Bimodal dielectric relaxation of electrolyte solutions in weakly polar solvents Tsuyoshi Yamaguchi and Shinobu Koda
Citation: The Journal of Chemical Physics 141, 244501 (2014); doi: 10.1063/1.4904276 View online: http://dx.doi.org/10.1063/1.4904276 View Table of Contents: http://aip.scitation.org/toc/jcp/141/24 Published by the American Institute of Physics
THE JOURNAL OF CHEMICAL PHYSICS 141, 244501 (2014)
Bimodal dielectric relaxation of electrolyte solutions in weakly polar solvents Tsuyoshi Yamaguchia) and Shinobu Koda Department of Molecular Design and Engineering, Graduate School of Engineering, Furo-cho B2-3 (611), Chikusa, Nagoya, Aichi 464-8603, Japan
(Received 2 October 2014; accepted 21 November 2014; published online 23 December 2014) The dielectric relaxation spectra of dilute electrolyte solutions in solvents of small dielectric constants are investigated both theoretically and experimentally. The theoretical calculation in our previous work [T. Yamaguchi, T. Matsuoka, and S. Koda, J. Chem. Phys. 135, 164511 (2011)] is reanalyzed, and it is shown that the dielectric relaxation spectra are composed of three components, namely, the relaxation of ionic atmosphere, the reorientational relaxation of ion pairs, and the collision between ions. The relaxation frequency of the slowest one increases with increasing the concentration, and the slower two relaxations, those of ionic atmosphere and ion pairs, merge into one at the concentration where the Debye length is comparable to the size of ions. Experimentally, the dielectric relaxation spectra of some electrolytes in two solvents, tetrahydrofuran and tetraglyme, are determined at frequencies from 300 kHz to 200 MHz, and the presence of the slower two relaxations was confirmed. The concentration dependence of the relaxation frequency is also in harmony with the theoretical calculation. The relationship between the dielectric relaxation spectra and the concentration dependence of the ionic conductivity is discussed. C 2014 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4904276]
I. INTRODUCTION
Dielectric spectroscopy is one of the important tools of physicochemical study on electrolyte solutions, which has been applied to various systems.1 In dielectric spectra of many electrolyte solutions, a novel relaxation is observed in the subGHz region, which is slower than the reorientational relaxation of polar solvents. The slow relaxation has traditionally been assigned to the reorientational relaxation of ion pairs, and the properties of ion pairs such as interionic distance and the association degree have been deduced from the dielectric spectra. We recently developed a theory on the frequency-dependent ionic conductivity of electrolyte solutions based on the generalized Langevin formalism.2 The theory calculates the frequency-dependent conductivity from the radial distribution function between ions, and it can treat the reorientational motion of ion pairs and the collective dynamics of ionic atmosphere simultaneously. The theory was then applied to the aqueous solution of NaCl, which is a typical strong electrolyte solution, and we found a relaxation in the sub-GHz region which was ascribed to the relaxation of the ionic atmosphere.3 The presence of the relaxation of the ionic atmosphere in the dielectric spectra of strong electrolyte solutions was then reported by experimental groups.4,5 We applied the same theory also to a model electrolyte solution in which the Bjerrum length was much larger than the ionic diameters, in order to clarify the molecular mechanism of the molar conductivity minimum.6 We succeeded in reproducing the concentration dependence of the equivalent conductivity qualitatively, and demonstrated that the collective a)E-mail: [email protected]
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dynamics of the charge density is responsible to the increase in the equivalent conductivity with concentration. In this work, we first analyze the dielectric aspect of our previous theoretical calculation on model electrolyte solutions of low polarity. Since the weak polarity of the solvent promotes ionic association, it is natural to expect the contribution of ion pairs to the dielectric spectra. In addition, the collective dynamics of ionic atmosphere, which elucidates the characteristic concentration dependence of equivalent conductivity, must be reflected in the dielectric spectra. We actually find that the dielectric spectra show bimodal relaxation, and that these two dynamics appear in the spectra separately. In the experimental part of this work, we determine the dielectric relaxation spectra of some electrolyte solutions whose solvents possess the dielectric constant as low as 7–8. The bimodal dielectric relaxation is confirmed for these solutions within the frequency range of 300 kHz–200 MHz. These spectra are approximated as the double Debye function, and the concentration dependence of the parameters is compared with theoretical predictions.
II. THEORETICAL CALCULATION A. Model and theory
The model electrolyte we analyzed in our previous work was described in detail in the literatures.6,7 Briefly, the ions are dissolved in solvent that is approximated as the dielectric continuum, whose relative dielectric constant is ε s. The dielectric constant of the solvent is assumed to be independent of both frequency and the concentration of ions for brevity. The ions are spherical, and the interionic interaction, u(r), is described as
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k BT l B ( σ ) 12 ( σ ) u(r) = ± , σ r r
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(1)
where kB and T stand for the Boltzmann constant and the absolute temperature, respectively. The sign of the second term is + for cation-cation and anion-anion interactions, and – for cation-anion one. The ionic radius, which is the same for the cation and the anion, is represented by σ. The Bjerrum length, l B, is a measure of the strength of the interionic Coulombic interaction, which is defined as lB ≡
e2 , 4πε 0ε s k BT
(2)
where e and ε 0 stand for the absolute value of the charge on both ions and the dielectric constant of the vacuum, respectively. In the calculations analyzed in this work, the value of l B is fixed to be 10σ, and the number density of each ion, ρ, is varied as 5 × 10−4 ≤ ρσ 3 ≤ 0.1. The random force on an ion from the solvent is approximated as a Gauss-Markov noise, and the hydrodynamic interaction between ions is neglected. The bare values of the self-diffusion coefficients of both ions are the same, which is denoted as D0. The frequency-dependent ionic conductivity was derived from the radial distribution function between ions using the theory we proposed based on the generalized Langevin theory.2 The static correlation functions between ions, which are required as input parameters, were taken from our Brownian dynamics simulation on the same system. B. Calculation of dielectric relaxation spectrum
The dielectric spectrum, ε(ν), is related to the frequencydependent electric conductivity, σ(ν), as σ(ν) = 2πiε 0ν[ε(ν) − ε s ].
(3)
Since the second term corresponds to the contribution of the electric polarization of the solvent, it is not included in the ionic conductivity, σ(ν). In the dielectric spectroscopy of electrolyte solutions, the low-frequency diverging part is usually subtracted as1 ∆ε(ν) ≡ ε(ν) −
σ(0) , 2πiε 0 ν
FIG. 1. The equivalent conductivity is plotted against the number density of each ion. The values of l B/σ are 2 (red), 5 (blue), 10 (green), and 20 (black). The filled symbols show the results of the Brownian dynamics simulation,7 while the open ones do that of the theoretical calculation.6 The arrows indicate the density at which the Debye lengths are equal to the ionic diameter.
value, Λ0, is given by Λ0 ≡
2e2 D0 . k BT
(5)
The simulation results in Fig. 1 are taken from those in Fig. 1 of Ref. 7, while the theoretical ones are from those in Fig. 6 of Ref. 6. As is demonstrated in Fig. 1, the increase in the equivalent conductivity with concentration is reproduced in our simulation of l B = 10σ. With the theoretical calculation, we succeeded in the concentration dependence of Λ(0), as is demonstrated also in Fig. 1. The frequency-dependent equivalent conductivity, Λ(ν), exhibited in Figs. 10 and 11 of Ref. 6, is converted into the dielectric spectrum, ∆ε(ν), and the real and imaginary parts are shown in Figs. 2 and 3, respectively. It is to be noted here that the longitudinal axis of Fig. 2 is in a linear scale, while that of Fig. 3 is in a log scale. In these figures and hereafter, the frequency is shown in the reduced unit in which D0/σ 2 is unity. The relaxation frequency increases with increasing concentration, as was demonstrated in Ref. 6. In the concentration region ρσ 3 < 0.001, in addition, the dielectric relaxation becomes bimodal, as is clearly seen in Fig. 3. The relaxation frequency of the faster one, ν ∼ 0.3, little depends on the concentration,
(4)
where σ(0) stands for the direct-current (DC) conductivity, and the remaining part of the spectrum, ∆ε(ν), is sometimes called simply the dielectric spectrum. In this work, ∆ε(ν) is calculated from σ(ν) using Eq. (4). C. Results and discussion on theoretical calculation
Figure 1 shows the concentration dependence of the equivalent DC conductivity calculated with Brownian dynamics simulation in our previous work.7 Although the theoretical analysis is performed solely on the solutions of l B = 10σ, the results of the systems of l B = 2σ, 5σ, and 20σ are also plotted for convenience. Here, the equivalent DC conductivity, Λ(0), is defined as Λ(0) = σ(0)/ρ, and its low-concentration limiting
FIG. 2. The real parts of the dielectric spectra of the model electrolyte solutions are exhibited. The values of the ionic density are, from left to right, ρσ 3 = 0.00005 (circles); 0.0001, 0.0002, 0.0005 (squares); 0.001, 0.002, 0.005 (diamonds); 0.01, 0.02, 0.05 (triangles); and 0.1.
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while that of the slower one increases with increasing concentration. The relative amplitude of the slower one to that of the faster one decreases with concentration. In order to assign these two relaxation modes, the frequency-dependent response of the pair distribution between a cation and an anion to the applied electric field, ρ˜ x (ν,r), is calculated, which is plotted in Fig. 4. The definition of the function ρ˜ x (ν,r) was given in Ref. 6. Since the angular dependence of the pair distribution is p-type, only the radial dependence of the distribution is plotted. The sign of the response function is defined so that the negative distribution means the drag of counter ions behind the electrophoretic motion of an ion. The calculation is performed on the lowest concentration, ρσ 3 = 0.00005, and the functions at two frequencies, ν = 0.0032 and 0.32, are exhibited. These two frequencies correspond to the relaxation frequency of the two modes in the dielectric spectrum, respectively. The slower relaxation accompanies the response of the long-range ionic pair distribution, r ∼ 10σ. This relaxation is thus naturally assigned to that of the ionic atmosphere. The increase in its relaxation frequency with concentration also supports the assignment, because the increase in the ionic concentration leads to the decrease in the Debye length, which results in the increase in the relaxation frequency. On the other hand, the response of the pair distribution is limited to the
contact region, r < 2σ, in the case of the faster relaxation. Therefore, the faster relaxation is ascribed to the reorientational relaxation of contact ion pair. Provided that the reorientational relaxation time of an isolated contact ion pair is independent of concentration, the small concentration dependence of the relaxation frequency of the faster relaxation is in harmony with the assignment above. With increasing the concentration of the salt, the Debye length becomes smaller, and it can be comparable or smaller than the interionic distance of the contact ion pair. We can thus no longer distinguish the relaxation of the ionic atmosphere and the reorientation of the ion pair, and these two relaxation modes are merged to form a coupled mode. The coupled mode inherits the concentration dependence of its relaxation frequency from the relaxation of the ionic atmosphere, so that its relaxation becomes faster with concentration. As was described in our previous work,6 the increase in the relaxation frequency of the coupled mode elucidates the increase in the DC equivalent conductivity with concentration. Figure 5 demonstrates the Debye length as the function of the concentration. The Debye length equals to 10σ at l B = 10σ and ρσ 3 = 0.00005, which is large enough for the separation of the relaxation modes of the ionic atmosphere and the ion pair as is exhibited in Fig. 3. The Debye length√however decreases with concentration in proportional to 1/ ρ, and it becomes smaller than σ at ρσ 3 > 0.004. The concentrations at which the Debye length equals to σ are indicated with arrows in Fig. 1. The concentrations of the minimum equivalent conductivity lie near the arrows in cases of l B = 10σ and 20σ, which supports our idea that the mergence of the relaxation modes of the ionic atmosphere and the ion pair occurs at the concentration where the increase in the equivalent conductivity with concentration begins. Although the theoretical calculation demonstrates the mergence of the relaxations of ion pairs and ionic atmospheres around the concentration of the minimum equivalent conductivity, it should be noted here that the presence of the bimodal relaxation is not the necessary condition of the increase in equivalent conductivity with concentration. In our previous work, we calculated the concentration dependence of the equivalent conductivity using the mean-spherical approximation and the mode-coupling theory, and succeeded
FIG. 4. The frequency-dependent responses of the ionic pair distribution at ρσ 3 = 0.00005 are demonstrated. The values of the frequencies are ν = 0.0032 (solid curve) and 0.32 (dotted curve), respectively, where the unit of the frequency is normalized to D/σ 2.
FIG. 5. The Debye lengths, l D, are plotted as the function of the number density of ions. The values of l B/σ are 2 (circles), 5 (squares), 10 (diamonds), and 20 (triangles).
FIG. 3. The imaginary parts of the dielectric spectra of the model electrolyte solutions are exhibited. The meanings of the symbols are the same as those in Fig. 2. It is to be noted that the vertical axis is drawn in a log scale.
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in reproducing the equivalent conductivity minimum.6 As we have demonstrated in our previous work, the mode-coupling theory is unable to describe the bound dynamics of ion pairs.2 Thus, the reorientational relaxation of ion pairs was absent in our mode-coupling theoretical calculation. The theory, nonetheless, reproduced the presence of the equivalent conductivity minimum, indicating that the reorientational relaxation of ion pairs does not play an essential role in the equivalent conductivity minimum. The important point is that the Debye screening length becomes smaller than the ionic size, as the mode-coupling theory suggested.6 We would like to comment here on the concentration dependence of the “static” permittivity, ε ′(0), which is defined as ε ′ (0) ≡ lim ε ′ (ν). ν→0
(6)
Based on the conventional assignment of the dielectric relaxation to the reorientational relaxation of ion pairs, the “static” permittivity has been used to obtain the concentration and the dipole moment of ion pairs. Since the concentration of ion pairs is usually an increasing function of ionic concentration, the “static” permittivity is also expected to increase with concentration. Experimentally, the “static” permittivity is a decreasing function of ionic concentration in highly polar solvents due to the dielectric saturation,1 while the increase is usually found in weakly polar solvents as will be shown in Sec. III. Since the value of ε s is assumed to be independent of ionic concentration, the dielectric saturation is not considered in our model. In Fig. 2, although ε ′(0) is essentially an increasing function of the concentration, a small decrease is found from ρσ 3 = 0.02 to 0.05. Such decrease in the “static” permittivity was already reported experimentally by Petrucci and coworkers, and they ascribed the decrease to the formation of a nonpolar quartet from two polar ion pairs.8 They also demonstrated the presence of an ultrasonic relaxation in the MHz region, which they assigned to the dimerization equilibrium of ion pairs. Considering that the reorientational relaxation of ion pairs is not isolated in such a concentrated regime, however, we consider that the decrease in the “static” permittivity may originate from the collective ionic dynamics represented as the coupled mode between the relaxations of ion pairs and ionic atmosphere.
± 0.2 ◦C, and the concentration was varied from 1 mmol/kg to 100 mmol/kg. The measurements were performed two or three times for a sample to test the reproducibility. The dielectric spectra of these solutions were determined using a parallel plate cell. The liquid solution was sandwiched with two circular electrodes made of stainless steel, whose diameter was 20 mm, and the frequency-dependent electric impedance between these two electrodes was obtained with a vector network analyzer (ZVL3/03, Rhode & Schwarz) through the measurement of S11(ν) parameter. The electric response of the sample cell including the sample solution, electrodes, and connectors was approximated as the equivalent circuit described in Ref. 10, and the parameters of the circuit were determined from the measurement of air and water. The distance between the two electrodes is estimated to be 240 µm according to the parameters of the equivalent circuit. The frequency range of the measurement was from 300 kHz to 200 MHz. The DC conductivity was determined from the plateau of σ(ν) around 1 MHz. The density of the solution was determined with a vibrating tube densitometer (DMA60/602, Anton Paar), which was used to calculate the molar DC conductivity. B. Results and discussion on dielectric spectrum
The real part of the frequency-dependent conductivity of TBATf/THF solutions, σ ′(ν), is exhibited in Fig. 6. The AC conductivity, σ ′(ν), of the neat polar solvent is proportional to ν 2 in the low-frequency limit due to the reorientational relaxation of the polar molecule, which is observed at ν > 100 MHz in Fig. 6. In the presence of the dissolved salt, σ ′(ν) shows a finite plateau value near 1 MHz, and it gradually increases with increasing frequency, reflecting both the frequency-dependent mobility of ions and the reorientational relaxation of the solvent. A small dip is observed in σ ′(ν) at 50 MHz when the salt concentration is low, which is an experimental artefact possibly originated from the insufficient ability of the equivalent circuit to describe the real sample cell. The molar DC conductivity determined from the lowfrequency plateau value is plotted as the function of the molar
III. DIELECTRIC SPECTROSCOPY A. Experimental
The dielectric spectra of three systems were investigated, namely, tetrabutylammonium trifluoromethanesulfonate (TBATf, Sigma-Aldrich) in tetrahydrofuran (THF, spectroscopic grade, Kishida Chemical co., Ltd.), lithium trifluoromethanesulfonate (LiTf, Kishida Chemical co., Ltd.) in THF, and lithium perchlorate (LiClO4, lithium battery grade, Kishida Chemical co., Ltd.) in tetraglyme (Sigma-Aldrich). The dielectric constants of these two solvents are about 7-8. We have investigated the dielectric spectra of these solutions at higher concentration regime in our previous works.9,10 Both solvents were dried with Molecular Sieves 3A (Kishida Chemical co., Ltd.) prior to use. The temperature was 25.0
FIG. 6. The real-parts of the AC conductivity of TBATf/THF solutions are shown as the functions of frequency. The concentrations are, from lower to upper, 0 (red), 0.85 (blue), 1.7 (light green), 4.2 (black), 8.5 (pink), 17 (aqua), 42 (deep green), and 86 mol/m3 (orange), respectively. The small dips near 50 MHz are the experimental artifacts (see text).
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concentration in Fig. 7. The experiment was performed three times for a given concentration, and all the results are plotted there. A minimum of the molar DC conductivity is clearly observed around 10 mol/m3 in Fig. 7. The values of the molar DC conductivity obtained in this work are consistent with those reported in literatures.9,11 The AC conductivity is converted into the salt-contribution of the dielectric spectrum, ∆∆ε(ν), which is defined as ∆∆ε(ν;c) ≡ ∆ε(ν;c) − ∆ε(ν;c = 0),
(7)
where c stands for the concentration of the salt. The contribution of the DC conductivity is already subtracted from ∆ε(ν;c) as is described in Eq. (4). The effects of ions on the dielectric response of solvent are neglected here. Figure 8 shows ∆∆ε(ν) of TBATf/THF solutions at various concentrations. The spectra at c = 0.85 and 1.7 mol/m3 are omitted here because their signal-to-noise ratios are low. All the spectra in Fig. 8 exhibit the bimodal relaxation, as the theoretical calculation predicts. The faster one lies in the 100 MHz region, and the slower one lies between 10 and 100 MHz. The relaxation frequency of the slower one increases with concentration, which is also in harmony with the theoretical calculation. Given the resemblance between the experimental and theoretical spectra, we can assign the faster and slower relaxations to those of ion pairs and ionic atmosphere, respectively. There have been some experimental works that report the presence of a small relaxation in the 100 MHz region in the dielectric spectra of the aqueous solutions of alkali halides.4,5 However, since the reorientational relaxation of ion pairs may also be observed in the same frequency region, there was no definite proof to assign the relaxation observed in these works to the relaxation of ionic atmosphere. In this work, therefore, we succeeded in measuring the dielectric relaxation modes of the ionic atmosphere and ion pairs separately for the first time to the best of our knowledge. The spectra are approximated as the double Debye function, which is given by ∆∆ε(ν) =
An . 1 + 2πiτn ν n=1,2
(8)
FIG. 7. The molar DC conductivity of TBATf/THF solution is exhibited as the function of concentration. The experiments were performed three times for a given concentration, and all the results are plotted together.
FIG. 8. The salt-contribution of the dielectric spectrum of TBATf/THF solution, ∆∆ε(ν), is demonstrated. The values of the concentrations are, from upper to lower, 86 (red), 42 (blue), 17 (green), 8.5 (black), and 4.2 mol/m3 (purple), respectively. The real and imaginary parts of the experimental values are plotted with filled and open symbols, respectively, and those of the fitting functions to the double-Debye function are with solid and dotted curves, respectively.
The fitting lines are also shown in Fig. 8. The double Debye function describes the spectra well at all the concentrations. One may consider that the effects of the dielectric saturation should be included in ∆∆ε(ν). Since the dielectric relaxation frequencies of neat solvents are higher than 10 GHz,9,10 the dielectric saturation is expected to appear as the negative constant contribution in the frequency region in our experiment. However, we succeeded in reproducing the spectra of TBATf/THF and LiClO4/tetraglyme solutions without such a constant negative term, which we consider is due to the following two reasons: one is that the concentrations of the salts are small enough so that the dielectric saturation is negligible, and the other is that it is effectively included in the higher relaxation because only the low-frequency side of the higher-frequency relaxation is measured here. The fitting parameters An and τn are plotted as the functions of concentration in Figs. 9(a) and 9(b), respectively. The total relaxation amplitude, Atot = A1 + A2, and the mean relaxation time, ⟨τ⟩ = (A1τ1 + A2τ2)/Atot, are also exhibited there. The decrease in τ1 with increasing the concentration, as is predicted by the theory, is clearly demonstrated here. The relatively small concentration dependence of τ2 is also in harmony with the theoretical calculation. However, given the large deviation of the parameters among independent measurements under the same condition, it is difficult to analyze these parameters quantitatively. In particular, the quantitative evaluation of τ2 suffers from the limited frequency range of our experiment. Figure 10 demonstrates the molar conductivity of LiTf/ THF solutions as the function of salt concentration. The results of two independent measurements are plotted together. Although the minimum of the molar conductivity like TBATf (Fig. 7) is observed as was reported in the literatures,9,12 the absolute value of the molar conductivity is about an order of magnitude smaller than that of TBATf. In addition, the concentration of the minimum molar conductivity is several times larger. We consider it is due to the stronger ionic association between Li+ and Tf − than that between TBA+ and Tf −, as was suggested in a literature based on vibrational spectroscopy.11 In our previous work on concentrated solutions, we found that
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FIG. 11. The salt-contribution of the dielectric spectrum of 88 mol/m3 LiTf/THF solution is exhibited. The filled and open symbols denote the real and imaginary parts, respectively, and the circles and the squares indicate the results of two independent measurements, respectively.
the relaxation frequency of the conductivity of LiTf solution do not increase with concentration contrary to the theoretical prediction, which was ascribed to the contributions of higherorder aggregates. Figure 11 exhibits ∆∆ε(ν) of LiTf/THF solution at 88 mol/m3. The measurements were performed twice, and both results are shown. Compared with ∆∆ε(ν) of TBATf/THF solution in Fig. 8, the relaxation amplitude is several times smaller. A small relaxation of LiTf/THF solution is found around 20 MHz, as is the case of TBATf/THF. However,
the relaxation amplitude is so small that the relaxation is hardly detectable at lower concentrations. Therefore, we do not perform further quantitative analyses in this work. The molar conductivity of LiClO4/tetraglyme solutions is shown in Fig. 12, which is consistent with those in the literatures.10,13 In spite that the small size of Li+ ion is in favour of ionic association, the magnitude of the molar conductivity is comparable with that of TBATf/THF solutions. The minimum of the molar conductivity is also observed at the similar concentration. The dielectric relaxation spectra, ∆∆ε(ν), of LiClO4/ tetraglyme solutions are demonstrated in Fig. 13. The bimodal relaxation is clearly observed as is the case of TBATf/THF solutions in Fig. 8. The relaxation is a little slower compared with the TBATf/THF system, which is in harmony with the higher viscosity of tetraglyme. We can therefore apply the same assignment as that of TBATf/THF solution, that is, the faster and the slower relaxations to the ion pairs and the ionic atmosphere, respectively. The spectra in Fig. 13 are also approximated with the double Debye function, Eq. (8), and the fitting parameters are exhibited in Fig. 14. These parameters also resemble those of TBATf/THF solutions in Fig. 9, except that the values of the relaxation times are larger. Although τ2 appears to decrease with increasing the concentration, we cannot judge definitely
FIG. 10. The molar DC conductivity of LiTf/THF solution is shown as the function of the concentration. The results of the two independent measurements are plotted together.
FIG. 12. The molar DC conductivity of LiClO4/tetraglyme solution is plotted as the function of concentration. The results of the two independent measurements are shown together.
FIG. 9. The fitting parameters in the double-Debye function for TBATf/THF solution are plotted as the function of the concentration. The amplitudes and the relaxation times are shown in panels (a) and (b), respectively. The values of A 1, A 2, and A 1 + A 2 are plotted with red, blue, and green symbols in panel (a), respectively, and those of τ 1, τ 2, and ⟨τ⟩ are with red, blue, and green symbols in panel (b), respectively. The three symbols for a given condition indicate the results of the fitting of three independent measurements.
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FIG. 13. The salt-contribution of the dielectric spectra of LiClO4/tetraglyme solution is exhibited. The concentrations of the salt are, from upper to lower, 100 (red), 50 (blue), 20 (green), 10 (black), and 5 mol/m3 (purple), respectively. The meanings of the symbols are the same as those in Fig. 8.
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Watanabe and coworkers demonstrated that the equimolar mixture of lithium salts and tetraglyme exhibits physicochemical properties common to ionic liquids. They considered it because a lithium ion solvated by a tetraglyme behaves as a large complex ion, and they classified the solution into the “solvate ionic liquid.”14 They also investigated the solutions of lithium salts in various ethers and showed that the solutions in THF are better regarded as an ordinary concentrated electrolyte solutions, rather than the solvate ionic liquids.15 We consider that the difference between LiClO4/tetraglyme and LiTf/THF can be understood in a similar way, that is, the lithium ion solvated with a tetraglyme behaves as a bulky cation as large as TBA+, so that the dielectric relaxation spectrum of LiClO4/tetraglyme is similar to that of TBATf/THF.
IV. SUMMARY
whether the increase is due to artefacts of the fitting procedure or not. The dielectric properties of LiClO4/tetraglyme are close to those of TBATf/THF rather than LiTf/THF, although both LiClO4/tetraglyme and LiTf/THF possess the small cation, Li+. Given that the larger size of Tf− than ClO4− disfavours the ionic association, the difference between LiClO4/tetraglyme and LiTf/THF is better ascribed to that in solvent rather than counter anion.
The dielectric relaxation spectrum of dilute electrolyte solutions in solvents of low polarity was investigated both theoretically and experimentally, and the presence of the bimodal relaxation in the MHz region was demonstrated. The faster and the slower relaxations were assigned to the relaxations of the ion pairs and the ionic atmosphere, respectively. The relaxation frequency of the former little depends on concentration in the low concentration regime, while the latter becomes faster with concentration, reflecting the decrease in the Debye length. As far as we know, we observed the relaxation of the ionic atmosphere separately from that of ion pairs for the first time. With increasing the concentration, the Debye length becomes smaller to be comparable with ionic size. The reorientational relaxation of ionic pair is indistinguishable from the relaxation of ionic atmosphere in such a concentration regime, and the two relaxations are coupled there. The relaxation frequency of the coupled mode increases with increasing the concentration, reflecting the property of the relaxation of the ionic atmosphere, which explains the increase in the molar DC conductivity with concentration. Although our theoretical calculation is limited to the general behaviours of a model system, the experimental results demonstrate that the details of the relaxation spectra depend on both ions and solvents. In particular, it is revealed that LiClO4/tetraglyme system is similar to TBATf/THF rather than to LiTf/THF because the lithium ion solvated by a tetraglyme molecule behaves as a bulky cation as TBA+. It is thus our future task to include the effects of the ion-ion and the ionsolvent interactions into the theoretical calculation in order to improve the description of real solutions.
ACKNOWLEDGMENTS
This work was partly supported by the Japanese Society for the Promotion of Science, KAKENHI Grant No. 24550019. 1R.
FIG. 14. The fitting parameters for LiClO4/tetraglyme solution are plotted as the functions of concentration. The panels (a) and (b) show the amplitudes and the relaxation times, respectively, and the meanings of the symbols are the same as those in Fig. 9.
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Citation: The Journal of Chemical Physics 141, 244501 (2014); doi: 10.1063/1.4904276 View online: http://dx.doi.org/10.1063/1.4904276 View Table of Contents: http://aip.scitation.org/toc/jcp/141/24 Published by the American Institute of Physics
THE JOURNAL OF CHEMICAL PHYSICS 141, 244501 (2014)
Bimodal dielectric relaxation of electrolyte solutions in weakly polar solvents Tsuyoshi Yamaguchia) and Shinobu Koda Department of Molecular Design and Engineering, Graduate School of Engineering, Furo-cho B2-3 (611), Chikusa, Nagoya, Aichi 464-8603, Japan
(Received 2 October 2014; accepted 21 November 2014; published online 23 December 2014) The dielectric relaxation spectra of dilute electrolyte solutions in solvents of small dielectric constants are investigated both theoretically and experimentally. The theoretical calculation in our previous work [T. Yamaguchi, T. Matsuoka, and S. Koda, J. Chem. Phys. 135, 164511 (2011)] is reanalyzed, and it is shown that the dielectric relaxation spectra are composed of three components, namely, the relaxation of ionic atmosphere, the reorientational relaxation of ion pairs, and the collision between ions. The relaxation frequency of the slowest one increases with increasing the concentration, and the slower two relaxations, those of ionic atmosphere and ion pairs, merge into one at the concentration where the Debye length is comparable to the size of ions. Experimentally, the dielectric relaxation spectra of some electrolytes in two solvents, tetrahydrofuran and tetraglyme, are determined at frequencies from 300 kHz to 200 MHz, and the presence of the slower two relaxations was confirmed. The concentration dependence of the relaxation frequency is also in harmony with the theoretical calculation. The relationship between the dielectric relaxation spectra and the concentration dependence of the ionic conductivity is discussed. C 2014 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4904276]
I. INTRODUCTION
Dielectric spectroscopy is one of the important tools of physicochemical study on electrolyte solutions, which has been applied to various systems.1 In dielectric spectra of many electrolyte solutions, a novel relaxation is observed in the subGHz region, which is slower than the reorientational relaxation of polar solvents. The slow relaxation has traditionally been assigned to the reorientational relaxation of ion pairs, and the properties of ion pairs such as interionic distance and the association degree have been deduced from the dielectric spectra. We recently developed a theory on the frequency-dependent ionic conductivity of electrolyte solutions based on the generalized Langevin formalism.2 The theory calculates the frequency-dependent conductivity from the radial distribution function between ions, and it can treat the reorientational motion of ion pairs and the collective dynamics of ionic atmosphere simultaneously. The theory was then applied to the aqueous solution of NaCl, which is a typical strong electrolyte solution, and we found a relaxation in the sub-GHz region which was ascribed to the relaxation of the ionic atmosphere.3 The presence of the relaxation of the ionic atmosphere in the dielectric spectra of strong electrolyte solutions was then reported by experimental groups.4,5 We applied the same theory also to a model electrolyte solution in which the Bjerrum length was much larger than the ionic diameters, in order to clarify the molecular mechanism of the molar conductivity minimum.6 We succeeded in reproducing the concentration dependence of the equivalent conductivity qualitatively, and demonstrated that the collective a)E-mail: [email protected]
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dynamics of the charge density is responsible to the increase in the equivalent conductivity with concentration. In this work, we first analyze the dielectric aspect of our previous theoretical calculation on model electrolyte solutions of low polarity. Since the weak polarity of the solvent promotes ionic association, it is natural to expect the contribution of ion pairs to the dielectric spectra. In addition, the collective dynamics of ionic atmosphere, which elucidates the characteristic concentration dependence of equivalent conductivity, must be reflected in the dielectric spectra. We actually find that the dielectric spectra show bimodal relaxation, and that these two dynamics appear in the spectra separately. In the experimental part of this work, we determine the dielectric relaxation spectra of some electrolyte solutions whose solvents possess the dielectric constant as low as 7–8. The bimodal dielectric relaxation is confirmed for these solutions within the frequency range of 300 kHz–200 MHz. These spectra are approximated as the double Debye function, and the concentration dependence of the parameters is compared with theoretical predictions.
II. THEORETICAL CALCULATION A. Model and theory
The model electrolyte we analyzed in our previous work was described in detail in the literatures.6,7 Briefly, the ions are dissolved in solvent that is approximated as the dielectric continuum, whose relative dielectric constant is ε s. The dielectric constant of the solvent is assumed to be independent of both frequency and the concentration of ions for brevity. The ions are spherical, and the interionic interaction, u(r), is described as
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k BT l B ( σ ) 12 ( σ ) u(r) = ± , σ r r
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(1)
where kB and T stand for the Boltzmann constant and the absolute temperature, respectively. The sign of the second term is + for cation-cation and anion-anion interactions, and – for cation-anion one. The ionic radius, which is the same for the cation and the anion, is represented by σ. The Bjerrum length, l B, is a measure of the strength of the interionic Coulombic interaction, which is defined as lB ≡
e2 , 4πε 0ε s k BT
(2)
where e and ε 0 stand for the absolute value of the charge on both ions and the dielectric constant of the vacuum, respectively. In the calculations analyzed in this work, the value of l B is fixed to be 10σ, and the number density of each ion, ρ, is varied as 5 × 10−4 ≤ ρσ 3 ≤ 0.1. The random force on an ion from the solvent is approximated as a Gauss-Markov noise, and the hydrodynamic interaction between ions is neglected. The bare values of the self-diffusion coefficients of both ions are the same, which is denoted as D0. The frequency-dependent ionic conductivity was derived from the radial distribution function between ions using the theory we proposed based on the generalized Langevin theory.2 The static correlation functions between ions, which are required as input parameters, were taken from our Brownian dynamics simulation on the same system. B. Calculation of dielectric relaxation spectrum
The dielectric spectrum, ε(ν), is related to the frequencydependent electric conductivity, σ(ν), as σ(ν) = 2πiε 0ν[ε(ν) − ε s ].
(3)
Since the second term corresponds to the contribution of the electric polarization of the solvent, it is not included in the ionic conductivity, σ(ν). In the dielectric spectroscopy of electrolyte solutions, the low-frequency diverging part is usually subtracted as1 ∆ε(ν) ≡ ε(ν) −
σ(0) , 2πiε 0 ν
FIG. 1. The equivalent conductivity is plotted against the number density of each ion. The values of l B/σ are 2 (red), 5 (blue), 10 (green), and 20 (black). The filled symbols show the results of the Brownian dynamics simulation,7 while the open ones do that of the theoretical calculation.6 The arrows indicate the density at which the Debye lengths are equal to the ionic diameter.
value, Λ0, is given by Λ0 ≡
2e2 D0 . k BT
(5)
The simulation results in Fig. 1 are taken from those in Fig. 1 of Ref. 7, while the theoretical ones are from those in Fig. 6 of Ref. 6. As is demonstrated in Fig. 1, the increase in the equivalent conductivity with concentration is reproduced in our simulation of l B = 10σ. With the theoretical calculation, we succeeded in the concentration dependence of Λ(0), as is demonstrated also in Fig. 1. The frequency-dependent equivalent conductivity, Λ(ν), exhibited in Figs. 10 and 11 of Ref. 6, is converted into the dielectric spectrum, ∆ε(ν), and the real and imaginary parts are shown in Figs. 2 and 3, respectively. It is to be noted here that the longitudinal axis of Fig. 2 is in a linear scale, while that of Fig. 3 is in a log scale. In these figures and hereafter, the frequency is shown in the reduced unit in which D0/σ 2 is unity. The relaxation frequency increases with increasing concentration, as was demonstrated in Ref. 6. In the concentration region ρσ 3 < 0.001, in addition, the dielectric relaxation becomes bimodal, as is clearly seen in Fig. 3. The relaxation frequency of the faster one, ν ∼ 0.3, little depends on the concentration,
(4)
where σ(0) stands for the direct-current (DC) conductivity, and the remaining part of the spectrum, ∆ε(ν), is sometimes called simply the dielectric spectrum. In this work, ∆ε(ν) is calculated from σ(ν) using Eq. (4). C. Results and discussion on theoretical calculation
Figure 1 shows the concentration dependence of the equivalent DC conductivity calculated with Brownian dynamics simulation in our previous work.7 Although the theoretical analysis is performed solely on the solutions of l B = 10σ, the results of the systems of l B = 2σ, 5σ, and 20σ are also plotted for convenience. Here, the equivalent DC conductivity, Λ(0), is defined as Λ(0) = σ(0)/ρ, and its low-concentration limiting
FIG. 2. The real parts of the dielectric spectra of the model electrolyte solutions are exhibited. The values of the ionic density are, from left to right, ρσ 3 = 0.00005 (circles); 0.0001, 0.0002, 0.0005 (squares); 0.001, 0.002, 0.005 (diamonds); 0.01, 0.02, 0.05 (triangles); and 0.1.
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while that of the slower one increases with increasing concentration. The relative amplitude of the slower one to that of the faster one decreases with concentration. In order to assign these two relaxation modes, the frequency-dependent response of the pair distribution between a cation and an anion to the applied electric field, ρ˜ x (ν,r), is calculated, which is plotted in Fig. 4. The definition of the function ρ˜ x (ν,r) was given in Ref. 6. Since the angular dependence of the pair distribution is p-type, only the radial dependence of the distribution is plotted. The sign of the response function is defined so that the negative distribution means the drag of counter ions behind the electrophoretic motion of an ion. The calculation is performed on the lowest concentration, ρσ 3 = 0.00005, and the functions at two frequencies, ν = 0.0032 and 0.32, are exhibited. These two frequencies correspond to the relaxation frequency of the two modes in the dielectric spectrum, respectively. The slower relaxation accompanies the response of the long-range ionic pair distribution, r ∼ 10σ. This relaxation is thus naturally assigned to that of the ionic atmosphere. The increase in its relaxation frequency with concentration also supports the assignment, because the increase in the ionic concentration leads to the decrease in the Debye length, which results in the increase in the relaxation frequency. On the other hand, the response of the pair distribution is limited to the
contact region, r < 2σ, in the case of the faster relaxation. Therefore, the faster relaxation is ascribed to the reorientational relaxation of contact ion pair. Provided that the reorientational relaxation time of an isolated contact ion pair is independent of concentration, the small concentration dependence of the relaxation frequency of the faster relaxation is in harmony with the assignment above. With increasing the concentration of the salt, the Debye length becomes smaller, and it can be comparable or smaller than the interionic distance of the contact ion pair. We can thus no longer distinguish the relaxation of the ionic atmosphere and the reorientation of the ion pair, and these two relaxation modes are merged to form a coupled mode. The coupled mode inherits the concentration dependence of its relaxation frequency from the relaxation of the ionic atmosphere, so that its relaxation becomes faster with concentration. As was described in our previous work,6 the increase in the relaxation frequency of the coupled mode elucidates the increase in the DC equivalent conductivity with concentration. Figure 5 demonstrates the Debye length as the function of the concentration. The Debye length equals to 10σ at l B = 10σ and ρσ 3 = 0.00005, which is large enough for the separation of the relaxation modes of the ionic atmosphere and the ion pair as is exhibited in Fig. 3. The Debye length√however decreases with concentration in proportional to 1/ ρ, and it becomes smaller than σ at ρσ 3 > 0.004. The concentrations at which the Debye length equals to σ are indicated with arrows in Fig. 1. The concentrations of the minimum equivalent conductivity lie near the arrows in cases of l B = 10σ and 20σ, which supports our idea that the mergence of the relaxation modes of the ionic atmosphere and the ion pair occurs at the concentration where the increase in the equivalent conductivity with concentration begins. Although the theoretical calculation demonstrates the mergence of the relaxations of ion pairs and ionic atmospheres around the concentration of the minimum equivalent conductivity, it should be noted here that the presence of the bimodal relaxation is not the necessary condition of the increase in equivalent conductivity with concentration. In our previous work, we calculated the concentration dependence of the equivalent conductivity using the mean-spherical approximation and the mode-coupling theory, and succeeded
FIG. 4. The frequency-dependent responses of the ionic pair distribution at ρσ 3 = 0.00005 are demonstrated. The values of the frequencies are ν = 0.0032 (solid curve) and 0.32 (dotted curve), respectively, where the unit of the frequency is normalized to D/σ 2.
FIG. 5. The Debye lengths, l D, are plotted as the function of the number density of ions. The values of l B/σ are 2 (circles), 5 (squares), 10 (diamonds), and 20 (triangles).
FIG. 3. The imaginary parts of the dielectric spectra of the model electrolyte solutions are exhibited. The meanings of the symbols are the same as those in Fig. 2. It is to be noted that the vertical axis is drawn in a log scale.
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in reproducing the equivalent conductivity minimum.6 As we have demonstrated in our previous work, the mode-coupling theory is unable to describe the bound dynamics of ion pairs.2 Thus, the reorientational relaxation of ion pairs was absent in our mode-coupling theoretical calculation. The theory, nonetheless, reproduced the presence of the equivalent conductivity minimum, indicating that the reorientational relaxation of ion pairs does not play an essential role in the equivalent conductivity minimum. The important point is that the Debye screening length becomes smaller than the ionic size, as the mode-coupling theory suggested.6 We would like to comment here on the concentration dependence of the “static” permittivity, ε ′(0), which is defined as ε ′ (0) ≡ lim ε ′ (ν). ν→0
(6)
Based on the conventional assignment of the dielectric relaxation to the reorientational relaxation of ion pairs, the “static” permittivity has been used to obtain the concentration and the dipole moment of ion pairs. Since the concentration of ion pairs is usually an increasing function of ionic concentration, the “static” permittivity is also expected to increase with concentration. Experimentally, the “static” permittivity is a decreasing function of ionic concentration in highly polar solvents due to the dielectric saturation,1 while the increase is usually found in weakly polar solvents as will be shown in Sec. III. Since the value of ε s is assumed to be independent of ionic concentration, the dielectric saturation is not considered in our model. In Fig. 2, although ε ′(0) is essentially an increasing function of the concentration, a small decrease is found from ρσ 3 = 0.02 to 0.05. Such decrease in the “static” permittivity was already reported experimentally by Petrucci and coworkers, and they ascribed the decrease to the formation of a nonpolar quartet from two polar ion pairs.8 They also demonstrated the presence of an ultrasonic relaxation in the MHz region, which they assigned to the dimerization equilibrium of ion pairs. Considering that the reorientational relaxation of ion pairs is not isolated in such a concentrated regime, however, we consider that the decrease in the “static” permittivity may originate from the collective ionic dynamics represented as the coupled mode between the relaxations of ion pairs and ionic atmosphere.
± 0.2 ◦C, and the concentration was varied from 1 mmol/kg to 100 mmol/kg. The measurements were performed two or three times for a sample to test the reproducibility. The dielectric spectra of these solutions were determined using a parallel plate cell. The liquid solution was sandwiched with two circular electrodes made of stainless steel, whose diameter was 20 mm, and the frequency-dependent electric impedance between these two electrodes was obtained with a vector network analyzer (ZVL3/03, Rhode & Schwarz) through the measurement of S11(ν) parameter. The electric response of the sample cell including the sample solution, electrodes, and connectors was approximated as the equivalent circuit described in Ref. 10, and the parameters of the circuit were determined from the measurement of air and water. The distance between the two electrodes is estimated to be 240 µm according to the parameters of the equivalent circuit. The frequency range of the measurement was from 300 kHz to 200 MHz. The DC conductivity was determined from the plateau of σ(ν) around 1 MHz. The density of the solution was determined with a vibrating tube densitometer (DMA60/602, Anton Paar), which was used to calculate the molar DC conductivity. B. Results and discussion on dielectric spectrum
The real part of the frequency-dependent conductivity of TBATf/THF solutions, σ ′(ν), is exhibited in Fig. 6. The AC conductivity, σ ′(ν), of the neat polar solvent is proportional to ν 2 in the low-frequency limit due to the reorientational relaxation of the polar molecule, which is observed at ν > 100 MHz in Fig. 6. In the presence of the dissolved salt, σ ′(ν) shows a finite plateau value near 1 MHz, and it gradually increases with increasing frequency, reflecting both the frequency-dependent mobility of ions and the reorientational relaxation of the solvent. A small dip is observed in σ ′(ν) at 50 MHz when the salt concentration is low, which is an experimental artefact possibly originated from the insufficient ability of the equivalent circuit to describe the real sample cell. The molar DC conductivity determined from the lowfrequency plateau value is plotted as the function of the molar
III. DIELECTRIC SPECTROSCOPY A. Experimental
The dielectric spectra of three systems were investigated, namely, tetrabutylammonium trifluoromethanesulfonate (TBATf, Sigma-Aldrich) in tetrahydrofuran (THF, spectroscopic grade, Kishida Chemical co., Ltd.), lithium trifluoromethanesulfonate (LiTf, Kishida Chemical co., Ltd.) in THF, and lithium perchlorate (LiClO4, lithium battery grade, Kishida Chemical co., Ltd.) in tetraglyme (Sigma-Aldrich). The dielectric constants of these two solvents are about 7-8. We have investigated the dielectric spectra of these solutions at higher concentration regime in our previous works.9,10 Both solvents were dried with Molecular Sieves 3A (Kishida Chemical co., Ltd.) prior to use. The temperature was 25.0
FIG. 6. The real-parts of the AC conductivity of TBATf/THF solutions are shown as the functions of frequency. The concentrations are, from lower to upper, 0 (red), 0.85 (blue), 1.7 (light green), 4.2 (black), 8.5 (pink), 17 (aqua), 42 (deep green), and 86 mol/m3 (orange), respectively. The small dips near 50 MHz are the experimental artifacts (see text).
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concentration in Fig. 7. The experiment was performed three times for a given concentration, and all the results are plotted there. A minimum of the molar DC conductivity is clearly observed around 10 mol/m3 in Fig. 7. The values of the molar DC conductivity obtained in this work are consistent with those reported in literatures.9,11 The AC conductivity is converted into the salt-contribution of the dielectric spectrum, ∆∆ε(ν), which is defined as ∆∆ε(ν;c) ≡ ∆ε(ν;c) − ∆ε(ν;c = 0),
(7)
where c stands for the concentration of the salt. The contribution of the DC conductivity is already subtracted from ∆ε(ν;c) as is described in Eq. (4). The effects of ions on the dielectric response of solvent are neglected here. Figure 8 shows ∆∆ε(ν) of TBATf/THF solutions at various concentrations. The spectra at c = 0.85 and 1.7 mol/m3 are omitted here because their signal-to-noise ratios are low. All the spectra in Fig. 8 exhibit the bimodal relaxation, as the theoretical calculation predicts. The faster one lies in the 100 MHz region, and the slower one lies between 10 and 100 MHz. The relaxation frequency of the slower one increases with concentration, which is also in harmony with the theoretical calculation. Given the resemblance between the experimental and theoretical spectra, we can assign the faster and slower relaxations to those of ion pairs and ionic atmosphere, respectively. There have been some experimental works that report the presence of a small relaxation in the 100 MHz region in the dielectric spectra of the aqueous solutions of alkali halides.4,5 However, since the reorientational relaxation of ion pairs may also be observed in the same frequency region, there was no definite proof to assign the relaxation observed in these works to the relaxation of ionic atmosphere. In this work, therefore, we succeeded in measuring the dielectric relaxation modes of the ionic atmosphere and ion pairs separately for the first time to the best of our knowledge. The spectra are approximated as the double Debye function, which is given by ∆∆ε(ν) =
An . 1 + 2πiτn ν n=1,2
(8)
FIG. 7. The molar DC conductivity of TBATf/THF solution is exhibited as the function of concentration. The experiments were performed three times for a given concentration, and all the results are plotted together.
FIG. 8. The salt-contribution of the dielectric spectrum of TBATf/THF solution, ∆∆ε(ν), is demonstrated. The values of the concentrations are, from upper to lower, 86 (red), 42 (blue), 17 (green), 8.5 (black), and 4.2 mol/m3 (purple), respectively. The real and imaginary parts of the experimental values are plotted with filled and open symbols, respectively, and those of the fitting functions to the double-Debye function are with solid and dotted curves, respectively.
The fitting lines are also shown in Fig. 8. The double Debye function describes the spectra well at all the concentrations. One may consider that the effects of the dielectric saturation should be included in ∆∆ε(ν). Since the dielectric relaxation frequencies of neat solvents are higher than 10 GHz,9,10 the dielectric saturation is expected to appear as the negative constant contribution in the frequency region in our experiment. However, we succeeded in reproducing the spectra of TBATf/THF and LiClO4/tetraglyme solutions without such a constant negative term, which we consider is due to the following two reasons: one is that the concentrations of the salts are small enough so that the dielectric saturation is negligible, and the other is that it is effectively included in the higher relaxation because only the low-frequency side of the higher-frequency relaxation is measured here. The fitting parameters An and τn are plotted as the functions of concentration in Figs. 9(a) and 9(b), respectively. The total relaxation amplitude, Atot = A1 + A2, and the mean relaxation time, ⟨τ⟩ = (A1τ1 + A2τ2)/Atot, are also exhibited there. The decrease in τ1 with increasing the concentration, as is predicted by the theory, is clearly demonstrated here. The relatively small concentration dependence of τ2 is also in harmony with the theoretical calculation. However, given the large deviation of the parameters among independent measurements under the same condition, it is difficult to analyze these parameters quantitatively. In particular, the quantitative evaluation of τ2 suffers from the limited frequency range of our experiment. Figure 10 demonstrates the molar conductivity of LiTf/ THF solutions as the function of salt concentration. The results of two independent measurements are plotted together. Although the minimum of the molar conductivity like TBATf (Fig. 7) is observed as was reported in the literatures,9,12 the absolute value of the molar conductivity is about an order of magnitude smaller than that of TBATf. In addition, the concentration of the minimum molar conductivity is several times larger. We consider it is due to the stronger ionic association between Li+ and Tf − than that between TBA+ and Tf −, as was suggested in a literature based on vibrational spectroscopy.11 In our previous work on concentrated solutions, we found that
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FIG. 11. The salt-contribution of the dielectric spectrum of 88 mol/m3 LiTf/THF solution is exhibited. The filled and open symbols denote the real and imaginary parts, respectively, and the circles and the squares indicate the results of two independent measurements, respectively.
the relaxation frequency of the conductivity of LiTf solution do not increase with concentration contrary to the theoretical prediction, which was ascribed to the contributions of higherorder aggregates. Figure 11 exhibits ∆∆ε(ν) of LiTf/THF solution at 88 mol/m3. The measurements were performed twice, and both results are shown. Compared with ∆∆ε(ν) of TBATf/THF solution in Fig. 8, the relaxation amplitude is several times smaller. A small relaxation of LiTf/THF solution is found around 20 MHz, as is the case of TBATf/THF. However,
the relaxation amplitude is so small that the relaxation is hardly detectable at lower concentrations. Therefore, we do not perform further quantitative analyses in this work. The molar conductivity of LiClO4/tetraglyme solutions is shown in Fig. 12, which is consistent with those in the literatures.10,13 In spite that the small size of Li+ ion is in favour of ionic association, the magnitude of the molar conductivity is comparable with that of TBATf/THF solutions. The minimum of the molar conductivity is also observed at the similar concentration. The dielectric relaxation spectra, ∆∆ε(ν), of LiClO4/ tetraglyme solutions are demonstrated in Fig. 13. The bimodal relaxation is clearly observed as is the case of TBATf/THF solutions in Fig. 8. The relaxation is a little slower compared with the TBATf/THF system, which is in harmony with the higher viscosity of tetraglyme. We can therefore apply the same assignment as that of TBATf/THF solution, that is, the faster and the slower relaxations to the ion pairs and the ionic atmosphere, respectively. The spectra in Fig. 13 are also approximated with the double Debye function, Eq. (8), and the fitting parameters are exhibited in Fig. 14. These parameters also resemble those of TBATf/THF solutions in Fig. 9, except that the values of the relaxation times are larger. Although τ2 appears to decrease with increasing the concentration, we cannot judge definitely
FIG. 10. The molar DC conductivity of LiTf/THF solution is shown as the function of the concentration. The results of the two independent measurements are plotted together.
FIG. 12. The molar DC conductivity of LiClO4/tetraglyme solution is plotted as the function of concentration. The results of the two independent measurements are shown together.
FIG. 9. The fitting parameters in the double-Debye function for TBATf/THF solution are plotted as the function of the concentration. The amplitudes and the relaxation times are shown in panels (a) and (b), respectively. The values of A 1, A 2, and A 1 + A 2 are plotted with red, blue, and green symbols in panel (a), respectively, and those of τ 1, τ 2, and ⟨τ⟩ are with red, blue, and green symbols in panel (b), respectively. The three symbols for a given condition indicate the results of the fitting of three independent measurements.
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FIG. 13. The salt-contribution of the dielectric spectra of LiClO4/tetraglyme solution is exhibited. The concentrations of the salt are, from upper to lower, 100 (red), 50 (blue), 20 (green), 10 (black), and 5 mol/m3 (purple), respectively. The meanings of the symbols are the same as those in Fig. 8.
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Watanabe and coworkers demonstrated that the equimolar mixture of lithium salts and tetraglyme exhibits physicochemical properties common to ionic liquids. They considered it because a lithium ion solvated by a tetraglyme behaves as a large complex ion, and they classified the solution into the “solvate ionic liquid.”14 They also investigated the solutions of lithium salts in various ethers and showed that the solutions in THF are better regarded as an ordinary concentrated electrolyte solutions, rather than the solvate ionic liquids.15 We consider that the difference between LiClO4/tetraglyme and LiTf/THF can be understood in a similar way, that is, the lithium ion solvated with a tetraglyme behaves as a bulky cation as large as TBA+, so that the dielectric relaxation spectrum of LiClO4/tetraglyme is similar to that of TBATf/THF.
IV. SUMMARY
whether the increase is due to artefacts of the fitting procedure or not. The dielectric properties of LiClO4/tetraglyme are close to those of TBATf/THF rather than LiTf/THF, although both LiClO4/tetraglyme and LiTf/THF possess the small cation, Li+. Given that the larger size of Tf− than ClO4− disfavours the ionic association, the difference between LiClO4/tetraglyme and LiTf/THF is better ascribed to that in solvent rather than counter anion.
The dielectric relaxation spectrum of dilute electrolyte solutions in solvents of low polarity was investigated both theoretically and experimentally, and the presence of the bimodal relaxation in the MHz region was demonstrated. The faster and the slower relaxations were assigned to the relaxations of the ion pairs and the ionic atmosphere, respectively. The relaxation frequency of the former little depends on concentration in the low concentration regime, while the latter becomes faster with concentration, reflecting the decrease in the Debye length. As far as we know, we observed the relaxation of the ionic atmosphere separately from that of ion pairs for the first time. With increasing the concentration, the Debye length becomes smaller to be comparable with ionic size. The reorientational relaxation of ionic pair is indistinguishable from the relaxation of ionic atmosphere in such a concentration regime, and the two relaxations are coupled there. The relaxation frequency of the coupled mode increases with increasing the concentration, reflecting the property of the relaxation of the ionic atmosphere, which explains the increase in the molar DC conductivity with concentration. Although our theoretical calculation is limited to the general behaviours of a model system, the experimental results demonstrate that the details of the relaxation spectra depend on both ions and solvents. In particular, it is revealed that LiClO4/tetraglyme system is similar to TBATf/THF rather than to LiTf/THF because the lithium ion solvated by a tetraglyme molecule behaves as a bulky cation as TBA+. It is thus our future task to include the effects of the ion-ion and the ionsolvent interactions into the theoretical calculation in order to improve the description of real solutions.
ACKNOWLEDGMENTS
This work was partly supported by the Japanese Society for the Promotion of Science, KAKENHI Grant No. 24550019. 1R.
FIG. 14. The fitting parameters for LiClO4/tetraglyme solution are plotted as the functions of concentration. The panels (a) and (b) show the amplitudes and the relaxation times, respectively, and the meanings of the symbols are the same as those in Fig. 9.
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