What Can We Learn from Ionic Conductivity Measurements in Polymer

Article pubs.acs.org/JPCB

What Can We Learn from Ionic Conductivity Measurements in Polymer Electrolytes? A Case Study on Poly(ethylene oxide) (PEO)−NaI and PEO−LiTFSI Nicolaas A. Stolwijk,* Manfred Wiencierz, Christian Heddier, and Johannes Kösters Institut für Materialphysik and Sonderforschungsbereich 458, University of Münster, D-48149 Münster, Germany ABSTRACT: We explore in detail what information on ionic diffusivity and ion pairing can be exclusively gained from combining accurate direct-current conductivity data in polymer electrolytes with a novel evaluation model. The study was performed on two prototype systems based on poly(ethylene oxide) (PEO) with known disparate ion-association properties, which are due to the dissimilar salt components being either sodium iodide (NaI) or lithium bis(trifluoromethane-sulfonyl)imide (LiN(CF3SO2)2 or LiTFSI). The temperature dependence of the conductivity can be described by an extended Vogel−Tammann−Fulcher (VTF) equation, which involves a Boltzmann factor containing the pair-formation enthalpy ΔHp. We find a distinct increase of the positive ΔHp values with decreasing salt concentration and similarly clear trends for the pertinent VTF parameters. The analysis further reveals that PEO−NaI combines a high pair fraction with a high diffusivity of the I− ion. By contrast, PEO−LiTFSI appears to be characterized by a low ion-pairing tendency and a relatively low mobility of the bulky TFSI− ion. The observed marked differences between PEO−NaI and PEO−LiTFSI complexes of homologous composition are most pronounced at high temperatures and low salt concentrations.

1. INTRODUCTION Since their discovery in 19731 solid-like polymer electrolytes (SPEs) have been extensively investigated. These complexes consisting of a polar polymer matrix and a dissolved metal salt with sufficiently low lattice energy exhibit a combination of properties which make them interesting for applications in batteries, dye-sensitized solar cells, and other electrochemical devices.2 Most published studies deal with poly(ethylene oxide) (PEO) as a base polymer because its excellent salt-solvating properties lead to fairly high ionic conductivies in the amorphous phase prevailing at elevated temperatures. Although simple PEO-based SPEs (i.e., without organic solvents acting as plasticizer) do not meet the technological requirements for most practical applications, they served for several decades as prototype systems in fundamental studies on ionic transport in salt-in-polymer complexes.3,4 Because of its technological relevance, the most frequently investigated physical property of SPEs is undoubtedly the ionic direct-current (DC) conductivity. For the same reasons, the overwhelming majority of studies focused on relatively salt-rich systems at ambient and slightly elevated temperatures. In PEObased electrolytes, temperatures above the PEO melting point at ca. 65 °C avoid the partial crystallization of the polymer and the concomitant significant decrease of the ionic conductivity. However, even at elevated temperatures sufficiently high salt concentrations Cs may induce crucial changes in the microstructure of the complex and/or promote the precipitation of a second salt-rich phase. Thus, for scientific purposes the regime of low Cs and high T may be of great interest, as it circumvents © 2012 American Chemical Society

the difficulties related to the conductivity analysis of two- or multiphase SPE structures. Commonly, the DC conductivity σ of dilute PEO-based electrolytes at T > 65 °C exhibits a temperature dependence that can be described by the Vogel−Tammann−Fulcher (VTF) equation. This type of behavior bears out the notion that ionic mobility may be intimately connected with the segmental motion of the polymer chains. Presumably, the occurrence of unambiguous VTF behavior in solvent-free SPEs may be promoted by the fact that usually one of the ionic species, almost without exception the anion, dominates in charge transfer processes. However, the magnitude of the conductivity is determined by the product of two key properties, that is, the ion mobility and the ion density. Hence, from a basic point of view, pure VTF behavior should not be expected in cases when ion association plays a major role, since processes such as ion pairing will commonly result in a T-dependent concentration of charge carriers. Conversely, information on pair formation has to be contained in the variation of σ with T. Based on these considerations, it should be possible to extract such specific ionpair-related information, if reliable σ data covering a sufficiently wide T-range are available. To explore the validity of our hypothesis we have measured the DC conductivity of two different amorphous PEO−salt systems over wide ranges of temperature and composition. One Received: November 21, 2011 Revised: January 23, 2012 Published: February 8, 2012 3065

dx.doi.org/10.1021/jp2111956 | J. Phys. Chem. B 2012, 116, 3065−3074

The Journal of Physical Chemistry B

Article

contrast to observations for PEO−MI complexes with larger alkali metal cations (M = K, Rb, or Cs).15 Glass transition temperatures Tg were determined on meltquenched specimens by a Perkin-Elmer Diamond DSC-9 calorimeter with liquid nitrogen cooling. Pertinent Tg values were taken from the point of inflection in heat-flow vs T curves showing an endothermic event at a heating rate of 10 °C/min. In situ quenching in the calorimeter and ex situ quenching using an oil-bath thermostat and liquid nitrogen as cooling medium yielded mutually consistent results. The same was true for different quenching temperatures (Tq) spanning the range from 100 to 215 °C within the contiguous amorphous phase field of the system under consideration. The electrical conductivity of the PEO−salt complexes was investigated by means of impedance spectroscopy using a HP Agilent 4192A LF impedance analyzer for alternating current (AC) frequencies between 5 Hz and 13 MHz. The electrolyte was enclosed in a cylindrical cavity between two coplanar stainless steel electrodes with a surface area of 1.00 cm2 and an accurately measured distance of typically 2−3 mm. Mechanical pressure could be adjusted by a spring system to achieve optimal contact between electrodes and electrolyte. Temperatures in the range of 67−220 °C were controlled by a siliconoil heating circulator connected to a double-walled glass or stainless-steel assembly encapsulating the measuring cell. The temperature was recorded by a Pt-100 sensor located close to the sample. The cell was operated under a continuous flow of pure nitrogen gas. Actual values of the DC conductivity σ were taken from the plateau region in plots of the real part of the complex conductivity versus the AC frequency, that is, at the location where the phase angle attains its minimum value close to zero.

system, PEO−NaI, has been the subject of earlier studies by our group, in which the DC conductivity was compared based on the Nernst−Einstein equationwith the radiotracer diffusivity of the individual ionic species.5−8 This system was chosen because it was found to reveal a strong tendency to form neutral ion pairs. The other system, PEO−LiTFSI, has been extensively investigated by different groups with a variety of experimental methods including Raman and infrared spectroscopy, pulsed-field-gradient nuclear magnetic resonance (PFGNMR), and impedance spectroscopy. 9−12 It is wellestablished now that pair formation in PEO−LiTFSI with EO/Li ratios near 20 only takes place to a minor extent, which relates to the delocalization of charge on the bulky N(CF3SO2)2− (TFSI−) anion.10,11 It should be emphasized, however, that this paper does not aim at revealing chemical trends in PEO/alkali-metal electrolytes or elucidating the nature of the ion−polymer and ion− ion interactions. Rather we focus on the question what can be learned from accurate and extensive conductivity measurements when they are analyzed within a suitable theoretical framework. Not unexpectedly, the conductivity data on PEO−NaI and PEO−LiTFSI complexes presented in this work show great disparities for homologous compositions covering EO/saltmolecule ratios that range from 20 to 1000. In all cases, the conductivity of PEO−LiTFSI exceeds that of the homologous PEO−NaI complex, the more so at high temperatures and low salt concentrations. The crucial feature of this work is the evaluation of the data within a newly developed, simplified model of charge transport that takes into account both the mobility and the volume density of the charge carriers. We show that the model is able to account for deviations of the VTF behavior that are observed in cases of strong ion pairing, thus providing a proof of concept. In particular, this makes it feasible to assess the pairformation enthalpy as a function of the salt concentration in the two different systems. It is further found that the PEO−NaI system combines a low fraction of “free” ions with a high diffusivity of the anion, whereas for PEO−LiTFSI the inverse situation appears to be established. The present results confirm and quantify by way of example the strong impact of ion association on charge transport in polymer electrolytes.

3. RESULTS AND ANALYSIS 3.1. Data on PEO−NaI Complexes. Figure 1 provides a survey of the conductivity data resulting from the measurements

2. EXPERIMENTAL SECTION Only brief descriptions of sample preparation and the employed experimental methods are given here. For more details the reader is referred to our earlier publications.6,8 To avoid contact with ambient moisture and air, all critical preparation steps and experimental procedures were carried out in closed containers or in a nitrogen-flushed glovebox. Appropriate amounts of PEO with a molecular weight of 8 × 106 g/mol (Aldrich) and NaI (Grüssing, purity >99%) or LiTFSI (SigmaAldrich, purity 99.95%) were dried under dynamic vacuum at elevated temperature and subsequently homogeneously dissolved in water-free acetonitrile. After evaporation of the solvent under dynamic vacuum, the so-obtained PEOyNaI complexes13 with y = 20, 30, 60, 120, 250, 500, or 1000 and PEOyLiTFSI complexes with y = 60, 250, or 1000 were characterized by mass density measurements based on hydrostatic weighing in dodecane and by differential scanning calorimetry (DSC) using Perkin-Elmer DSC-7 equipment. In DSC analysis from room temperature to 200 °C, the only feature in second and multiple heating/cooling cycles was the endothermic peak near 67 °C due to the melting of PEO. We found no indications of precipitation, in agreement with literature data on PEO−NaI (see, e.g., Fauteux at al.14) and PEO−LiTFSI9 but in

Figure 1. DC conductivity of PEOyNaI electrolytes with different EO/ Na ratios y as a function of inverse temperature. The solid lines are least-squares fits based on the weak-electrolyte approximation (cf. eq 15). 3066



Article

on PEOyNaI complexes of different composition, with y ranging from 20 to 1000. For each composition, the data monitored during heating-up and cooling-down cycles were found to coincide. The reliability of the data was further confirmed by the employment of at least two different specimens of the same composition showing mutual agreement within an error tolerance of about ±10%. In some cases, nominally equal specimens were additionally taken from different production batches. It is worth noting that the neat PEO material exhibits a conductivity that is lower by 2 orders of magnitude over the whole T-range (data not shown) than the PEO1000NaI conductivity depicted in Figure 1. The solid lines in Figure 1 represent virtually perfect fits of the experimental data. It should be emphasized already at this stage that the temperature dependence of σ cannot be welldescribed with the VTF equation. This relates to the observation that the σ(T) data show a maximum near the high end of the investigated T-range for the more dilute complexes. Therefore, the solid lines in Figure 1 arise from a conductivity model that accountsunlike the VTF equationfor the Tdependent ion-pairing effect (see Section 4). Simply reasoning, the conductivity is made up from the product of the ion diffusivity and ion concentration. In the present polymer−electrolyte case, either the mean effect of cations and anions is considered, or one assumes the predominance of a single ionic species in charge transport (commonly the anion). It seems plausible that below a certain degree of dilution the ion diffusivity no longer depends on a further decreasing salt concentration Cs. In this situation, σ at a fixed temperature should directly scale with the composition parameter y = CEO/Cs (CEO = monomer concentration). In other words, one would expect factorof-two differences in σ between consecutive composition steps differing by a factor of 2 in y. This ideal case is indicated by the length of the two-headed arrows in Figure 1. However, the gap between two consecutive sets of data appears to be larger than a factor of 2 for y ≥ 60. This finding may point to changes in the extent of ion pairing as a function of salt concentration (and temperature). More specifically, the data are compatible with an increase of the pair fraction with decreasing Cs, as will be outlined below. Another way of analyzing the data involves the Nernst− Einstein equation to introduce the charge diffusivity Dσ as

Dσ ≡

Figure 2. Charge diffusivity Dσ of PEOyNaI (open symbols) and PEOyLiTFSI (closed symbols) electrolytes as a function of inverse temperature. The following compositions are displayed: y = 20 (up triangles), y = 60 (down triangles), y = 250 (circles), and y = 1000 (squares). The solid line fitting the data for y = 20 serves to guide the eye.

value. In accordance with our preceding conjecture, the remaining discrepancies may be (partly) due to the formation of uncharged ion pairs. At low salt concentrations, the Dσ curves in Figure 2 seem to converge. This may be rationalized by the notions that in strongly diluted complexes the mobility tends to a constant value and the pair fraction approaches a limiting value less than unity. Additional evidence for ion association may be given by the comparison of Dσ with the individual diffusion coefficients of cations and anions. Specifically, Figure 3 displays Dσ of

σkBT Cse 2

(1)

where kB denotes Boltzmann's constant, T is temperature, and e is the elementary charge. According to this definition, Dσ signifies the charge diffusivity per salt molecule, so it comprises the joint effect of one cation and one anion. Obviously, Dσ closely relates to the specific conductivity σ/Cs and provides a measure of the mean cation−anion mobility. Figure 2 shows the Dσ results for some of the PEOyNaI complexes that are associated with the conductivity data in Figure 1, that is, for y = 20, 60, 250, and 1000. We note that the corresponding Dσ-data of PEOyLiTFSI also plotted in Figure 2 will be discussed in Section 3.2. After having eliminated the concentration effect by deriving Dσ from σ, Figure 2 still shows differences among the examined compositions. However, the dynamic range at high T has been reduced from a factor of 200 in σ to about a factor of 4 in Dσ. This reduction in the magnitude range by a factor of 50 corresponds to the ratio 1000:20 of the highest to the lowest y

Figure 3. Various ion-related diffusion coefficients in PEO250NaI as a function of inverse temperature: charge diffusivity Dσ (circles), 22Na radiotracer diffusivity D*Na (down triangles), and 125I radiotracer diffusivity D*I (up triangles). The solid lines arise from simultaneous fitting of all three data sets within a comprehensive model of ion transport see refs 8 and 16. The dashed line represents the sum of the cation and anion diffusivity, as indicated.

PEO250NaI along with the corresponding tracer diffusivities D*Na and D*I , which resulted from diffusion experiments with the radioisotopes 22Na and 125I. Details about the radiotracer 3067



Article

technique can be found in earlier publications by our group.6−8 Figure 3 shows that the anion moves faster than the cation over the entire T-interval. The difference between D*Na and D*I amounts to a factor of 3 at the lowest temperature investigated, but it monotonically reduces to a factor of 1.25 at the highest. The solid curves arise from simultaneous fitting of all three sets of data within a comprehensive model of ion transport published elsewhere.8,16 It is worth noting that ion pairing constitutes a key concept in this transport model. In Figure 3, the crucial feature pointing to strong ion association is the fact that Dσ falls distinctly below the sum of D*Na and D*I (dashed line), the more so at higher temperatures. Obviously, ion pairing increases with increasing temperature, in agreement with earlier findings.5,16,17 3.2. Comparison with PEO−LiTFSI Data. Figure 4 compares the conductivities of three different PEO−NaI complexes

The solid curves in Figure 4 are fits based on the VTF equation σ=

Σ0σ ⎛ Bσ ⎞ exp⎜ − ⎟ T ⎝ T − T0σ ⎠

(2)

Here, Σ0σ is a pre-exponential factor, Bσ acts as pseudo activation energy, and T0σ denotes the temperature where the conductivity goes to zero. It should be noted that the factor 1/T enters eq 2 for consistency reasons, as will be elucidated in Section 4.3, but otherwise does not play a significant role. Least-squares fitting based on eq 2 involved the optimization of the three free parameters, that is, Σ0σ, Bσ, and T0σ. However, close inspection of Figure 4 reveals that the PEO−NaI data are not optimally adjusted by the VTF function (solid lines). In particular, the local maximum appearing in the Arrhenius plots for PEO250NaI and PEO500NaI (not shown in Figure 4) is not reproduced by the VTF fits. More importantly, the values obtained for the VTF parameters are not meaningful (see Section 5.2). Therefore, we derive an extension of the VTF expression in the next section, which does take the ion-pairing effect explicitly into account.

4. MODEL OF IONIC CONDUCTIVITY INCLUDING PAIR FORMATION 4.1. Basic Equations. The starting point for the following theoretical treatment is a comprehensive ion transport model (ITM) that was used before for the simultaneous evaluation of conductivity and ionic diffusion data in polymer electrolytes containing MX as a salt component.5,7,8,16 The model is based on the assumption that besides charged single cations (M+) and anions (X−) also uncharged ion pairs (MX0) may play a crucial role, as expressed by the reaction M+ + X − ⇌ MX 0

(3)

Within this model, the charge diffusivity Dσ is written as

Figure 4. Comparison of the ionic conductivity in PEOyNaI (open symbols) and PEOyLiTFSI (closed symbols) electrolytes for y = 60 (down triangles), y = 250 (circles), and y = 1000 (squares). The solid lines are fits of the VTF equation with freely adjustable T0σ (cf. eq 2).

Dσ = (1 − rp)(D M + + D X −)

(4)

where rp = Cp/Cs denotes the fraction of neutral pairs. DM+ and DX− are the diffusivities of M+ and X−, respectively. Equation 4 reflects that only unpaired ions with their relative abundance 1 − rp contribute to charge transfer. Within the ITM framework, the individual ionic diffusion coefficients obey the VTF equation, that is,

with the corresponding compositions of the PEO−LiTFSI system. It is seen that for each of the depicted y values, 60, 250, and 1000, σ of PEOyNaI falls below that of PEOyLiTFSI over the entire T-range of interest. The differences are most pronounced at the highest temperatures, where they amount to about a factor of 3 for y = 60 and more than a factor of 4 for y = 250 and 1000. In contrast to, for example, PEO250NaI, none of the PEO−LiTFSI complexes shows a local maximum in the σ versus 1/T plot. It is well-established in the literature that the PEO−LiTFSI system reveals, if at all, only little ion association. This technologically useful property relates to the bulky character of the TFSI− ion and the concomitant delocalization of the negative charge. Published work on PEO−LiTFSI comprises conductivity measurements, PFG-NMR analysis, and Raman/infrared spectroscopy.9−12 Altogether, the experimental evidence for the high degree of ion dissociation in PEO−LiTFSI is quite impressive and, moreover, supported by molecular dynamics simulations.18,19 Therefore, the higher conductivity of PEO− LiTFSI complexes with respect to their homologous PEO−NaI counterparts provides a further argument in favor of substantial ion-pair formation in the latter system.

D M + = D 0 + exp(−B+ /(T − T0)) M

(5)

and D X − = D X0 − exp(−B−/(T − T0))

(6)

In general, the VTF parameters B+ and B− may differ from each other, whereas T0 is a common reference temperature characterizing the overall flexibility of the salt-in-polymer complex on a microscopic scale. It should be emphasized that DM+ and DX− must be distinguished from the directly measured PFG-NMR (or radiotracer) diffusion coefficients, which are given by5,16 NMR * ) = (1 − rp)D + + rpDp DM (or DM M

(7)

and * ) = (1 − rp)D X − + rpDp D XNMR (or D X 3068

(8)



Article

respectively. In these equations, Dp denotes the diffusivity of the MX0 pairs. The law of mass action applied to the ion-pairing reaction given by eq 3 yields an expression for the “free-ion” fraction reading 1 − rp = ( 1 + 4k p − 1)/2k p

Using the definition of Dσ given by eq 1, we finally obtain an approximate equation for the DC ionic conductivity, that is, σ=

Σ0 =

(11)

where ΔSp stands for the entropy of pair formation. It is interesting to note that in polymer electrolytes the respective enthalpy and entropy differences ΔHp and ΔSp are usually of positive sign. This reflects the strong binding of the cation to the polymer chains leading to substantial conformational constraints.7,20 The application of the mass action law leading to eq 9 requires dilute electrolytes, so that activities may be replaced by concentrations. Under the same conditions, the EO-to-salt ratio y is large, which avoids a competition among the cations for attracting enough polyether oxygens to construct their solvation cells. In complexes with Li- or Na-salts, the cations are coordinated to typically 4 or 5 oxygen atoms while the variation of the coordination number within one complex (nc = 1 to 9) seems to be greater than than the differences between Li and Na.18,19,21,22 In the present study y ≫ nc generally holds, which justifies not only the above assumptions but also our approach to directly compare PEO−NaI and PEO−LiTFSI complexes of the same salt concentration which each other. 4.2. Weak-Electrolyte Case. Electrolytes with strong ionassociation behavior are characterized by large values of kp (≫1). In this case, the fraction of “free” ions 1 − rp may be approximated as 1 − rp ≈

1 = kp

⎛ ΔHp ⎞ 1 ⎟ exp⎜ k p0 ⎝ 2kBT ⎠

kB k p0

(16)

σ = σid(1 − rp)

(17)

where σid designates the “ideal conductivity” that would prevail without the occurrence of ion-pairing, that is, σid =

Σ0id exp(−B−/(T − T0)) T

In this equation, according to

Σ0id

e 2CsD X0 − id Σ0 = kB

(18)

combines several factorial constants

(19)

Indeed, in the absence of ion-pairing (rp = 0) eq 17 reduces to the general VTF form (cf. eqs 2 and 18). In general cases, however, the factor 1 − rp cannot be omitted or reduced and thus must be taken fully into account in data fitting, that is, in the form of eq 9 using eqs 10 and 11. We further note that σid essentially reflects the product of salt concentration Cs and diffusivity of free anions DX−. Strong electrolytes exhibit a high degree of salt dissociation, which may imply that the ion-pair fraction is small but not negligible over the temperature range under consideration, that is, rp ≪ 1. In this instant, we may utilize the Taylor expansion rp ≈ kp − 2kp2 + ···. Practically, this leads to the replacement of the factor 1 − rp in eq 17 by 1 − kp. For fitting purposes, however, this reduced form of eq 17 turned out to be of little use since it does not significantly differs from σid representative of rp = 0.

(12)

Furthermore, in SPE systems the anion is usually much more mobile than the cation. In particular, for PEO−NaI, PEO− LiTFSI, and closely related polyether−salt complexes it has been reported that the conductivity is almost entirely due to the anions, that is, DX− ≫ DM+.7,12,16,23,24 This relates to a partly decoupling of the anions from the segmental motion of the polymer. Under these conditions, the sum of the ionic diffusivities reduces to a good approximation to D M + + D X − ≈ D X0 − exp(−B−/(T − T0))

e 2CsD X0 −

Compared to the conventional VTF form given by eq 2, the newly derived expression contains the additional factor exp(ΔHp/2kBT). It should be emphasized that this Boltzmann-like factor has an exponent with a positive sign, since ΔHp > 0 generally holds for SPE systems.7,20 Obviously, this very factor may explain a decrease of the conductivity above the threshold temperature Tmax, at which σ reaches its maximum value. It is further noteworthy that Σ0 contains an additional factor 1/(kp0)1/2 compared to the ideal case without ion pairing (see eq 19, next section). 4.3. Strong-Electrolyte and General Cases. Assuming as before the predominance of anions in charge transport, a general expression for σ can be readily derived within the present concepts as

(10)

where ΔHp represents the pair formation enthalpy. The dimensionless prefactor kp0, which contains a proportionality factor Cs, may be written as

k p0 = exp(ΔSp/kB)

(15)

where the pre-exponential factor is given by

(9)

which is the complement of the MX0 pair fraction rp . Here, the (reduced) reaction constant rp usually adopts the Boltzmann form written as k p = k p0 exp( −ΔHp/kBT )

⎛ ΔHp ⎞ ⎛ Σ e 2Cs B− ⎞ ⎟exp⎜ − Dσ ≈ 0 exp⎜ ⎟ kBT T ⎝ 2kBT ⎠ ⎝ T − T0 ⎠

5. EVALUATION OF MODEL PARAMETERS 5.1. Relationship between T0 and Tg. Fitting of experimental data by eqs 15 and 17 introduces one (ΔHp) and two (ΔHp, ln kp0 = ΔSp/kB) additional free parameters, respectively. However, the validity and reliability of the present concepts will be the higher, the smaller the number of required adjustable variables are. Therefore, we made use of the common

(13)

Substituting eqs 12 and 13 in eq 4 yields the following expression for the charge diffusivity: Dσ ≈ D X0 − exp(ΔHp/2kBT )exp(−B−/(T − T0))/ k p0 (14) 3069



Article

empirical finding that the VTF parameter T0 characterizing atomic/ionic mobility or viscosity is linked to the glass transition temperature Tg of the substance under consideration.25,26 More specifically, it has been found in numerous studies on polymer electrolytes that T0 ≈ Tg − 50 K complies with the experimental observations.27,28 Recently, Seki et al.29 reported for P(EO/PO)−LiTFSI complexes T0 values that are 40− 50 °C below Tg, depending on composition. In our group, a recent combined evaluation of the radiotracer diffusivity and conductivity in the PEO−NaI system, including the data presented in Figures 1 and 3, yielded T0 = Tg − 44 K over a wide range of salt concentrations.30 It is this particular result that will be used in the fitting of the present conductivity data. Specifically, for PEO−NaI we obtain T0/K = 173.5 + 55.8·[1 − exp( −43.5/y)]

lines in Figure 1. Unlike the proper VTF expression given by eq 2, the extended equation is capable of reproducing the retrograde σ-behavior for high y values as a function of temperature. This feature is more clearly illustrated for PEO250NaI in Figure 6. The solid line representing the three-parameter

(20)

which results from a corresponding empirical fitting expression describing Tg as a function of the composition parameter y. The Tg values of PEO−LiTFSI were found to differ distinctly from those of comparable PEO−NaI complexes. Accordingly, the composition dependence of T0 for PEO−LiTFSI requires a specific expression reading T0/K = 172.2 + 865/y

(21)

Figure 6. Comparison of different evaluation models for the ionic conductivity data (open circles) in PEO250NaI. Coarse-dashed line: VTF fit with T0σ fixed (cf. eq 2). Fine-dashed line: VTF fit with T0σ freely adjusted (cf. eq 2). Solid line: extended VTF fit with T0 fixed (cf. eq 15).

Equations 20 and 21 are depicted in Figure 5 by the dashed and solid lines, respectively. An extensive report on the measurement and evaluation of Tg will be published elsewhere.

(Σ0, B−, ΔHp) fit by eq 15 intersects all data points (open circles) right through the middle. By contrast, the coarsedashed line depicting the two-parameter (Σ0σ, Bσ) VTF fit with eq 2 does not remotely reflect the experimental behavior. In these cases, T0 and T0σ were fixed at 182.4 K, as specified by eq 20. Alternatively, a three-parameter VTF fit, thus including T0σ as freely adjustable parameter, leads to a fair match, as illustrated by the fine-dashed line, but cannot reproduce the σ-decrease at high T. Moreover, this particular fit results in a best estimate for T0σ of 286 ± 3 K, which is hard to reconcile with the pertaining Tg value of 182.4 K. Specifically, it would imply that T0 were higher than Tg by about 100 K in contrast to the common experience that T0 is lower than Tg by 40−50 K. Also the B parameters greatly differ among the distinct fitting approaches. The weak-electrolyte and free VTF case yield B− = 1501 K and Bσ = 87 K, respectively, whereas the VTF fit with T0 fixed produces an intermediate value of 395 K. The above findings also apply to the other PEO−NaI compositions. Figure 5 illustrates that the freely adjusted T0σ values (open circles) from the standard VTF fit with eq 2 lie clearly above the dashed line representing the empirical (fixed) T0 composition dependence. These observations support the validity of our approach to make allowance for (strong) ionpairing. It is remarkable, however, that the differences between the individual fitting models become smaller with increasing salt concentration, both for the overall standard error and the distinct parameter values. For instance, whereas T0σ (free) exceeds T0 (fixed) by 103 K for y = 1000, the surplus reduces to about 10 K for y = 20 (see Figure 5). Similarly, the difference between the B values is lowered from 1469 K for y = 1000 (1570 vs 101 K) to about 164 K for y = 20 (769 vs 605 K). These converging trends indicate that ion pairing becomes less predominant at higher salt concentration.

Figure 5. VTF zero-mobility temperatures in PEOyNaI (open symbols, dashed line) and PEOyLiTFSI (closed symbols, solid line) electrolytes as a function of the reduced salt concentration 1/y. The dashed and solid line represent eqs 20 and 21, respectively, which result from measured Tg data as T0 = Tg − 44 K. Open and closed circles are freely adjusted T0σ data obtained from fits of the VTF equation to the temperature dependence of the ionic conductivity (cf. eq 2).

5.2. Parameters Characterizing PEO−NaI Complexes. The conductivity data on the different PEOyNaI complexes have been fitted by eq 15 using the predetermined T0 values given by eq 20. The choice of the “weak-electrolyte” equation for σ is justified by the evidence given above that ion-pairing in PEO−NaI is extremely strong, at least in the complexes more dilute in salt. Indeed, the application of eq 15 leads to virtually perfect fits for all compositions, as demonstrated by the solid 3070



Article

The parameter values of ΔHp, B−, and Σ0 resulting from fitting by eq 15 are displayed in Figures 7, 8, and 9 as open

Figure 9. VTF parameter Σ0 in PEOyNaI (open symbols, dashed line) and PEOyLiTFSI (closed symbols, solid line) electrolytes as a function of the reduced salt concentration 1/y. The data are obtained within the weak-electrolyte approximation represented by eq 15. The parallel dashed and solid line serve to guide the eye.

Figure 7. Pair formation enthalpy ΔHp in PEOyNaI (open symbols, dashed line) and PEOyLiTFSI (closed symbols, solid line) electrolytes as a function of the reduced salt concentration 1/y. The data are obtained within the weak-electrolyte approximation represented by eq 15. The parallel dashed and solid line serve to guide the eye.

concentration itself, since Σ0 contains Cs as proportionality factor (cf. eq 16). 5.3. Parameters Characterizing PEO−LiTFSI Complexes. The conductivity of the PEO−LiTFSI complexes was evaluated in a similar fashion to those of PEO−NaI. The corresponding results for T0σ, ΔHp, B−, and Σ0, resulting from fitting by eq 15, are plotted in Figures 5, 7, 8, and 9, respectively, as closed symbols. In Figures 7, 8, and 9, the solid line fitted to the PEO−LiTFSI data runs parallel to the dashed line representative of PEO−NaI to guide the eye but has no deeper meaning. It is remarkable that for PEO60LiTFSI the use of the conventional and extended VTF expressions given by eqs 2 and 15, respectively, lead to closely similar results. This is demonstrated by the fact that in Figure 5 T0σ (free) = 186.3 ± 1.5 (closed circle) falls upon the solid line representing eq 21 (T0 = 186.6). Consistently, ΔHp = −0.002 ± 0.008 does not significantly differ from zero (cf. Figure 7), so that eq 15 virtually coincides with eq 2. This may be taken as evidence that ion pairing in PEO60LiTFSI may be neglected, in accordance with studies on slightly more concentrated systems (y ≤ 20).10,11,19 Conversely, the results for PEO60LiTFSI validate the method of analysis suggested in this paper.31 By contrast, the evaluation of the PEO250LiTFSI and PEO1000LiTFSI data does yield clear indications for ion-pair formation. The differences between T0σ (free) and T0 amount to 31 and 44 K, respectively, as shown by Figure 5. For the same compositions, ΔHp takes values of 0.17 and 0.25 eV, as displayed in Figure 7. Furthermore, all PEO−LiTFSI parameters exhibit similar compositional trends to the PEO− NaI ones, as revealed by Figures 5, 7, 8, and 9. This suggests that in both systems the same ion-pairing mechanisms are operating but that quantitative differences arise from the specific ionic properties. In particular, the impact of pair formation is found to be distinctly less in PEO−LiTFSI.

Figure 8. VTF parameter Bσ in PEOyNaI (open symbols, dashed line) and PEOyLiTFSI (closed symbols, solid line) electrolytes as a function of the reduced salt concentration 1/y. The data are obtained within the weak-electrolyte approximation represented by eq 15. The parallel dashed and solid lines serve to guide the eye.

symbols. It is seen in Figure 7 that ΔHp monotonically increases with decreasing salt concentration (1/y) up to y = 250. This is demonstrated by the dashed line representing a threeparameter fit (i.e., ΔHp/eV = −0.110 + 0.918 exp(−35.1/y)) through the data for the five highest concentrations. This line serves to guide the eye and has no deeper meaning. However, Figure 7 suggests that the PEO−NaI system enters another regime of ion properties for very dilute compositions. A similar impression is obtained from the plot of B− versus 1/y in Figure 8, also showing increasing values of this anion-related VTF parameter with decreasing salt concentration. We further observe an opposite trend for Σ0, which monotonically decreases with diminishing 1/y over 4 orders of magnitude. Only about a factor of 50 of this total decrease is due to the salt

6. DISCUSSION Commonly, the VTF equation is used to describe the temperature dependence of the DC conductivity in polymer electrolytes, since it is usually able to reproduce the curvature 3071



Article

case may be encountered for PEO30NaI at high T, where DM+/ DX− ≈ 0.15 was obtained.30 The second assumption in the derivation of eq 15 was that kp ≫ 1 should hold. For PEOyNaI it can be deduced from the accompanying study30 that this requirement is well-fulfilled for all compositions and temperatures, except for y = 30 at low T, where kp ≈ 7. However, for the PEO−LiTFSI system such additional information is lacking hitherto. In the case of PEO60NaI, we deduced in Section 5.3 that ion pairing is of minor importance. In contrast, for the more dilute PEOyLiTFSI complexes with y = 250 and 1000 a stronger ion-pairing tendency is evidenced by the parameter values in Figures 5, 7, 8, and 9. More specifically, PEO250LiTFSI may be compared with PEO30NaI in terms of pair formation, as indicated by the similar results of ΔHp for these complexes (0.167 versus 0.174; cf. Figure 7). This suggests that kp ≫ 1 may be approximately fulfilled for dilute PEO−LiTFSI electrolytes as well. For PEO−NaI the effects of the approximations made in this work may be estimated by a comparison with the more extended study, which also includes radiotracer diffusion experiments.30 It turns out that the present ΔHp data tend to be too high, with a maximum deviation of ∼0.1 eV at low Cs. In addition, the B− values are generally too high but not more than by ∼15%. Moreover, despite a much smaller database (no diffusion measurements) and a more simple evaluation, the present treatment correctly predicts the compositional trends of ΔHp and B− (cf. Figures 7 and 8 with Figures 5 and 7 in ref 30, respectively). Mathematically, a ΔHp value significantly different from zero means that a “correction” to the VTF equation (with fixed T0) is necessary to fit the experimental data. As discussed above, in the weak-electrolyte case it is effectively −ΔHp/2 that acts as activation energy through a Boltzmann factor and thus allows for additional flexibility in fitting curvatures of Arrhenius plots of σ . In a rigorous treatment, the correction factor is given by 1 − rp (cf. eqs 9 and 17), which involves kp0 as a fourth fitting parameter.34 However, it can be shown that the use of the general expression given by eq 17 instead of its weak-electrolyte approximation eq 15 first does not lead to better fits and second produces essentially the same results in ΔHp and B−. This relates to the fact that the factor 1 − rp has hardly any influence for low values of rp, whereas for high rp values eq 17 reduces to eq 15. Moreover, in the narrow intermediate range 0.3 < rp < 0.7 the difference between these two equations is small. The general expression of eq 17 may be employed to obtain estimates for ΔSp (cf. eq 11) and to decompose σ into the ideal conductivity σ id and the fraction of free ions 1 − rp. This decomposition is displayed for PEO250NaI in Figure 10. For this complex, the weak-electrolyte approximation certainly holds, so that we may take the previous results ΔHp = 0.685 ± 0.004 eV and B− =1501 ± 7 K from fitting with eq 15 as unbiased values (cf. Figures 7 and 8). Inserting these values as fixed parameters in eq 17 enables us to screen the acceptable range of ΔSp by data fitting. In fact, only a lower threshold ΔSpmin can be estimated from comparisons of the best-fit standard deviation with that from fitting with 15 pertaining to infinitely large ΔSp values. For PEO250NaI, ΔSpmin = 33kB is obtained as minimum value when a 10% increase of the minimum standard deviation is tolerated. Simultaneously, the prefactor of eq 18 results as ln(Σ0id/S·cm−1·K) = 10.3.34 Altogether, Figure 10 shows that a potentially high conductivity given by σ id is counteracted by a low fraction of free ions 1 − rp.

exhibited by the corresponding Arrhenius plot. Then fitting of the experimental σ data yields best estimates for the three VTF parameters Bσ, T0σ, and Σ0σ. However, we have shown in this paper that these parameter values may not be very meaningful. In the case of strong ion pairing, as observed in salt-poor PEO−NaI complexes, the shortcoming of the VTF equation is obvious, as it does not fit the experimentally observed behavior. Under moderate ion-pairing conditions, however, excellent VTF fits are obtained. Nevertheless, the results for T0σ exceed the acceptable range that relates to the pertinent Tg value of the electrolyte under consideration. Under the same conditions, Bσ tends to be too small. Therefore, it should be emphasized that the mere quality of the VTF fit provides no evidence for the validity of the adjusted parameter values. This is even true for highly accurate σ data covering a wide temperature interval. Meaningful results from VTF fitting can only be expected if ion pairing plays a negligible role. In any case, it is advantageous to determine Tg to assess T0σ independently. Making allowance for ion pairing leads to the extended VTF expression of eq 15, which contains the additional Boltzmann factor exp(−Q/kBT) with activation energy Q = −ΔHp/2. Similar expressions have been proposed previously based on an extension of the free-volume theory.26,32,33 In this early work, the activation energy Q appearing in the additional Boltzmann factor included beside the pair dissociation energy (which equals −ΔHp/2) also contributions related to the displacement of a polymer segment and to the jump of an ion. Implicitly it was assumed, however, that all energy contributions including the dissociation energy are positive implying Q > 0. By contrast, in the present approach, the energetic terms for the correlated motion of ions and polymer segments are contained in the mobility parameter B of the VTF equation. Moreover, it turns out that in polymer electrolytes the ionic dissociation energy is a negative quantity, or in other words, that ΔHp > 0 holds. Thus based on enthalpy considerations alone, pair formation is an unfavorable process; its impact rather relies on entropy effects8,20 (see below). Altogether, some basic assumptions made in the early formalism of the T-dependent conductivity may be questioned,32,33 and straightforward applications to the analysis of experimental data on salt-in-polymer systems are hard to find. Figures 6, 7, and 8 reveal the compositional trends in ΔHp, B−, and Σ0σ that result from the fitting by eq 15 derived in Section 4.2 for the weak-electrolyte case. This raises the question whether the investigated PEO−NaI and PEO−LiTFSI complexes fulfill the conditions for the validity of eq 15. In fact, two major assumptions are involved. The first assumption, DM+/DX− ≪ 1, will be fulfilled for PEO−LiTFSI, since NMR ≤ published PFG-NMR diffusion data show that DLiNMR/DTFSI 0.1 for solid-like complexes based on high-molecular-weight NMR / PEO.12,23 It should be mentioned that DM+ /DX− ≤ DM DXNMR because of a possible nonzero pair contribution that NMR equally contributes to both DM and DXNMR. This may be recognized from eqs 7 and 8. Our own data for PEO20LiTFSI and PEO60LiTFSI (both not shown here) indicate that DLiNMR/ NMR DTFSI ≈ 0.035 and ≈ 0.06 over the entire T-range, respectively. Also for the PEOyNaI complexes the condition DM+/DX− ≪ 1 largely holds. This can be concluded from a recent comprehensive evaluation of both conductivity and tracer diffusion data, such as shown in Figure 3 for PEO250NaI.30 The finding that for this composition and others D*Na is only slightly smaller than D*I at high temperatures relates to the fact that the pair contribution predominates for both ionic species. A borderline 3072



Article

the two different complexes with equal molar salt concentration. Consequently, the σ id curves are close to the lower borderline of the respective ideal conductivities. It can be safely stated, however, that PEO250LiTFSI (cf. Figure 11) exhibits a lower ideal conductivity but a higher fraction of free ions than its PEO250NaI counterpart (cf. Figure 10). These properties may be primarily explained by the bulky nature of the TFSI anion, which reduces both its mobility and pairing propensity with regard to the small I− ion. The above analysis provides additional evidence that saltin-PEO complexes are characterized by positive values of both the enthalpy and the entropy of ion-pair formation. The property ΔHp > 0 relies on the strong coordinative binding of the cation to the oxygen atoms in the PEO chains. The same binding characteristics lead to a substantial loss of conformational freedom of the polymer segments, which is then partly regained upon ion pairing, in accordance with ΔSp > 0. Taken together, these two effects give rise to a negative free enthalpy of pair formation ΔGp = ΔHp − TΔSp. This can be verified for PEO250NaI and PEO250LiTFSI using the parameter values derived above and choosing T = 400 K as a central temperature within the range of interest. We actually obtain ΔGp = −0.45 and −0.056 eV, respectively, which does confirm the impact of ion pairing upon charge transport in polymer electrolytes. ΔHp is found to increase with decreasing (reduced) salt concentration 1/y, which suggests a scaling with the polymersegment length per cation given by y = CEO/Cs. Very likely, large segment lengths available to the cation allow for its optimal coordination and the formation of strong polymer− cation bonds. Then the (partly) release of the cation upon ion pairing requires a high positive ΔHp value. By contrast, for small segment lengths steric hindrance effects may cause ΔHp to be comparatively low. Similar arguments relate the magnitude of ΔSp > 0 to the gain in polymer entropy upon pair formation.30 Finally, the paper does not claim to have predictive power. It rather provides a useful framework to analyze conductivity data on polymer electrolytes. Conversely, the results of such an analysis applied to a multitude of related systems could reveal chemical trends and provide information on the nature of the underlying ion−polymer and ion−ion interactions.

Figure 10. Decomposition of the measured ionic conductivity in PEO250NaI (open circles, solid line) into the ideal conductivity σ id (coarse-dashed line) and the free-ion fraction 1 − rp (fine-dashed line) according to eq 17.

For known Cs, σ id directly relates to the diffusivity of free anions DX− (cf. eqs 18 and 19). A similar analysis was applied to PEO250LiTFSI, which is characterized by ΔHp = 0.167 ± 0.006 eV and B− =1072 ± 11 K in the weak-electrolyte approximation (cf. Figures 7 and 8). For this complex, the extent of ion-pairing may be only moderate, so that slightly other parameter values from fitting with eq 17 seemed possible. Exploring the parameter space under the constraint of at most 10% increase in standard deviation yielded 0.14 eV as minimum value for ΔHp . Concomitantly, we obtained B− = 970 ± 110 K, ΔSp/kB = 5.7 ± 4.2, and ln(Σ0id/S·cm−1·K) = 3.1 ± 2.2. This set of parameter values gives rise to the situation depicted in Figure 11. Indeed, the

7. SUMMARY In summary, we have shown that accurate conductivity measurements on amorphous polymer electrolytes covering wide ranges of temperature and salt concentration yield pertinent information on ionic transport that can be extracted within a proper theoretical framework. Specifically, we have derived approximate expressions for the DC conductivity that take into account the effect of ion pairing and thus go beyond the conventional VTF approach. With these expressions and independent estimates of the zero mobility temperature T0 based on Tg measurements it was feasible to estimate relevant iontransport parameters of PEO−NaI and PEO−LiTFSI complexes that were chosen as already well-investigated prototype systems. Distinct trends with temperature and composition in the conductivity behavior of these electrolytes were also disclosed. Altogether, the following important conclusions can be drawn from this case study: (I) Although the commonly employed simple VTF equation may fit the ionic conductivity in polymer electrolytes as a function of temperature, that is, in cases of strong and

Figure 11. Decomposition of the measured ionic conductivity in PEO250LiTFSI (closed circles, solid line) into the ideal conductivity σ id (coarse-dashed line) and the free-ion fraction 1 − rp (fine-dashed line) according to eq 17.

resulting 1 −rp values between about 0.3 and 0.5 are indicative of moderate ion pairing. The 1 − rp curves in Figures 10 and 11 should be considered as the highest estimates of the T-dependent free-ion fraction in 3073



Article

(12) Orädd, G.; Edman, L.; Ferry, A. Solid State Ionics 2002, 152− 153, 131−136. (13) As composition parameter, the symbol y was preferred over the conventional n because the latter symbol suggests that the EO-to-salt ratio must be an integer, which is not generally true. (14) Fauteux, D.; Lupien, M. D.; Robitaille, C. D. J. Electrochem. Soc. 1987, 134, 2761−2766. (15) Bastek, J.; Stolwijk, N. A.; Köster, Th. K.-J.; van Wüllen, L. Electrochim. Acta 2010, 55, 1289−1297. (16) Fögeling, J.; Kunze, M.; Schönhoff, M.; Stolwijk, N. A. Phys. Chem. Chem. Phys. 2010, 12, 7148−7161. (17) Ferry, A.; Orädd, G.; Jacobsson, P. J. Chem. Phys. 1998, 108, 7426−7434. (18) Borodin, O.; Smith, G. D.; Geiculescu, O.; Creager, S. E.; Hallac, B.; DesMarteau, D. J. Phys. Chem. B 2006, 110, 24266−24274. (19) Diddens, D.; Heuer, A.; Borodin, O. Macromolecules 2010, 43, 2028−2036. (20) Ratner, M. A.; Nitzan, A. Faraday Discuss. Chem. Soc. 1989, 88, 19−42. (21) Neyertz, S.; Brown, D. J. Chem. Phys. 1996, 104, 3797−3809. (22) Siqueira, L. J. A.; Ribeiro, M. C. C. J. Chem. Phys. 2005, 122, 194911−194919. (23) Suarez, S.; Abbrent, S.; Greenbaum, S. G.; Shin, J. H.; Passerini, S. Solid State Ionics 2004, 166, 407−415. (24) Hayamizu, K.; Sugimoto, K.; Akiba, E.; Aihara, Y.; Bando, T.; Price, W. S. J. Phys. Chem. B 2002, 106, 547−554. (25) Angell, C. A. Solid State Ionics 1983, 9,10, 3−16. (26) Ratner, M. A. In Polymer Electrolyte Reviews; MacCallum, J. R., Vincent, C. A., Eds.; Elsevier: London, 1987; pp 173−236. (27) Ratner, M. A.; Shriver, D. F. Chem. Rev. 1988, 88, 109−124. (28) Baril, D.; Michot, C.; Armand, M. Solid State Ionics 1997, 94, 35−47. (29) Seki, S.; Susan, M. A. B. H.; Kaneko, T.; Tokuda, H.; Noda, A.; Watanabe, M. J. Phys. Chem. B 2005, 109, 3886−3829. (30) Wiencierz, M.; Stolwijk, N. A. Solid State Ionics, DOI: 10.1016/ j.ssi.2012.02.002. (31) The extent of ion pairing also manifests itself by modified Arrhenius plots, in which σ (or Dσ) is displayed against 1/(T − T0) instead of 1/T, provided that T0 can be independently assessed. For weak ion pairing the data will almost fall onto a straight line, whereas in the opposite case the downward curvature, also seen in the conventional Arrhenius plot, will remain. Indeed, checking the present data it was found that the “apparent deviation” from a straight-line plot scales with the degree of pair formation. Of the complexes investigated, only the PEO20NaI and PEO60LiTFSI data shows virtually linear behavior in the modified Arrhenius plot. (32) Miyamoto, T.; Shibayama, K. J. Appl. Phys. 1973, 44, 5372− 5376. (33) Cheradame, H.; LeNest, J. F. In Polymer Electrolyte Reviews; MacCallum, J. R., Vincent, C. A., Eds.; Elsevier: London, 1987; pp 103−137. (34) In the fitting procedures, ln Σ0 and ln kp0 = ΔSp/kB are chosen as free or fixed parameters rather than kp0 and Σ0, respectively, that is, for numerical-technical reasons.

moderate ion pairing, the resulting parameter values have little or no significance. (II) The pair-formation enthalpy is a positive quantity that increases with decreasing salt concentration. Together with a positive pair-formation entropy this leads to a negative Gibbs free enthalpy in the temperature range of interest. (III) If ion-pair formation is strong enough, the measured conductivity can be decomposed into the product of an “ideal” conductivity (representative of full dissociation) and the fraction of “free” ions (not bound in ion pairs). (IV) The VTF mobility parameter B−, representative of the fast diffusing charged single anions, increases with decreasing salt concentration. (V) The mobility of I− ions appears to be much higher than that of TFSI− ions in PEO-based electrolytes of equally dilute salt concentration. In terms of charge transport, the reverse situation applies due to the much stronger pairing tendency of I− with respect to TFSI−. Further findings of this work not only confirm some previously established knowledge on polymer electrolytes, but also provide quantitative estimates for the systems investigated. (VI) Ion pairing increases with increasing temperature and decreasing salt concentration. (VII) Ion pairing is much more pronounced in PEO−NaI than in PEO−LiTFSI complexes with the same salt concentration. (VIII) Pair formation is mainly driven by changes in the polymer entropy.

■

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Phone: +49 (0)251 8339013. Fax: +49 (0)251 8338346. Notes

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS The authors thank Manuel Reschke and Shahmahmood Obeidi for help in experiment and data analysis. Financial support by the Deutsche Forschungsgemeinschaft within the collaborative research centre SFB 458 is gratefully acknowledged.

■

REFERENCES

(1) Fenton, D. E.; Parker, J. M.; Wright, P. Polymer 1973, 14, 589− 594. (2) Wright, P. V. Mater. Res. Soc. Bull. 2000, 27, 597. (3) Gray, F. M. Solid Polymer Electrolytes; VCH: New York, 1991. (4) Chandrasekhar, V. Advances in Polymer Science; Springer: Heidelberg, 1998; pp 139−205. (5) Stolwijk, N. A.; Obeidi, Sh. Phys. Rev. Lett. 2004, 93, 125901− 125904. (6) Obeidi, Sh.; Zazoum, B.; Stolwijk, N. A. Solid State Ionics 2004, 173, 77−82. (7) Stolwijk, N. A.; Wiencierz, M.; Obeidi, Sh. Faraday Discuss. 2007, 134, 157−169. (8) Stolwijk, N. A.; Wiencierz, M.; Fögeling, J.; Bastek, J.; Obeidi, Sh. Z. Phys. Chem. 2010, 224, 1707−1733. (9) Gorecki, W.; Jeannin, M.; Belorizky, E.; Roux, C.; Armand, M. J. Phys.: Condens. Matter 1995, 7, 6823−6832. (10) Johansson, A.; Gogoll, A.; Tegenfeldt, J. Polymer 1996, 37, 1387−1393. (11) Edman, L. J. Phys. Chem. B 2000, 104, 7254−7258. 3074


What Can We Learn from Ionic Conductivity Measurements in Polymer

Recommend Documents