (KΛ) of Li+ with PF6- in PC. However, this value would be overestimated for the following reasons: One is ignoring the viscosity change of the solutions. No correction of the viscosity increase gives a value of KΛ larger than the real one. The other reason was using the theoretical conductivity equation derived by Fuoss.12,13 Fuoss assumed two states for ion pairs, conducting pairs and nonconducting pairs. The latter corresponds to solvent-shared or solvent-separated ones whose formation constant was given by Fuoss’ ion-
Because the Walden’s product (Λη) maintains a constant value as described in the previous section, we can obtain the following equation:
κ ) Λc/103 ) constant c/η 103
(10)
The maximum value of the ionic conductivity is due to an exponential increase in η against c. The viscosity η can be expressed by the following equation with additional higher order terms to the Jones-Dole equation:16
η ) η0{1 + Ac1/2 + Bc + Cc2 + O(c3)}
(11)
where O(c3) represents all the terms of a higher order more than c3, that is, O(c3) = O1c3 + O2c4 + O3c5 + .... The coefficient A is theoretically given by the Debye-Huckel theory10,16 and was estimated to be 0.0115 M-1/2 using the value of η0(PC), r(PC), and Λ∞(LiPF6 in PC). The values of B17 and C were estimated to be 0.54 M-1 and 1.65 M-2, respectively, using the values of η in the concentration range below 1.0 M and ignoring the O(c3) term. On the other hand, the conductivity κ is transformed using eqs 4 and 11; eq 9 becomes
κ ) (η0/η)Λ∞Q(c)c/103 ) Λ∞Q′(c)c × 10-3{1 + Ac1/2 + Bc + Cc2 + O(c3)}-1 (12) where Q′(c) corresponds to Q(c) or RQ(Rc). Because the maximum conductivity κmax should be found at dκ/dc ) 0 and Q′(c) ()Λη/Λ∞η0) is almost constant, we obtain a relationship as follows:
1 + Acmax1/2/2 - Ccmax2 + O(cmax3) cmax dO(cmax3)/dcmax ) 0 (13) The Acmax1/2 term is negligibly small compared with the other terms. When the higher terms are also negligible in the concentration, cmax, at κmax, eq 12 can be approximately
1 - Ccmax2 ) 0
(14)
Thus, the value of cmax turns out to be 0.778 M with C ) 1.65 M-2, and it is in fair agreement with both the maximum position in Figure 1 and the value of 0.787 M previously reported.4 Further, we can deduce κmax ) 6.4 mS cm-1 using cmax ) 0.778
Conductivity and Solvation of Li+ Ions of LiPF6
J. Phys. Chem. B, Vol. 104, No. 20, 2000 5043
Figure 3. Raman spectra of PC-LiPF6 as a function of salt concentration.
Figure 5. 1H, 7Li, and 31P NMR chemical shifts of PC-LiPF6 as a function of salt concentration.
TABLE 2: Raman Intensities in LiPF6-PC normalized Raman amplitude concn c/ Li+:PC mol dm-3 molar ratio I712
Figure 4. Normalized Raman intensity of each band at 712, 721, and 741 cm-1 of PC-LiPF6 as a function of salt concentration.
M and Q′(c) ) 0.8, which is very close to our experimental value of 6.1 mS cm-1.4 Raman Spectra and NMR spectra in Concentrated Solutions. Figure 3 shows Raman spectra of LiPF6 solutions from 710 to 740 cm-1. Raman spectra in the region vary as the LiPF6 concentration increases, as seen in Figure 4. The signal at 712 cm-1 is assigned to the ring-deformation band of free PC molecules in the pure solvent or in solutions, and its intensity decreases as the LiPF6 concentration increases, while the intensity around 721 cm-1 increases to the contrary. The new band centered at 721 cm-1 is most certainly assigned as the ring deformation of PC molecules bound to the Li+ ion. The assignment is also in accordance with the results of Battisti et al.18 To estimate the ratio of free to bound PC molecules, Raman bands in the range 705-725 cm-1 were separated into two components of 712 and 721 cm-1 by a deconvolution, and then their intensities (I712 and I721) were normalized by a comparison with the band at 848 cm-1, which is attributable to a vibration of a methyl group of PC molecules and negligibly dependent on the salt concentration. The values of I712 and I721, and their sum I712 + I721, are shown in Figure 4 and in Table 2. The values of I712, due to free PC molecules, decrease with increasing LiPF6 concentration, while those of I721, due to the bound PC molecules, increase with a bending point at a concentration around 2 M. The sum of I712 and I721 is nearly constant at
0.0 0.118 0.565 0.992 1.60 2.07 2.50 2.91 3.29
1:100 1:20 1:11 1:6.7 1:5 1:4 1:3.3 1:2.85
0.70 0.71 0.61 0.51 0.37 0.24 0.17 0.09 0.04
I721
I712 + I721
I741
0.00 0.00 0.09 0.16 0.29 0.45 0.49 0.53 0.56
0.70 0.71 0.70 0.67 0.66 0.69 0.66 0.62 0.60
0.000 0.014 0.067 0.122 0.201 0.268 0.332 0.417 0.507
I741I712 + I721 + broken line I741-broken line 0.000 0.000 0.000 0.000 0.001 0.008 0.018 0.052 0.094
0.70 0.71 0.70 0.67 0.66 0.70 0.68 0.67 0.69
concentrations lower than 2 M but gradually decreases at higher concentrations: the decrease at 3.29 M is about 15%. The decrease suggests the appearance of PC molecules bound to lithium ions in a different way, and these types of PC molecules have a new Raman band outside 705-725 cm-1, probably superimposing on the band of 741 cm-1 assiged to a streching vibration of PF6-. The intensity at 741 cm-1, I741, shown in Figure 4, proportionally increases with the increasing concentration of LiPF6 but deviates upward from the straight line at concentrations higher than 2 M. This increment would be caused by the overlapping of a new band attributed to the differently bound PC molecules. The intensity difference (I741-broken line) is shown in Table 2 with the values of I712, I721, I712 + I721, and I741. The sum of I712 + I721 and the I741-broken line remain almost constant, as shown in Table 2. Therefore, the band around 741 cm-1 originates from both PF6- ions and PC molecules differently bound to the Li+ ion from the Li+-PC at 721 cm-1. NMR chemical shifts of 1H, 7Li, and 31P are plotted against concentrations of LiPF6 in Figure 5. The PC molecule has four kinds of hydrogen atoms, as shown in Figure 5. The NMR signals of 1Ha, 7Li, and 31P shift to higher field with the salt concentration, while those of 1Hb, 1Hc, and 1Hd scarcely change in the concentration range below 2 M. All the chemical shifts
5044 J. Phys. Chem. B, Vol. 104, No. 20, 2000 except for 1Ha show a significant change of patterns at about c ) 2 M, which is the same concentration as found in the Raman intensity change. Li+-PC Interactions and Cluster Formation. The Stokes radius (rs) of the Li+ ion in PC is estimated as 4.4 Å, using a value of 7.3 S cm-1 mol-1 for a limiting molar conductivity of Li+.19 The effective volume of solvated lithium ions, Vef(Li+), is determined as 357 Å3 by calculation from the equation, Vef(Li+) ) 4/3πr3. This suggests that a lithium ion is solvated by 4.3 PC molecules in a dilute solution, because the volume of a PC molecule is estimated to be 82.8 Å3 using the values of the van der Waals increments of atoms by Edward.20 This means that the coordination number of a lithium ion in PC is 4-5, as in water. Gutmann’s donor number (DN)21 of PC is 15.1 and smaller than the 18.0 of water but slightly larger than the 14.1 of acetonitrile. A PC molecule has three plausible oxygen atoms as donors and will usually use a carbonyl oxygen atom to bond to a metal ion because of a greater electron density than those of the other oxygen atoms.22,23 However, the orientation of PC molecules in the first coordination sphere of Li+ is somewhat disordered at concentrated solutions. For example, at 2.5 M concentration where the PC/Li+ ratio is 4, only about 25% of the PC molecules is regarded as free, as deduced from the Raman intensity at 712 cm-1 (I712) shown in Figure 4. The remakable increase in the solution viscosity beyond about 2 M is presumed to be closely related to the specific change in both Raman intensities and NMR chemical shifts appearing at the same concentration. In concentrated solutions higher than 2 M, most PC molecules probably exist in contact with lithium ions, and then their molecules are shared by Li+ ions in the solution. One of the most typical ways of sharing is by bridging two lithium ions with the two oxygen atoms. The Raman spectrum changes observed at the signals of 712, 721, and 741 cm-1 reflect the increase in PC molecule restriction by direct interactions of one or two oxygen atoms with lithium ions. The PC molecule showing the Raman signal centered at 741 cm-1 may be involved in the bridging. There may also be some PC molecules binding to the same lithium ion with two oxygen atoms of the molecule, or there may be PC molecules only in contact with two lithium ions without bridging. However, the remarkable increase in the viscosity of the solution will be predominantly due to some aggregation or cluster formation by bridging between lithium ions and PC molecules. In the most concentrated solution (3.29 M) where the PC/Li+ ratio is 2.85, we can deduce, from the result that the fraction of (I741-broken line) to I712 + I721 + I741broken line is about 0.15, that about 50% of the PC molecules should be shared and about 30% of them should be involved in bridging. Even in concentrated solutions, lithium ions would
Kondo et al. not be in stable contact with the PF6- ions but would be surrounded by PC molecules, as seen from the constant values of the Walden’s product as well as the great solubility of LiPF6 in PC. Solvated lithium ions or their clusters interact with unspecified PF6- ions existing in their vicinity. The change in the NMR chemical shift of 31P of PF6- at around 2.0 M would have something to do with the formation of the clusters, because they somewhat affect the electronic environment around the PF6- ions in a different manner from the simply solvated lithium ions. The concentration dependence of Dη shown in Figure 2 apparently correlates to the viscosity change approximately expressed by the equation Dη ) D0η0{1 + κ log(η/η0)}. The change in Dη at a concentration around 1.6 M would be involved in the cluster formation, which changes the diffusion kinetics of the solvent. References and Notes (1) Ue, M. Lithium Ion Secondary Batteries; Yoshio, M., Kozawa, A., Eds.; Nikkan Kogyo: Tokyo, 1996; Chapter 6. (2) Barthel, J.; Gores, H. J. Chemistry of Nonaqueous Solutions, Current Progress; Mamantov, G., Popov, A. I., Eds.; VCH Publishers: New York, 1994; Chapter 1. (3) Webber, A. J. Electrochem. Soc. 1991, 138, 2586. Ue, M. J. Electrochem. Soc. 1994, 141, 3336. Croce, F.; D’Aprano, Nanjundiah, C.; Koch, V. R.; Walker, C. W.; Salomon, M. J. Electrochem. Soc. 1996, 143, 154. (4) Christie, A. M.; Vincent, C. A. J. Phys. Chem. 1996, 100, 4618. (5) Mills, R. J. Phys. Chem. 1973, 77, 685. (6) Stejskal, E. O. J. Chem. Phys. 1965, 43, 3597. (7) Werblan, L.; Balkowska, A. J. Electroanal. Chem. 1993, 354, 25. (8) (a) Fernandez-Prini, R.; Prue, J. E. Z. Phys. Chem. 1965, 228, 373. (b) Fernandez-Prini, R. Trans. Faraday Soc. 1969, 65, 3311. (c) Renard, E.; Justice, J.-C. J. Solution Chem. 1974, 3, 633. (d) Yokoyama, H. Electrochemistry (Denki Kagaku) 1997, 65, 926. (9) Fuoss, R. M.; Accascina, F. Electrolyic Conductance; Interscience: New York, 1959. (10) Stokes, R. H.; Mills, R. Viscosity of Electrolytes and Related Properties; Pergamon Press: Oxford, U.K., 1965. (11) Robinson, R. A.; Stokes, R. H. Electrolyte Solutions, 2nd ed.; Butterworth: London, 1959. (12) Fuoss, R. M. J. Phys. Chem. 1978, 82, 2427. (13) Fuoss, R. M. Proc. Natl. Acad. Sci. U.S.A. 1978, 75, 16. (14) Fuoss, R. M. J. Am. Chem. Soc. 1958, 80, 5059. (15) Yokoyama, H.; Yamatera, H. Bull. Chem. Soc. Jpn. 1975, 48, 1770. (16) Horvath, A. L. Handbook of Aqueous Electrolyte Solutions; Ellis Horwood: NewYork, 1985. (17) Gurney, R. W. Ionic Processes in Solution; McGraw-Hill: New York, 1953. (18) Battisti, D.; Nazri, G. A.; Klassen, B.; Aroca, R. J. Phys. Chem. 1993, 97, 5826. (19) Mukherjee, L. M.; Boden, D. B.; Lindauer, R. J. Phys. Chem. 1970, 74, 1942. (20) Edward, J. T. J. Chem. Educ. 1970, 47, 261. (21) Gutmann, V. The Donor-Acceptor to Molecular Interactions; Plenum Press: New York, 1978. (22) Yeager, H. L.; Fedyk, J. D.; Parker, R. J. J. Phys. Chem. 1973, 77, 2407. (23) Matsui, T.; Takeyama, K. Matsusita Technol. J. 1998, 44, 71.
