Article pubs.acs.org/JPCB
Ion Diffusion Coefficients Model and Molar Conductivities of Ionic Salts in Aprotic Solvents Leoncio Garrido,* Alberto Mejía, Nuria García, Pilar Tiemblo, and Julio Guzmán* Departamento de Química-Física, Instituto de Ciencia y Tecnología de Polímeros, Consejo Superior de Investigaciones Científicas (ICTP-CSIC), Juan de la Cierva 3, E-28006 Madrid, Spain S Supporting Information *
ABSTRACT: In the study of the electric properties of electrolytes, the determination of the diffusion coefficients of the species that intervene in the charge transport process is of great importance, particularly that of the free ions (D+ and D−), the only species that contribute to the conductivity. In this work we propose a model that allows, with reasonable assumptions, determination of D+ and D−, and the degree of dissociation of the salt, α, at different concentrations, using the diffusion coefficients experimentally obtained with NMR. Also, it is shown that the NMR data suffice to estimate the conductivity of the electrolytes. The model was checked by means of experimental results of conductivity and NMR diffusion coefficients obtained with solutions of lithium triflate in ethylene and propylene carbonates, as well as with other results taken from the literature.
1. INTRODUCTION The molar conductivity of electrolytes, Λ, defined as the ratio between the intrinsic conductivity σ and the molar concentration c, for ideal electrolytes where the ionic species are completely dissociated, depends solely on the temperature and on the cation and anion diffusion coefficients. However, an additional variable must be considered in real systems due to the incomplete dissociation of the salt in the medium, that is, the degree of dissociation, α, at different concentrations. Thus, the Nernst−Einstein equation that relates the molar conductivity to the diffusion coefficients can be expressed in the form1 Λ=
F2 σ α(D+ + D−) = c RT +
requires the experimental determination of at least two magnitudes: conductivity and viscosity in the former case, and conductivity and ion diffusion coefficients in the latter. Concerning the diffusion coefficients, it should be pointed out that the values of the diffusion coefficients DNMR determined by NMR spectroscopy do not correspond to the values of D+ and D− in eq 1. This is due to the fact that, in addition to free ions, nondissociated species also contribute to the diffusion coefficients measured by NMR, but they do not contribute to the conductivity12 and, consequently, the value of α cannot be obtained directly from the experimental Λ and DNMR. In this work, we propose a model that relates the conductivities of the electrolytes with the NMR diffusion coefficients. It will be shown that with only the NMR diffusion coefficients, we can determine the conductivities of various electrolytes at different concentrations without prior knowledge of the values of the dissociation coefficients α. Furthermore, with some reasonable assumptions, it will be possible to determine not only the values of α but also the diffusion coefficients of the ions D+ and D− at variable concentrations. In this manner, it will be possible to estimate the true contribution of the ionic diffusion coefficients to the conductivity during the charge transport process. To prove and support the adequacy of experimental results to the proposed model, measurements of conductivities and diffusion coefficients have been carried out. The system chosen for the experiments was lithium trifluoromethanesulfonate (lithium triflate, LiTf) in ethylene carbonate (EC) and
(1)
−
where D and D represent the variable diffusion coefficients of the ions for each electrolyte solution, F is the Faraday number, R is the gas constant, and T is the absolute temperature. Experimentally, it is easy to determine Λ at different concentrations and the measurement of diffusion coefficients could be carried out by means of NMR spectroscopy.2−8 Then, a value of α may be obtained from eq 1 and compared to the Walden product α = Λη/Λ0η0 that also can be used as an approximate measure of α, and where η and η0 are, respectively, the viscosity of the solution and solvent.9,10 On the other hand, if we consider that the Stokes−Einstein equation applies,11 then the Walden product may be equivalent to the relation ΛD0/ Λ0D which could be derived from eq 1, and where D and D0 are the sum of the diffusion coefficients D+ + D− at a given concentration and at infinite dilution, respectively. Obviously with these considerations in mind, it is evident that the value of α, approximate or true (according to eq 1), © XXXX American Chemical Society
Received: November 5, 2014 Revised: January 17, 2015
A
DOI: 10.1021/jp511075q J. Phys. Chem. B XXXX, XXX, XXX−XXX
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The Journal of Physical Chemistry B
Figure 1. Plot of normalized (A) 7Li and (B) 19F signal intensities vs b in solutions of LiTf in EC at concentrations of (○) 12; (Δ) 111; (□) 540; (◇) 1154, and (×) 2115 mol m−3. The solid lines represent the fits to eq 2.
to 20 h. The decay of the echo amplitude was monitored typically to at least 50% of its initial value, and the apparent diffusion coefficient was calculated by fitting a monoexponential function to the decay curve. Previously, the magnetic field gradient was calibrated as described elsewhere.19 Electric Measurements. The electrolytes were placed in Novocontrol sample cell BDS 1307 for accurate measurements of the dielectric properties of liquids. The complex impedance measurements were carried out isothermally by means of a NOVOCONTROL GmbH Concept 40 broadband dielectric spectrometer in the frequency interval 10−1−107 Hz at 25 °C and at a voltage amplitude of 20 mV. The temperature was controlled to ±0.1 °C with a nitrogen jet during the sweep in frequency.
propylene carbonate (PC). Different lithium salts solutions in aprotic solvents of high dielectric permittivity, low viscosity, and high boiling point, such as cyclic carbonates like EC, PC, and its mixtures with other solvents, are used as ionic electrolytes in the development of lithium batteries. Therefore, a significant amount of work has been dedicated to study their main properties such as thermal stability, toxicity, sensitivity to ambient humidity, resistance to oxidation, and, especially, ionic conductivity.13,14 Despite its low water resistance, the lithium salt conventionally used in lithium batteries is LiPF6 due to the high conductivity of its solutions and safety. In contrast, the lithium salt derived from the trifluoromethane sulfonic is not considered a good choice due to its lower conductivity compared to other salts,13,14 but especially to the aluminum corrosion in the cells, where that metal is habitually used as a substrate.13,14 However, we have chosen LiTf as the salt because of its high resistance to humidity that allows the preparation of solid-like polymer electrolytes by meltcompounding which, currently, is a subject of investigation in our laboratory.15−17 Nevertheless, we have also used published data corresponding to solutions of other salts, such as the aforementioned LiPF6 and LiBF4, to prove the model.
3. RESULTS AND DISCUSSION Diffusion Coefficients and Conductivity. According to Stejskal and Tanner,18 the diffusion coefficients can be determined by using the following equation: A(g ) = A(0) exp[−bD]
(2)
where b is defined as b = (γNMR gδ)2 (Δ − δ /3)
2. EXPERIMENTAL SECTION Materials. Ethylene Carbonate (EC), propylene carbonate (PC), and lithium trifluoromethanesulfonate (LiTf) were battery quality grade from Aldrich and were used as received. Solutions of LiTf in EC and PC at variable concentrations between 0.001 and 2 mol L−1 were prepared in a humidity-free atmosphere. 7 Li and 19F PFG NMR Measurements. The NMR measurements were performed in a Bruker Avance 400 spectrometer equipped with a 89 mm wide bore, 9.4 T superconducting magnet (Larmor frequencies of 7Li and 19F at 155.51 and 376.51 MHz, respectively). The 7Li and 19F diffusion reported data were acquired at 25 ± 0.2 °C with a Bruker diffusion probe head, Diff60, using 90° radiofrequency (rf) pulse lengths of 11.0 μs. In the diffusion experiments, a pulsed field gradient stimulated spin echo pulse sequence was used.18 The time between the first two 90° rf pulses (the echo time), τ1, was 3.12 ms, and the self-diffusion coefficients of 7Li and 19F, DLi and DF, respectively, were measured varying the amplitude of the gradient pulse between 0 and 460 G cm−1. The diffusion time, Δ, and length of the gradient pulses, δ, were 50 and 2 ms, respectively. The repetition rate was always five times the spin− lattice relaxation time, T1, of the nuclei being observed. The total acquisition time for these experiments varied from 10 min
(3)
A(g) and A(0) are the intensity of the resonance signals corresponding to the echo amplitude in the presence of pulse gradients of amplitude g and 0, respectively, and γNMR is the gyromagnetic ratio of the observed nuclei. The 7Li and 19F NMR spectra corresponding to solutions of LiTf in EC and PC at different values of the applied magnetic field gradient are illustrated in Figures 1S and 2S (Supporting Information), and the attenuation of the NMR signals is observed. The line width of the resonance signals is narrow, as it corresponds to the high mobility of the ionic species in these low viscosity solvents. Graphical representations of these intensities, normalized in each diffusion experiment as A(g)/ A(0), and fitted according to eq 2, are shown in Figure 1. A good fit of the experimental results to the single exponential function is obtained. This allows determination of the values of the corresponding diffusion coefficients D7Li and D19F, which are shown in Table 1 for all the solutions studied. The variation of the diffusion coefficients in the liquid electrolytes and the total diffusion coefficient is represented as a function of the concentration of LiTf in Figure 2. The data depicted in this figure indicate that the values of D7Li and D19F decrease with the concentration but the reduction is greater for the anion than for the cation, both reaching almost the same B
DOI: 10.1021/jp511075q J. Phys. Chem. B XXXX, XXX, XXX−XXX
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The Journal of Physical Chemistry B Table 1. Diffusion Coefficients of 7Li and 19F, Total Diffusion Coefficient (DNMR = D7Li + D19F) and Apparent Transport Number (βapp) in Solutions of LiTf in EC and PC from Measurements by PFG NMR, and Conductivity at 25 °C [LiTf] (mol m−3)
σDC (μS cm−1)
2115 1154 540 111 10.2
1700 2999 3338 1647 211
1931 1053 493 93.8 9.3 0.93
1080 1740 2110 1110 204 23
D7Li × 1010 (m2 s−1)
D19F × 1010 (m2 s−1)
EC solutions 0.53 0.55 1.02 1.20 2.23 2.92 3.01 4.43 2.45 4.55 PC solutions 0.37 0.38 0.83 0.98 1.38 1.86 1.98 3.21 2.20 4.07 − −
βappa
DNMR × 1010 (m2 s−1)
0.49 0.46 0.43 0.40 0.35
1.08 2.26 5.15 7.44 7.00
0.49 0.46 0.42 0.38 0.35 −
0.75 1.81 3.24 5.19 6.27 −
Figure 3. Nyquist diagrams showing the variation of the imaginary part of the impedance (−Z″) against the real part (Z′) at 25 °C for the electrolytes with molar concentrations of LiTf in PC of 0.93, 9.3, 93.8, and 493 mol m−3 (from higher to lower Rb). Graph is in double logarithm form to show the differences in Rb.
a
The apparent transport number (βapp) is defined here as the ratio D7Li/DNMR.
Z′ =
Rb 1 + R b2C b2ω 2
Z″ =
(5)
R b2C bω 1 + R b2C b2ω 2
(6)
with Cb as the bulk capacitance. Due to the fact that in the high frequency region quasiperfect semicircles (Nyquist diagram in usual form) are obtained for all the studied electrolytes, eqs 5 and 6 are representative of their electrical behavior. At low frequencies (higher real impedances) almost straight lines are observed due to ionic diffusion and electrode polarization. The values of Rb obtained in the analysis of the Nyquist diagrams enable the determination of the direct current conductivity values σDC:
σDC =
Figure 2. Variation of the experimental NMR self-diffusion coefficients of D7Li (□), D19F (△), and total DNMR (○) for solutions of lithium triflate (LiTf) in PC at 25 °C. The closed symbols represent calculated values at infinite dilution.
(7)
where l and A are the thickness and area of the measurement cell, respectively. The corresponding results for the electrolytes at 25 °C are included in the second column of Table 1. The influence of the concentration on σ can be derived from the data in Table 1. In Figure 3S (Supporting Information) σ is plotted as a function of the concentration of LiTf in EC and PC. The figure illustrates the strong dependence of σ on the concentration of LiTf, at the lower concentration range, increasing steeply with increasing concentration of the ionic species, c, up to c ≈ 0.5 M. Above c ≈ 0.5 M, σ tends to decrease very slowly with further increasing of the concentration. The maximum σ is attained when the electrolyte concentration reaches between 0.5 and 1 M, values that seem to be in the expected range for lithium salts in aprotic solvents of high dielectric permittivity.22 In Figure 4, the variation of the molar conductivity Λ, with the square root of the concentration is shown. The observed trend is the usual decrease of Λ in a nonlinear form due to, among other causes, the effects of ionic association. Only at very low concentrations the empirical equation first postulated by Kohlrausch and further developed theoretically by Debye− Hückel−Onsager (eq 8) can be applied.
value for concentrations of around 2 mol L−1, i.e. with apparent transport numbers close to 0.5. To determine the σ values of the different electrolyte solutions, the impedance results are usually represented in the form of Nyquist diagrams, −Z″ vs Z′, where Z′ and Z″ are, respectively, the real and imaginary parts of the complex impedance Z*: Z * = Z ′ − jZ ″
1 l Rb A
(4)
The Nyquist diagrams corresponding to different electrolytes are shown in Figure 3, and the data may be analyzed considering an equivalent circuit composed by the bulk resistance (Rb) of the electrolyte in parallel with a capacitor (ideal case) or a constant phase element (CPE) more characteristic of the behavior of real electrolytes.20,21 In the former case, the values of the real and imaginary parts of the impedance are given by C
DOI: 10.1021/jp511075q J. Phys. Chem. B XXXX, XXX, XXX−XXX
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The Journal of Physical Chemistry B D+ + D− =
Λ=
(D19F − D 7Li ) γ + 1 α γ−1
(12)
γ + 1 F2 F2 (D19F − D 7Li ) α(D+ + D−) = RT γ − 1 RT
(13)
A graphical representation according to eq 13 is shown in Figure 5 for solutions of LiTf in PC and EC. Straight lines with
Figure 4. Variation of the molar conductivity Λ with the square root of the concentration of LiTf in EC (■) and in PC (red ●) at 25 °C.
σDC = Λ = Λ0 − (A + BΛ0)c1/2 c
(8)
where A and B are two parameters depending upon the temperature, viscosity, solvent dielectric constant, and ionic activity. The value of the molar conductivity at infinite dilution for the LiTf solution in PC was taken from the literature Λ0 = 25.32 S cm2 mol−1, a value that was determined experimentally by Ue23 and calculated theoretically by He et al.24 Proposed Diffusion Model. The Nernst−Einstein equation that relates the molar conductivity to the diffusion coefficients can be expressed in the form shown in eq 1.1 As indicated earlier, the diffusion coefficients determined by NMR include the contribution of dissociated and nondissociated species, the latter not contributing to σ.12 Thus, the experimental NMR diffusion coefficients D7Li and D19F could be given by eqs 9 and 10: D7Li = αD+ + (1 − α)D±
(9)
D19F = αD− + (1 − α)D±
(10)
Figure 5. Plot according to eq 13 of experimental molar conductivities and NMR diffusion coefficients at 25 °C for solutions of LiTf in EC (squares) and PC (circles). The open symbol represents the value at infinite dilution corresponding to the lithium triflate in PC.
similar slopes are obtained in a wide range of concentrations, indicating that this treatment seems to be appropriated to establish the true relation between the experimental NMR diffusion coefficients on the molar conductivities of electrolyte solutions. The fact that the relationship between Λ and (D19F − D7Li) is linear in the range of concentrations studied implies that γ (or β, as they are directly related) is constant throughout that range for the studied electrolytes. The value of γ can be obtained from the slope of the straight line, leading to an experimental value of γ ≈ 2 for LiTf in PC, practically identical to the value of γ0 ≈ 2.003 = D−0 /D+0 , with D−0 and D+0 obtained from eq 13 with α = 1 and using, in each case, the individual ion molar conductivities at zero concentration given in the literature.23,24 In the case of LiTf in EC γ0 and Λ0 are unknown although the value of γ0 could be calculated from the slope of the straight line shown in Figure 5 assuming that γ = γ0. Thus, a value of γ = 2.21 was obtained, similar to that determined for LiTf in PC. In the same way and assuming that the molar conductivity at infinite dilution of the Li cation is equal to 8.43 × 10−4 S m2 mol−1, a value that is practically constant for almost all the Li salts in different solvents,24 we can directly determine the limiting molar conductivity of the anion (18.63 × 10−4 S m2 mol−1) and, consequently, the value of Λ0. On the other hand, α cannot be obtained unequivocally by these methods of measurement of the diffusion coefficients and conductivities, because in the equations the product α (D+ + D−) cannot be separated. However, if γ is known and once the values of α(D+ + D−) and (1 − α)D± are known, it is possible to obtain approximate values of α making use of eqs 9, 10, and 13, assuming that the radius of the ionic pair is the sum of the ionic radii of the cation and anion. This assumption implies that
±
with D as the diffusion coefficient corresponding to nondissociated species, i.e. such as ionic pairs or higher aggregates, those not contributing to σ. These equations are similar or equal to those indicated by several authors25−28 for the mutual diffusion coefficient in the case of electrolytes exhibiting association. On the other hand, because ions are affected in the same manner by the medium viscosity, the ratio γ = D−/D+ may be considered almost constant or, in other words, that the true transport number defined as D+/(D+ + D−) does not change with concentration. We believe that this is a reasonable assumption, although it might not hold in some cases. For example, if the solvation sphere changes with the salt concentration, the value of γ might not be constant. This is a question to be addressed in each case by comparing the experimental data of diffusion coefficients and conductivity, as we will consider below. Determination of the Transport Number β and Estimation of α, D+, and D−. The following eqs 11, 12, and 13 are straightforwardly deduced from eqs 1, 9, and 10: D19F − D 7Li = α(γ − 1)D+
(11) D
DOI: 10.1021/jp511075q J. Phys. Chem. B XXXX, XXX, XXX−XXX
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The Journal of Physical Chemistry B the corresponding values of D± will be given, provided that the Stokes−Einstein equation applies, by D± =
D+γ D+D− − = D +D 1+γ +
values of both dependent variables must be equal at each concentration. As it could be observed, this is the case not only for our results, in which it has already been proven that γ is constant, but also for those by other authors.29 Only slight differences between Λ/Λ0 and (DF − DLi)/(D0F − D0Li) vs the concentration of salt are observed that could be attributed to errors inherent to the experimental measurements. Hence, this method allows values of Λ to be obtained which are very approximate to the experimental ones and, consequently, the values of α, D+, D−, and D±. From the results shown in Figure 5, it is clearly shown that the transport number β = 1/(γ + 1) does not change for LiTf dissolved in PC and EC in a wide range of concentrations, i.e, β = 1/(γ0 + 1) and, therefore, the values of DF and DLi determined by NMR are the only coefficients necessary to obtain values of Λ very close to the experimental ones. In the case of LiBF4 and LiPF6 the values of γ0 in PC must be 2.513 and 2.119 according to literature data.23,24 In Figure 7, Λ
(14)
In this manner, we can determine not only the values of α but also those of the diffusion coefficients of the ions D+ and D− at variable concentrations. Some authors consider D± equal to D−,27,28 but this assumption implies that the diffusion coefficient of the free anion is always equivalent to that determined by NMR, something that in our opinion is difficult to accept. Nevertheless, with this assumption, similar equations relating α with the diffusion coefficients could be obtained. Now, it is necessary to consider two hypothetical cases in eq 13. The first one is for γ = 1 since the value of Λ diverges, and the second one is when the experimental values of the NMR diffusion coefficients D7Li and D19F are equal, which occurs when the molar conductivity takes a zero value due to the impossibility of ions to diffuse, i.e. α = 0. However, we cannot disregard the possibility in which γ = 1 with a nonzero α value, and in this case, eq 13 does not apply. This could be solved using eq 1 with D+ = D− and, together with eqs 14 and 9, leads to eq 15 that would permit the calculation of α. Λ=
F2 4α D 7Li RT 1+α
(15)
Nevertheless, as we have seen above, the case in which γ = 1, where the transport number β = 0.5, seems that it does not occur, at least for the analyzed electrolytes. Estimation of Λ from NMR Diffusion Coefficients. In electrolytes where Λ is unknown, it is still possible to estimate this magnitude by using the model described above, as shown in Figure 6. In this figure, the experimental results of Takeuchi et al.29 for LiBF4 and LiPF6 solutions in PC and our own are shown in the form of Λ/Λ0 and (DF − DLi) /(D0F − D0Li) vs the concentration of salt. D0F and D0Li were calculated from literature data corresponding to the individual molar ionic conductivities.23,24 If γ is equal to D0F/D0Li for each electrolyte in all the intervals of concentration and eq 13 holds, then the
Figure 7. Plot according to eq 15 of the experimental molar conductivities and NMR diffusion coefficients at 25 °C for solutions of LiPF6 (squares) and LiBF4 (triangles) in PC. Open symbols represent the corresponding values at infinite dilution for both lithium salts.
is plotted as a function of (DF − DLi) where, in addition to the experimental data (closed symbols taken from ref 29), values of Λ at c → 0 (Λ0) corresponding to solutions of LiPF6 and LiBF4 in PC at infinite dilution23,24 are also shown. These points must determine the slope (γ0 + 1)/(γ0 − 1) of the straight lines shown in Figure 7 and according to which the molar conductivity must vary with the difference of diffusion coefficients following the predictions of the proposed model. It is important to highlight in this figure that the results corresponding to LiBF4 are in good agreement with those calculated with γ = 2.485 ≈ γ0 whereas in the case of Li PF6 higher differences were observed with γ = 2.318. Other published data1,30,31 are not included in Figure 7 to avoid complexity in the graph, although all of them follow the same trend. We believe that the model is adequate, but very accurate experimental data must be obtained to prove its validity for very different ionic salts in different media.
Figure 6. Variation of the reduced molar conductivities (Λ/Λ0, open symbols) and reduced diffusion coefficient differences ((DF−DLi)/ (D0F− D0Li), closed symbols), as a function of the molar concentration of LiTf (circles), LiBF4 (triangles), and LiPF6 (squares) in propylene carbonate at 25 °C. The experimental data for LiBF4 and LiPF6 were taken from ref 29.
4. CONCLUSIONS A simple model has been developed to separate the contribution of dissociated and nondissociated species to the diffusion coefficients of lithium and fluorine, determined by E
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NMR for different solutions of lithium triflate in ethylene and propylene carbonate. Analysis of the experimental results seems to indicate that the proposed model, with certain reasonable assumptions, permits the determination of the degree of dissociation and all the diffusion coefficients that intervene in the transport of charge. The model was also tested with published data, and it was shown that the transport of charge in lithium electrolytes dissolved in aprotic solvents is adequately described. Thus, the model proposed using only the diffusion coefficients measured with NMR serves as a general procedure for the determination of the conductivity of lithium salts in these solvents.
■
ASSOCIATED CONTENT
S Supporting Information *
Spin−lattice relaxation times (T1) and 7Li and 19F NMR spectra of lithium triflate solutions, and plot of the conductivity of these solutions at various concentrations. This material is available free of charge via the Internet at http://pubs.acs.org.
■
AUTHOR INFORMATION
Corresponding Authors
*E-mail: [email protected]. *E-mail: [email protected]. Notes
The authors declare no competing financial interest.
■
ACKNOWLEDGMENTS
■
REFERENCES
The authors thank E. Benito for performing the impedance measurements and the anonymous reviewers for their helpful comments. This work was supported by the Spanish Ministry of Economy and Competitiveness (Project MAT 2011-29174C02-02) and the Consejo Superior de Investigaciones ́ Cientificas.
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DOI: 10.1021/jp511075q J. Phys. Chem. B XXXX, XXX, XXX−XXX
Ion Diffusion Coefficients Model and Molar Conductivities of Ionic Salts in Aprotic Solvents Leoncio Garrido,* Alberto Mejía, Nuria García, Pilar Tiemblo, and Julio Guzmán* Departamento de Química-Física, Instituto de Ciencia y Tecnología de Polímeros, Consejo Superior de Investigaciones Científicas (ICTP-CSIC), Juan de la Cierva 3, E-28006 Madrid, Spain S Supporting Information *
ABSTRACT: In the study of the electric properties of electrolytes, the determination of the diffusion coefficients of the species that intervene in the charge transport process is of great importance, particularly that of the free ions (D+ and D−), the only species that contribute to the conductivity. In this work we propose a model that allows, with reasonable assumptions, determination of D+ and D−, and the degree of dissociation of the salt, α, at different concentrations, using the diffusion coefficients experimentally obtained with NMR. Also, it is shown that the NMR data suffice to estimate the conductivity of the electrolytes. The model was checked by means of experimental results of conductivity and NMR diffusion coefficients obtained with solutions of lithium triflate in ethylene and propylene carbonates, as well as with other results taken from the literature.
1. INTRODUCTION The molar conductivity of electrolytes, Λ, defined as the ratio between the intrinsic conductivity σ and the molar concentration c, for ideal electrolytes where the ionic species are completely dissociated, depends solely on the temperature and on the cation and anion diffusion coefficients. However, an additional variable must be considered in real systems due to the incomplete dissociation of the salt in the medium, that is, the degree of dissociation, α, at different concentrations. Thus, the Nernst−Einstein equation that relates the molar conductivity to the diffusion coefficients can be expressed in the form1 Λ=
F2 σ α(D+ + D−) = c RT +
requires the experimental determination of at least two magnitudes: conductivity and viscosity in the former case, and conductivity and ion diffusion coefficients in the latter. Concerning the diffusion coefficients, it should be pointed out that the values of the diffusion coefficients DNMR determined by NMR spectroscopy do not correspond to the values of D+ and D− in eq 1. This is due to the fact that, in addition to free ions, nondissociated species also contribute to the diffusion coefficients measured by NMR, but they do not contribute to the conductivity12 and, consequently, the value of α cannot be obtained directly from the experimental Λ and DNMR. In this work, we propose a model that relates the conductivities of the electrolytes with the NMR diffusion coefficients. It will be shown that with only the NMR diffusion coefficients, we can determine the conductivities of various electrolytes at different concentrations without prior knowledge of the values of the dissociation coefficients α. Furthermore, with some reasonable assumptions, it will be possible to determine not only the values of α but also the diffusion coefficients of the ions D+ and D− at variable concentrations. In this manner, it will be possible to estimate the true contribution of the ionic diffusion coefficients to the conductivity during the charge transport process. To prove and support the adequacy of experimental results to the proposed model, measurements of conductivities and diffusion coefficients have been carried out. The system chosen for the experiments was lithium trifluoromethanesulfonate (lithium triflate, LiTf) in ethylene carbonate (EC) and
(1)
−
where D and D represent the variable diffusion coefficients of the ions for each electrolyte solution, F is the Faraday number, R is the gas constant, and T is the absolute temperature. Experimentally, it is easy to determine Λ at different concentrations and the measurement of diffusion coefficients could be carried out by means of NMR spectroscopy.2−8 Then, a value of α may be obtained from eq 1 and compared to the Walden product α = Λη/Λ0η0 that also can be used as an approximate measure of α, and where η and η0 are, respectively, the viscosity of the solution and solvent.9,10 On the other hand, if we consider that the Stokes−Einstein equation applies,11 then the Walden product may be equivalent to the relation ΛD0/ Λ0D which could be derived from eq 1, and where D and D0 are the sum of the diffusion coefficients D+ + D− at a given concentration and at infinite dilution, respectively. Obviously with these considerations in mind, it is evident that the value of α, approximate or true (according to eq 1), © XXXX American Chemical Society
Received: November 5, 2014 Revised: January 17, 2015
A
DOI: 10.1021/jp511075q J. Phys. Chem. B XXXX, XXX, XXX−XXX
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The Journal of Physical Chemistry B
Figure 1. Plot of normalized (A) 7Li and (B) 19F signal intensities vs b in solutions of LiTf in EC at concentrations of (○) 12; (Δ) 111; (□) 540; (◇) 1154, and (×) 2115 mol m−3. The solid lines represent the fits to eq 2.
to 20 h. The decay of the echo amplitude was monitored typically to at least 50% of its initial value, and the apparent diffusion coefficient was calculated by fitting a monoexponential function to the decay curve. Previously, the magnetic field gradient was calibrated as described elsewhere.19 Electric Measurements. The electrolytes were placed in Novocontrol sample cell BDS 1307 for accurate measurements of the dielectric properties of liquids. The complex impedance measurements were carried out isothermally by means of a NOVOCONTROL GmbH Concept 40 broadband dielectric spectrometer in the frequency interval 10−1−107 Hz at 25 °C and at a voltage amplitude of 20 mV. The temperature was controlled to ±0.1 °C with a nitrogen jet during the sweep in frequency.
propylene carbonate (PC). Different lithium salts solutions in aprotic solvents of high dielectric permittivity, low viscosity, and high boiling point, such as cyclic carbonates like EC, PC, and its mixtures with other solvents, are used as ionic electrolytes in the development of lithium batteries. Therefore, a significant amount of work has been dedicated to study their main properties such as thermal stability, toxicity, sensitivity to ambient humidity, resistance to oxidation, and, especially, ionic conductivity.13,14 Despite its low water resistance, the lithium salt conventionally used in lithium batteries is LiPF6 due to the high conductivity of its solutions and safety. In contrast, the lithium salt derived from the trifluoromethane sulfonic is not considered a good choice due to its lower conductivity compared to other salts,13,14 but especially to the aluminum corrosion in the cells, where that metal is habitually used as a substrate.13,14 However, we have chosen LiTf as the salt because of its high resistance to humidity that allows the preparation of solid-like polymer electrolytes by meltcompounding which, currently, is a subject of investigation in our laboratory.15−17 Nevertheless, we have also used published data corresponding to solutions of other salts, such as the aforementioned LiPF6 and LiBF4, to prove the model.
3. RESULTS AND DISCUSSION Diffusion Coefficients and Conductivity. According to Stejskal and Tanner,18 the diffusion coefficients can be determined by using the following equation: A(g ) = A(0) exp[−bD]
(2)
where b is defined as b = (γNMR gδ)2 (Δ − δ /3)
2. EXPERIMENTAL SECTION Materials. Ethylene Carbonate (EC), propylene carbonate (PC), and lithium trifluoromethanesulfonate (LiTf) were battery quality grade from Aldrich and were used as received. Solutions of LiTf in EC and PC at variable concentrations between 0.001 and 2 mol L−1 were prepared in a humidity-free atmosphere. 7 Li and 19F PFG NMR Measurements. The NMR measurements were performed in a Bruker Avance 400 spectrometer equipped with a 89 mm wide bore, 9.4 T superconducting magnet (Larmor frequencies of 7Li and 19F at 155.51 and 376.51 MHz, respectively). The 7Li and 19F diffusion reported data were acquired at 25 ± 0.2 °C with a Bruker diffusion probe head, Diff60, using 90° radiofrequency (rf) pulse lengths of 11.0 μs. In the diffusion experiments, a pulsed field gradient stimulated spin echo pulse sequence was used.18 The time between the first two 90° rf pulses (the echo time), τ1, was 3.12 ms, and the self-diffusion coefficients of 7Li and 19F, DLi and DF, respectively, were measured varying the amplitude of the gradient pulse between 0 and 460 G cm−1. The diffusion time, Δ, and length of the gradient pulses, δ, were 50 and 2 ms, respectively. The repetition rate was always five times the spin− lattice relaxation time, T1, of the nuclei being observed. The total acquisition time for these experiments varied from 10 min
(3)
A(g) and A(0) are the intensity of the resonance signals corresponding to the echo amplitude in the presence of pulse gradients of amplitude g and 0, respectively, and γNMR is the gyromagnetic ratio of the observed nuclei. The 7Li and 19F NMR spectra corresponding to solutions of LiTf in EC and PC at different values of the applied magnetic field gradient are illustrated in Figures 1S and 2S (Supporting Information), and the attenuation of the NMR signals is observed. The line width of the resonance signals is narrow, as it corresponds to the high mobility of the ionic species in these low viscosity solvents. Graphical representations of these intensities, normalized in each diffusion experiment as A(g)/ A(0), and fitted according to eq 2, are shown in Figure 1. A good fit of the experimental results to the single exponential function is obtained. This allows determination of the values of the corresponding diffusion coefficients D7Li and D19F, which are shown in Table 1 for all the solutions studied. The variation of the diffusion coefficients in the liquid electrolytes and the total diffusion coefficient is represented as a function of the concentration of LiTf in Figure 2. The data depicted in this figure indicate that the values of D7Li and D19F decrease with the concentration but the reduction is greater for the anion than for the cation, both reaching almost the same B
DOI: 10.1021/jp511075q J. Phys. Chem. B XXXX, XXX, XXX−XXX
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The Journal of Physical Chemistry B Table 1. Diffusion Coefficients of 7Li and 19F, Total Diffusion Coefficient (DNMR = D7Li + D19F) and Apparent Transport Number (βapp) in Solutions of LiTf in EC and PC from Measurements by PFG NMR, and Conductivity at 25 °C [LiTf] (mol m−3)
σDC (μS cm−1)
2115 1154 540 111 10.2
1700 2999 3338 1647 211
1931 1053 493 93.8 9.3 0.93
1080 1740 2110 1110 204 23
D7Li × 1010 (m2 s−1)
D19F × 1010 (m2 s−1)
EC solutions 0.53 0.55 1.02 1.20 2.23 2.92 3.01 4.43 2.45 4.55 PC solutions 0.37 0.38 0.83 0.98 1.38 1.86 1.98 3.21 2.20 4.07 − −
βappa
DNMR × 1010 (m2 s−1)
0.49 0.46 0.43 0.40 0.35
1.08 2.26 5.15 7.44 7.00
0.49 0.46 0.42 0.38 0.35 −
0.75 1.81 3.24 5.19 6.27 −
Figure 3. Nyquist diagrams showing the variation of the imaginary part of the impedance (−Z″) against the real part (Z′) at 25 °C for the electrolytes with molar concentrations of LiTf in PC of 0.93, 9.3, 93.8, and 493 mol m−3 (from higher to lower Rb). Graph is in double logarithm form to show the differences in Rb.
a
The apparent transport number (βapp) is defined here as the ratio D7Li/DNMR.
Z′ =
Rb 1 + R b2C b2ω 2
Z″ =
(5)
R b2C bω 1 + R b2C b2ω 2
(6)
with Cb as the bulk capacitance. Due to the fact that in the high frequency region quasiperfect semicircles (Nyquist diagram in usual form) are obtained for all the studied electrolytes, eqs 5 and 6 are representative of their electrical behavior. At low frequencies (higher real impedances) almost straight lines are observed due to ionic diffusion and electrode polarization. The values of Rb obtained in the analysis of the Nyquist diagrams enable the determination of the direct current conductivity values σDC:
σDC =
Figure 2. Variation of the experimental NMR self-diffusion coefficients of D7Li (□), D19F (△), and total DNMR (○) for solutions of lithium triflate (LiTf) in PC at 25 °C. The closed symbols represent calculated values at infinite dilution.
(7)
where l and A are the thickness and area of the measurement cell, respectively. The corresponding results for the electrolytes at 25 °C are included in the second column of Table 1. The influence of the concentration on σ can be derived from the data in Table 1. In Figure 3S (Supporting Information) σ is plotted as a function of the concentration of LiTf in EC and PC. The figure illustrates the strong dependence of σ on the concentration of LiTf, at the lower concentration range, increasing steeply with increasing concentration of the ionic species, c, up to c ≈ 0.5 M. Above c ≈ 0.5 M, σ tends to decrease very slowly with further increasing of the concentration. The maximum σ is attained when the electrolyte concentration reaches between 0.5 and 1 M, values that seem to be in the expected range for lithium salts in aprotic solvents of high dielectric permittivity.22 In Figure 4, the variation of the molar conductivity Λ, with the square root of the concentration is shown. The observed trend is the usual decrease of Λ in a nonlinear form due to, among other causes, the effects of ionic association. Only at very low concentrations the empirical equation first postulated by Kohlrausch and further developed theoretically by Debye− Hückel−Onsager (eq 8) can be applied.
value for concentrations of around 2 mol L−1, i.e. with apparent transport numbers close to 0.5. To determine the σ values of the different electrolyte solutions, the impedance results are usually represented in the form of Nyquist diagrams, −Z″ vs Z′, where Z′ and Z″ are, respectively, the real and imaginary parts of the complex impedance Z*: Z * = Z ′ − jZ ″
1 l Rb A
(4)
The Nyquist diagrams corresponding to different electrolytes are shown in Figure 3, and the data may be analyzed considering an equivalent circuit composed by the bulk resistance (Rb) of the electrolyte in parallel with a capacitor (ideal case) or a constant phase element (CPE) more characteristic of the behavior of real electrolytes.20,21 In the former case, the values of the real and imaginary parts of the impedance are given by C
DOI: 10.1021/jp511075q J. Phys. Chem. B XXXX, XXX, XXX−XXX
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The Journal of Physical Chemistry B D+ + D− =
Λ=
(D19F − D 7Li ) γ + 1 α γ−1
(12)
γ + 1 F2 F2 (D19F − D 7Li ) α(D+ + D−) = RT γ − 1 RT
(13)
A graphical representation according to eq 13 is shown in Figure 5 for solutions of LiTf in PC and EC. Straight lines with
Figure 4. Variation of the molar conductivity Λ with the square root of the concentration of LiTf in EC (■) and in PC (red ●) at 25 °C.
σDC = Λ = Λ0 − (A + BΛ0)c1/2 c
(8)
where A and B are two parameters depending upon the temperature, viscosity, solvent dielectric constant, and ionic activity. The value of the molar conductivity at infinite dilution for the LiTf solution in PC was taken from the literature Λ0 = 25.32 S cm2 mol−1, a value that was determined experimentally by Ue23 and calculated theoretically by He et al.24 Proposed Diffusion Model. The Nernst−Einstein equation that relates the molar conductivity to the diffusion coefficients can be expressed in the form shown in eq 1.1 As indicated earlier, the diffusion coefficients determined by NMR include the contribution of dissociated and nondissociated species, the latter not contributing to σ.12 Thus, the experimental NMR diffusion coefficients D7Li and D19F could be given by eqs 9 and 10: D7Li = αD+ + (1 − α)D±
(9)
D19F = αD− + (1 − α)D±
(10)
Figure 5. Plot according to eq 13 of experimental molar conductivities and NMR diffusion coefficients at 25 °C for solutions of LiTf in EC (squares) and PC (circles). The open symbol represents the value at infinite dilution corresponding to the lithium triflate in PC.
similar slopes are obtained in a wide range of concentrations, indicating that this treatment seems to be appropriated to establish the true relation between the experimental NMR diffusion coefficients on the molar conductivities of electrolyte solutions. The fact that the relationship between Λ and (D19F − D7Li) is linear in the range of concentrations studied implies that γ (or β, as they are directly related) is constant throughout that range for the studied electrolytes. The value of γ can be obtained from the slope of the straight line, leading to an experimental value of γ ≈ 2 for LiTf in PC, practically identical to the value of γ0 ≈ 2.003 = D−0 /D+0 , with D−0 and D+0 obtained from eq 13 with α = 1 and using, in each case, the individual ion molar conductivities at zero concentration given in the literature.23,24 In the case of LiTf in EC γ0 and Λ0 are unknown although the value of γ0 could be calculated from the slope of the straight line shown in Figure 5 assuming that γ = γ0. Thus, a value of γ = 2.21 was obtained, similar to that determined for LiTf in PC. In the same way and assuming that the molar conductivity at infinite dilution of the Li cation is equal to 8.43 × 10−4 S m2 mol−1, a value that is practically constant for almost all the Li salts in different solvents,24 we can directly determine the limiting molar conductivity of the anion (18.63 × 10−4 S m2 mol−1) and, consequently, the value of Λ0. On the other hand, α cannot be obtained unequivocally by these methods of measurement of the diffusion coefficients and conductivities, because in the equations the product α (D+ + D−) cannot be separated. However, if γ is known and once the values of α(D+ + D−) and (1 − α)D± are known, it is possible to obtain approximate values of α making use of eqs 9, 10, and 13, assuming that the radius of the ionic pair is the sum of the ionic radii of the cation and anion. This assumption implies that
±
with D as the diffusion coefficient corresponding to nondissociated species, i.e. such as ionic pairs or higher aggregates, those not contributing to σ. These equations are similar or equal to those indicated by several authors25−28 for the mutual diffusion coefficient in the case of electrolytes exhibiting association. On the other hand, because ions are affected in the same manner by the medium viscosity, the ratio γ = D−/D+ may be considered almost constant or, in other words, that the true transport number defined as D+/(D+ + D−) does not change with concentration. We believe that this is a reasonable assumption, although it might not hold in some cases. For example, if the solvation sphere changes with the salt concentration, the value of γ might not be constant. This is a question to be addressed in each case by comparing the experimental data of diffusion coefficients and conductivity, as we will consider below. Determination of the Transport Number β and Estimation of α, D+, and D−. The following eqs 11, 12, and 13 are straightforwardly deduced from eqs 1, 9, and 10: D19F − D 7Li = α(γ − 1)D+
(11) D
DOI: 10.1021/jp511075q J. Phys. Chem. B XXXX, XXX, XXX−XXX
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The Journal of Physical Chemistry B the corresponding values of D± will be given, provided that the Stokes−Einstein equation applies, by D± =
D+γ D+D− − = D +D 1+γ +
values of both dependent variables must be equal at each concentration. As it could be observed, this is the case not only for our results, in which it has already been proven that γ is constant, but also for those by other authors.29 Only slight differences between Λ/Λ0 and (DF − DLi)/(D0F − D0Li) vs the concentration of salt are observed that could be attributed to errors inherent to the experimental measurements. Hence, this method allows values of Λ to be obtained which are very approximate to the experimental ones and, consequently, the values of α, D+, D−, and D±. From the results shown in Figure 5, it is clearly shown that the transport number β = 1/(γ + 1) does not change for LiTf dissolved in PC and EC in a wide range of concentrations, i.e, β = 1/(γ0 + 1) and, therefore, the values of DF and DLi determined by NMR are the only coefficients necessary to obtain values of Λ very close to the experimental ones. In the case of LiBF4 and LiPF6 the values of γ0 in PC must be 2.513 and 2.119 according to literature data.23,24 In Figure 7, Λ
(14)
In this manner, we can determine not only the values of α but also those of the diffusion coefficients of the ions D+ and D− at variable concentrations. Some authors consider D± equal to D−,27,28 but this assumption implies that the diffusion coefficient of the free anion is always equivalent to that determined by NMR, something that in our opinion is difficult to accept. Nevertheless, with this assumption, similar equations relating α with the diffusion coefficients could be obtained. Now, it is necessary to consider two hypothetical cases in eq 13. The first one is for γ = 1 since the value of Λ diverges, and the second one is when the experimental values of the NMR diffusion coefficients D7Li and D19F are equal, which occurs when the molar conductivity takes a zero value due to the impossibility of ions to diffuse, i.e. α = 0. However, we cannot disregard the possibility in which γ = 1 with a nonzero α value, and in this case, eq 13 does not apply. This could be solved using eq 1 with D+ = D− and, together with eqs 14 and 9, leads to eq 15 that would permit the calculation of α. Λ=
F2 4α D 7Li RT 1+α
(15)
Nevertheless, as we have seen above, the case in which γ = 1, where the transport number β = 0.5, seems that it does not occur, at least for the analyzed electrolytes. Estimation of Λ from NMR Diffusion Coefficients. In electrolytes where Λ is unknown, it is still possible to estimate this magnitude by using the model described above, as shown in Figure 6. In this figure, the experimental results of Takeuchi et al.29 for LiBF4 and LiPF6 solutions in PC and our own are shown in the form of Λ/Λ0 and (DF − DLi) /(D0F − D0Li) vs the concentration of salt. D0F and D0Li were calculated from literature data corresponding to the individual molar ionic conductivities.23,24 If γ is equal to D0F/D0Li for each electrolyte in all the intervals of concentration and eq 13 holds, then the
Figure 7. Plot according to eq 15 of the experimental molar conductivities and NMR diffusion coefficients at 25 °C for solutions of LiPF6 (squares) and LiBF4 (triangles) in PC. Open symbols represent the corresponding values at infinite dilution for both lithium salts.
is plotted as a function of (DF − DLi) where, in addition to the experimental data (closed symbols taken from ref 29), values of Λ at c → 0 (Λ0) corresponding to solutions of LiPF6 and LiBF4 in PC at infinite dilution23,24 are also shown. These points must determine the slope (γ0 + 1)/(γ0 − 1) of the straight lines shown in Figure 7 and according to which the molar conductivity must vary with the difference of diffusion coefficients following the predictions of the proposed model. It is important to highlight in this figure that the results corresponding to LiBF4 are in good agreement with those calculated with γ = 2.485 ≈ γ0 whereas in the case of Li PF6 higher differences were observed with γ = 2.318. Other published data1,30,31 are not included in Figure 7 to avoid complexity in the graph, although all of them follow the same trend. We believe that the model is adequate, but very accurate experimental data must be obtained to prove its validity for very different ionic salts in different media.
Figure 6. Variation of the reduced molar conductivities (Λ/Λ0, open symbols) and reduced diffusion coefficient differences ((DF−DLi)/ (D0F− D0Li), closed symbols), as a function of the molar concentration of LiTf (circles), LiBF4 (triangles), and LiPF6 (squares) in propylene carbonate at 25 °C. The experimental data for LiBF4 and LiPF6 were taken from ref 29.
4. CONCLUSIONS A simple model has been developed to separate the contribution of dissociated and nondissociated species to the diffusion coefficients of lithium and fluorine, determined by E
DOI: 10.1021/jp511075q J. Phys. Chem. B XXXX, XXX, XXX−XXX
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The Journal of Physical Chemistry B
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NMR for different solutions of lithium triflate in ethylene and propylene carbonate. Analysis of the experimental results seems to indicate that the proposed model, with certain reasonable assumptions, permits the determination of the degree of dissociation and all the diffusion coefficients that intervene in the transport of charge. The model was also tested with published data, and it was shown that the transport of charge in lithium electrolytes dissolved in aprotic solvents is adequately described. Thus, the model proposed using only the diffusion coefficients measured with NMR serves as a general procedure for the determination of the conductivity of lithium salts in these solvents.
■
ASSOCIATED CONTENT
S Supporting Information *
Spin−lattice relaxation times (T1) and 7Li and 19F NMR spectra of lithium triflate solutions, and plot of the conductivity of these solutions at various concentrations. This material is available free of charge via the Internet at http://pubs.acs.org.
■
AUTHOR INFORMATION
Corresponding Authors
*E-mail: [email protected]. *E-mail: [email protected]. Notes
The authors declare no competing financial interest.
■
ACKNOWLEDGMENTS
■
REFERENCES
The authors thank E. Benito for performing the impedance measurements and the anonymous reviewers for their helpful comments. This work was supported by the Spanish Ministry of Economy and Competitiveness (Project MAT 2011-29174C02-02) and the Consejo Superior de Investigaciones ́ Cientificas.
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