Fundamental Limitations of Ionic Conductivity in Polymerized Ionic

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Fundamental Limitations of Ionic Conductivity in Polymerized Ionic Liquids Eric W. Stacy,*,† Catalin P. Gainaru,*,§ Mallory Gobet,∥ Zaneta Wojnarowska,⊥,# Vera Bocharova,*,⊥ Steven G. Greenbaum,∥ and Alexei P. Sokolov‡,⊥

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†

Department of Physics and Astronomy and ‡Department of Chemistry, University of Tennessee, Knoxville, Tennessee 37996, United States § Fakultät Physik, Technische Universität Dortmund, D-44221 Dortmund, Germany ∥ Department of Physics and Astronomy, Hunter College of The City University of New York, New York, New York 10065, United States ⊥ Chemical Sciences Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, United States # Institute of Physics, University of Silesia, SMCEBI, 75 Pulku Piechoty 1A, 41-500 Chorzow, Poland ABSTRACT: We present detailed studies of ionic conductivity in several polymerized ionic liquids (PolyILs) with different size of mobile ions. Presented analysis revealed that charge diffusion in PolyILs is about 10 times slower than ion diffusion, suggesting strong ion−ion correlations that reduce ionic conductivity. The activation energy for the ion diffusion shows a nonmonotonous dependence on the mobile ion size, indicating a competition between Coulombic and elastic forces controlling ion transport in PolyILs. The former dominates mobility of small ions (e.g., Li), while the latter controls mobility of large ions (e.g., TFSI). We propose a simple qualitative model describing the activation energy for the ion diffusion. It suggests that an increase in dielectric constant of PolyILs should lead to a significant enhancement of conductivity of small ions (e.g., Li and Na).

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INTRODUCTION Solid polymer electrolytes are one of the most promising materials for the next generation of batteries.1−3 Replacing traditional liquid electrolytes with solid polymer electrolytes will significantly improve battery performance and safety.1,4−6 However, low ionic conductivity in polymer electrolytes at ambient temperature remains the major obstacle to their widespread applications.1,2,7−10 Thus, understanding fundamental parameters controlling ionic conductivity is critical in the development of solid polymer electrolytes. Polymerized ionic liquids (PolyILs) present a subclass of polymer electrolytes, where one ion remains mobile while a counterion is attached to a polymer chain. As a result, they are essentially single ion conductors which is beneficial for many applications. This also makes them good model systems for analysis of mechanisms controlling ionic conductivity in polymers. A classical description of polymer electrolytes relates the ion mobility to the polymer segmental mobility.11,12 Thus, one of the traditional ways of enhancing ionic conductivity is to design polymer electrolytes with lower glass transition temperature, Tg. Decreasing Tg leads to a faster segmental mobility and correspondingly higher ionic conductivity at ambient conditions. Recent studies revealed13,14 that PolyIL’s Tg depends on chain rigidity, dielectric constant, and the volume of a structural unit Vm (monomer + mobile ion).15 In particular, an increase in the dielectric constant leads to © XXXX American Chemical Society

screening of electrostatic interactions, resulting in faster polymer segmental mobility (lower Tg). This analysis, however, suggested that it would be difficult (if not impossible) to design a PolyIL with Tg below 200 K.13 This provides a clear limit to how much we can decrease Tg of PolyILs, and Tg ∼ 200 K is not sufficient to reach the required segmental mobility at ambient conditions. It was demonstrated, however, that in many polymers,16−18 and especially in PolyILs,19,20 ionic conductivity can be decoupled from segmental dynamics, and this decoupling degree can easily reach 4−7 orders of magnitude.19 This enables an alternative way to enhance ionic conductivity in PolyILs. However, it requires a detailed fundamental understanding of microscopic parameters controlling ion conductivity and its decoupling from segmental relaxation in polymers. To unravel these details, we analyzed ionic conductivity in several PolyILs with different chemical structure and different size of mobile ions. Using broadband dielectric spectroscopy (BDS), nuclear magnetic resonance (NMR), and the recently developed model of conductivity relaxation,21 we analyzed ion diffusion and the so-called Haven ratio in studied PolyILs. The latter revealed strong correlations in ionic motions that Received: June 8, 2018 Revised: October 3, 2018

A

DOI: 10.1021/acs.macromol.8b01221 Macromolecules XXXX, XXX, XXX−XXX

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Macromolecules Scheme 1. Chemical Structures of the Analyzed PolyILs with Mobile Ionsa

X = Li+, K+, and Cs+; A = Br−, hexafluorophosphate (PF6−), and bis(trifluoromethane)sulfonimide (TFSI−).

a

Table 1. Information about Cations and Anions, Mobile Ion Radii Taken from Refs 25 and 26, Measured Densities of the Samples at Room Temperature, Monomer + Mobile Ion Molecular Weight, n the Mobile Ion Concentration, Tg Estimated from DSC, Eσ the Activation Energy of Conductivity at T < Tg, and Eτ the Activation Energy of the Conductivity Relaxation Time at T < Tg cation +

Li K+ Cs+ PolyEtVIm PolyEGVIm PolyEGVIm PolyEGVIm

anion

mobile ion radius [nm]

PolySTF PolySTF PolySTF TFSI− Br− PF6− TFSI−

24

0.073 0.15224 0.18124 0.32723 0.18224 0.25423 0.32723

density [g/cm3]

M [g/mol]

n [nm−3]

Tg(DSC) [K]

1.6 1.78 2.33 1.61 1.52 1.48 1.53

320.94 353.1 446.9 403.15 320.9 385.97 521.15

2.99 3.02 3.13 2.40 2.84 2.30 1.76

507 500 470 344 348 296 261

155 105 85 110 90 120 130

± ± ± ± ± ± ±

10 10 10 10 10 10 10

Eτ [kJ/mol] 155 103 81 104 83 110 120

± ± ± ± ± ± ±

10 10 10 10 10 10 10

Dielectric Measurements. Dielectric spectra were measured in the frequency range of 10−1−107 Hz using a Novocontrol sytem, which includes an Alpha-A impedance analyzer. The polyEtVIm and polyEGVIm samples were measured using a parallel-plate configuration dielectric cell made of invar and sapphire. Separation between the electrodes was 47 μm, which yielded a geometrical capacitance of 21 pF. The samples were placed inside the cryostat with a dry nitrogen atmosphere (Novocontrol Quattro system) and equilibrated at 380 K for at least 2 h before any measurements. The measurements were performed from high to low temperatures. The polySTF samples were measured using a stainless steel capacitor with the separation between electrodes of 0.1 mm provided by fused silica fibers. All polySTF samples were equilibrated at T = Tg + 20 K for at least 45 min before any measurements. Also in this case the dielectric spectra were recorded during cooling of sample. All the samples were equilibrated for at least 15 min to achieve thermal stabilization within 0.1 K after each temperature step. After reaching the lowest temperature, the samples were measured at several temperatures on heating back to the highest temperature to verify reproducibility of the data. The experimental results for all the samples were reproducible, which indictated they did not degrade and contained no significant residual solvent. Herein it should be noted that the polySTF samples were additionally dried at 393 K for at least 48 h under the vacumm conditions before any measurement. To perform detailed quantitative analysis of conductivity values, the dielectric spectra were normalized to have the high frequency dielectric constant ε∞ ∼ 3. Calorimetric Measurements. The glass transition temperature of PolyILs was measured using the TA Instruments Q1000. The heating and cooling cycles were performed with a rate of 10 K/min and repeated three times to verify reproducibility. Tg was estimated as the midstep of the transition in the heat flow on cooling cycle. No signs of crystallization were noticed in the temperature range from

significantly reduce ionic conductivity. These correlations are not attributable to motion of ion pairs and require further studies. Analysis of ionic conductivity in glassy PolyILs (below Tg) reveals two competing mechanisms characterizing decoupled ion transport in frozen polymer structure: (i) Coulombic interactions that dominate only at very small ion size (e.g., for Li) and (ii) elastic force that increases with the increase in the ion size but can be weakened by a frustration in chain packing (free volume). The presented analysis provides a clear suggestion for enhancing conductivity of small ions (e.g., Li and Na) in PolyILs: design a polymer with high dielectric constant that not only will enhance segmental dynamics and reduce Tg but also will enhance the decoupling of small ions mobility from segmental dynamics.

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Eσ [kJ/mol]

EXPERIMENTAL SECTION

In this work we used three polymerized ionic liquids with several different mobile ions. Their chemical structures are presented in Scheme 1. All the polymers were synthesized in our group, and a detailed description of their synthesis and characterization has been presented previously.13 X-ray scattering studies of these PolyILs (will be published elsewhere) revealed their amorphous structure, in agreement with multiple earlier studies of similar polymers.22,23 Special care has been taken to remove solvents and especially water from the samples before any measurements, and these procedures were also previously described.13 The mass density of the samples was measured at room temperature using pycnometry and was also described in ref 13. A critical parameter in our studies is the mobile ion size. There are several computational approaches to calculate sizes of ions in ionic liquids.24−26 To compare our data with the data reported in the literature, we selected sizes reported in Table 1.25,26 B

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Macromolecules 473 K down to temperatures well below Tg. Also, first indications of irreversible degradations occurred for temperatures well beyond the range of this study. Additional details were published previously.13 PFG NMR Measurements. Pulsed-field gradient NMR (PFG NMR) was used to measure diffusion of fluorine nuclei (PF6 and TFSI anions) at high temperatures. For the diffusivity measurements PolyEtVIm samples were smashed to a rough powder and placed in sealed NMR glass tubes. PolyEGVIm samples were initially dissolved in a small quantity of dried acetone. The solution was then transferred in unsealed NMR tubes which were placed overnight under a nitrogen atmosphere in an oven at around 50 °C to remove most of solvent. The remaining solvent traces were finally removed by heating the samples at 85 °C, under vacuum, for ∼12 h. The measurements monitoring the dynamics of fluorine nuclei (hence the diffusion of anions only) were performed on a 7.05 T Varian-S direct drive wide bore spectrometer equipped with a z-gradient DOTY Scientific, Inc., probe and using a stimulated-echo PFG sequence. Sixteen field gradient values that linearly increased from 2 up to 1000 G/cm as needed were used. Field gradient pulse durations δ were 1−3 ms, and diffusion delays Δ were 0.2−0.6 s. Gradient stabilization delays of 1 ms and spoiler gradient pulses of 2 ms at 70 G/cm were used. Sixteen transients for each setting were recorded.

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RESULTS Examples of measured BDS spectra for selected PolyILs are presented in Figure 1. To reveal details of conductivity and conductivity relaxation processes in studied PolyILs, we used several representations of the dielectric spectra, such as real part of conductivity σ′(ν) (Figure 1a), dielectric loss modulus M″(ν) (Figure 1b), and real ε′(ν) and imaginary ε″(ν) parts of the permittivity spectra (Figures 1c and 1d, respectively). The conductivity spectra (Figure 1a) show three regimes: (i) the power-law behavior at high frequency represents the ac conductivity, when ions are moving in some confined space; (ii) the frequency-independent plateau corresponding to the dc conductivity, σ0, when ions start their drift, appears in the intermediate frequency range; and (iii) the drop at lower frequency corresponds to the electrode polarization effect, when ions reach electrodes and cannot drift any longer. Changes of the σ′(ν) spectra with temperature (Figure 1a) reflect slowing down of ion dynamics and decrease of σ0(T) upon cooling. The dc conductivity can be obtained directly from the dc plateau value, and the results for all materials studied here are shown in Figure 2. Another presentation of the dielectric response is the complex electrical modulus M*(ν) = M′(ν) + iM″(ν). The peak in the imaginary part of the M″(ν) (Figure 1b) and the characteristic step in the real part of the dielectric permittivity spectra ε′(ν) (Figure 1c) represent the conductivity relaxation ascribed to ionic rearrangements within the polymer matrix,21 which roughly coincides with the frequency of the crossover from the ac to dc regime in the conductivity spectra (Figure 1a). It is interesting that the amplitude of the conductivity relaxation process in many samples decreases with cooling (Figure 3), which is opposite to the usual dipolar relaxation where amplitude increases on cooling, following the Curie law. 27 The conductivity relaxation process is not visible in ε″(ν) spectra because it is obscured by the dc conductivity dominating the ε″(ν) spectra at these frequencies (Figure 1d). The conductivity relaxation time, τσ, can be estimated from the M″(ν) maximum peak and is inversely proportional to σ0, τσ = ε0εs/σ0.27 The conductivity spectra can be described using the random barrier model (RBM)28,29 (Figure 1a). According to RBM the ions jump in a potential energy landscape with random barrier

Figure 1. Dielectric spectra of PolySTF-Li+ sample at a few selected temperatures: (a) real part of complex conductivity, (b) electric loss modulus M″, and (c) real and (d) imaginary parts of permittivity. The symbols are experimental data, the lines in (a) and (d) are fits to the random barrier model (RBM) predictions (eq 1), and lines in (c) are fits to a Havriliak−Negami function plus electrode polarization (eq 2).

heights contributing to ac conductivity. However, once they overcome the largest energy barrier, they drift and enter the dc conductivity regime. This largest energy barrier determines the characteristic time scale for the crossover from ac- to dc regimes, τRBM. RBM predicts for the conductivity spectra Ä É−1/3 i 2πντRBMσ0 ÅÅÅÅ i 2πντRBMσ0 ÑÑÑÑ σ *(ν) ÅÅ1 + 2.66 ÑÑ = ln σ0 σ *(ν) ÅÅÅÇ σ *(ν) ÑÑÑÖ (1) Equation 1 describes well the conductivity spectra with two free fit parameters, τRBM and σ0 (Figure 1a). Moreover, it also describes well the conductivity relaxation process in the permittivity spectra (e.g., Figure 1d). However, the highfrequency permittivity plateau, ε∞, unrelated to ion dynamics, needs to be additionally taken into account. The τRBM obtained from the RBM fit of conductivity spectra is approximately the same as the conductivity relaxation time estimated from the M″(ν) maximum, τRBM ∼ τσ, while the value of the RBM parameter σ0 is consistent with the value of dc conductivity estimated directly from the plateau level. The permittivity spectra were independently analyzed using the Havriliak−Negami (HN) function (Figures 1c and 3) describing the conductivity relaxation process, dc conductivity, and electrode polarization (EP) contributions: C

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difference in the value of conductivity at Tg, σ0(Tg). The crossover from VFT-like behavior to Arrhenius behavior at T ∼ Tg is also obvious in the temperature dependence of the conductivity relaxation time obtained from the RBM fits (Figure 2b,d). These data also reveal that the rate of ion rearrangements at Tg differs widely between the studied PolyILs (Figure 2d). This emphasizes strong variation in the decoupling of ion mobility and segmental dynamics (the latter has the same rate at Tg) among those PolyILs. This decoupling reaches more than 10 orders of magnitude in some cases (Figure 2d).

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DISCUSSION Microscopic Picture for the Random Barrier Model. We start the discussion with analysis of the conductivity relaxation spectra that provides important information about ion dynamics in PolyILs, which can be well described by the RBM eq 1 (Figure 1). However, the original RBM does not present a clear microscopic picture of the conductivity relaxation process. In concentrated ionic systems each ion is usually surrounded by several counterions (Figure 4). For

Figure 2. Arrhenius plots of dc conductivity (a) and conductivity relaxation time (b) of all PolyIL systems as a function of 1000/T: polyEGVIm−TFSI (black squares); polyEGVIm−PF6 (filled red circles); polyEGVIm−Br (open circles); polyEtVIm−TFSI (filled blue triangles); polySTF−Li (open triangles); polySTF−K (filled stars); polySTF−Cs (open stars). (c) and (d) present the same data vs Tg/T. The vertical dashed line in (c) and (d) marks Tg estimated from DSC. The error bars are on the order of the symbol size.

Figure 4. Schematic illustration of the mobile ion (brown circle) escaping the Coulombic cage formed by the surrounding counterions (blue circles). λ is the average size of the Coulombic cage.

example, atomistically detailed simulations of a PolyIL revealed that each mobile PF6 ion is surrounded on average by four imidazolium counterions attached to a polymer.32 Therefore, essentially the mobile ion is trapped in the Coulombic cage created by the surrounding counterions (Figure 4). At short times this mobile ion can move inside this Coulombic cage, thus providing a contribution to the ac conductivity. However, according to RBM considerations, at longer times the mobile ion will overcome the largest energy barrier in the system by escaping from the Coulombic cage and then will contribute to the dc conductivity. This relatively naive picture suggests a connection between ion diffusion and the characteristic RBM relaxation time, τRBM. We can estimate the ion diffusion constant from the classical equation

Figure 3. Real part permittivity spectra of polyEGVIm−TFSI at a few selected temperatures (symbols), their fits to HN function (dashed lines), and fits including EP contribution (eq 2, solid lines). The data clearly show a decrease in the amplitude of the relaxation process upon cooling.

ε*(ω) = ε∞ +

σ Δε + i 0 + A ω −γ α β ε0ω [1 + (iωτHN) ]

(2)

where Δε is the amplitude of the relaxation process, τHN is the characteristic relaxation time, α and β are the shape parameters, γ is the electrode polarization slope, and A is the amplitude. The temperature dependence of the dc conductivity (Figure 2a) shows a crossover from a Vogel−Fulcher−Tammann (VFT) behavior above the glass transition temperature Tg to an Arrhenius-like behavior at T < Tg. This crossover has been reported for many ionic systems.30,31 Scaling the temperature dependence of conductivity by Tg (Figure 2c), where polymer segmental dynamics by definition has comparable relaxation time for all polymers, τα ∼ 102−103 s, reveals significant

D=

λ2 6τRBM

(3)

with λ = (3/4πn) being the radius of the Coulombic cage (Figure 4). Here n is the mobile ion concentration (Table 1). Comparison of the estimated diffusion coefficient of mobile ions using eq 3 with the data from the pulsed-field gradient NMR measurements (Figure 5a) shows a very good quantitative agreement, as was also demonstrated for several other systems in our earlier paper.21 1/3

D

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Figure 5. (a) Diffusion coefficients of mobile ions in polyEGVIm− TFSI (black squares), polyEGVIm−PF6 (red circles), and polyEtVIm−TFSI (blue triangles) from PFG NMR measurements (open symbols) and from the conductivity relaxation (filled symbols). (b) Diffusion coefficients of mobile ions for all studied here PolyILs using the conductivity relaxation: polyEGVIm−TFSI (black squares), polyEGVIm−PF6 (filled red circles), polyEGVIm−Br (open circles), polyEtVIm−TFSI (filled blue triangles), polySTF−Li (open triangles), polySTF−K (filled stars), and polySTF−Cs (open stars). The error bars are on the order of the symbol size.

Figure 6. Inverse Haven ratio for all studied samples as a function of (a) 1000/T and (b) Tg/T estimated using NMR data (∗ symbols) and conductivity relaxation data (all other symbols). The vertical dashed line in (b) is the glass transition temperature from DSC measurements. The error bars are on the order of the symbol size.

single-particle diffusion constant, D, is expressed through Green−Kubo formalism38,39 1 ∞ D= ⟨vi⃗(0)vi⃗(t )⟩ dt (6) 3 0

∫

Thus, the RBM provides a direct estimate of the mobile ion diffusion without any adjustable parameters, just from the knowledge of the size of the Coulombic cage λ (Figure 5). For simplicity, we estimated it from the averaged ion concentration. This value can be more accurately estimated from analysis of X-ray diffraction, as has been previously reported.33,34 This microscopic picture also explains the appearance of the conductivity relaxation process as the dipolar relaxation in permittivity spectra (Figure 3). Escape of the mobile ion from the Coulombic cage leads to a fluctuation of the global dipole moment and appears as a dipolar relaxation in the permittivity spectra (Figures 1 and 3). As previously discussed,21 this picture also provides a clear microscopic justification for the empirical Barton−Nakajama−Namikawa (BNN) relationship: σ0 = pε0Δεωc, where ωc = 1/τσ and p is the temperature-independent numerical constant, p ∼ 1. Correlations in the Ion Motions. Having estimates of the ion diffusion from the NMR and conductivity relaxation data, we can now compare this diffusion with “averaged charge diffusion” Dσ estimated from the Nernst−Einstein equation:27

in which ⟨vi⃗(0)vi⃗(t )⟩ is the particle velocity self-correlation function; i.e., it depends only on velocity correlation of the same ion. The conductivity σ0 is defined through the current− current correlation function ⟨J(⃗ 0)J(⃗ t)⟩:38,39 σdc =

1 3VkBT

∫0

∞

⟨J ⃗ (0)J ⃗ (t )⟩ dt

(7)

Here V is the volume of the sample. The current−current correlation function can be also expressed through the ion velocity correlation function: N

N

⟨J ⃗ (0)J ⃗ (t )⟩ = q2⟨∑ [sgn(qi)vi⃗(0)]∑ [sgn(qj)vj⃗(t )]⟩ i=1

= Nq2H⟨vi⃗(0)vj⃗(t )⟩

j=1

(8)

where sgn is the ion charge sign function, H is the inverse Haven ratio, and N is the total number of mobile ions in the volume V. Thus, the main difference between ion diffusion and ion conductivity is that the latter includes not only selfcorrelations but also correlations of the velocity of the given ion with velocities of all the other ions in the system. Only in a dilute regime we can neglect distinct ions correlations, and this is the basis of the Nernst−Einstein equation for a dilute solution. In this case, the charge and the ion diffusion will be the same. However, in concentrated ionic systems ion−ion correlations are not negligible and should be taken explicitly into account. The most common explanation for the inverse Haven ratio in ionic liquids being smaller than 1 is that not all the ions contribute to the charge transport.34,40,41 In particular, diffusion of ion pairs (ion + counterion) does not transport charge but still contributes to the measured ion diffusion.40−42

nq2 Dσ (4) kT where n is the total concentration of the mobile ions. We emphasize that the Nernst−Einstein equation is valid for dilute ion solutions where ion−ion correlations are negligible and all ions contribute to conductivity equally. The ratio of the ion diffusion D to the charge diffusion Dσ is called the Haven ratio.35−37 The inverse Haven ratio H = Dσ/D reveals significant difference between the two diffusion coefficients, and it decreases upon cooling for the PolyILs studied here (Figure 6). It clearly indicates that the charge transport in PolyILs is significantly slower than the actual ions diffusion. To understand this observation, we need to go back to the basic definitions of the diffusion and ionic conductivity. The σ0 =

E

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(Figure 7) leads to no change of the dipole moment of the system. Thus, the same ions that contribute to the ion diffusion, but do not contribute to conductivity, do not contribute to the permittivity spectra. Indeed, direct comparison of the inverse Haven ratio and of the amplitude of the conductivity relaxation Δε reveals very similar behavior (Figure 8). Thus, the developed model of conductivity

Formation of other ion aggregates (e.g., triplets and quadruplets) will also reduce the inverse Haven ratio, as has been discussed e.g. in ref 42. In the case of superionic glasses and crystals the inverse Haven ratio is often larger than 1, indicating faster charge diffusion than actual ion diffusion.36,43,44 This is possible in the case of correlated chainlike ion jumps where each ion jumps only a small distance while the effective charge moves much a larger distance.36,43,44 In contrast to ionic liquids, we can exclude ion pair (or triples and any other ion aggregates) diffusion as the mechanism of reduction of the inverse Haven ratio in PolyILs (Figure 6) because the counterions are attached to a polymer chain and cannot diffuse on a large distance with the mobile ions. We can also exclude a role of small local motions of ions attached to the chain because both diffusion (measured e.g. by NMR) and conductivity are measured on macroscopic length scale far exceeding the scale of the local fluctuations. The only plausible mechanism of the ion−ion correlations we can find is the correlation between the mobile ions. Figure 7 presents a

Figure 8. Inverse Haven ratio (filled symbols) and the strength of the conductivity relaxation process (open symbols) as a function of temperature for polyEGVIm−TFSI and polySTF−Li. Δε is shown in absolute values, while the inverse Haven ratio is multiplied by the value presented in the plot. Figure 7. Schematic presentation of the proposed ion−ion correlated motions without charge transport through the polymer medium. Brown circles denote mobile ions, and blue circles present counterions attached to polymer chains.

relaxation not only provides a good description of the dielectric spectra but also provides direct estimates of the ion diffusion and ion−ion correlations. Decoupling of Ionic and Segmental Dynamics at Tg. We now turn to the discussion of the decoupling of ionic conductivity from segmental dynamics. The strength of the decoupling can be analyzed from the conductivity at Tg, σ(Tg) (Figure 2). In concentrated ionic systems with ion motions strongly coupled to structural (segmental for polymers) dynamics the conductivity at Tg is expected to be σ(Tg) ∼ 10−14−10−15 S/cm.45 This led several groups46−48 to introduce the decoupling coefficient defined as R = log[σ(Tg)] + 15. This definition can be explained using simple estimates of the conductivity in ideal ionic systems, where all ions contribute to the conductivity and the ion rate jump is coupled to the rate of structural relaxation. We can rewrite the Nernst−Einstein equation (eq 4) through the rate of ion jumps 1/τ:

cartoon of possible mobile ion correlations, which can be described as a “backflow”; i.e., there are ringlike motions of ions where all mobile ions diffuse, but there is no charge transport. This idea is similar to the mechanisms proposed earlier for the Haven ratio observed for proton transport in oxides.37 We do not exclude other possible microscopic mechanisms that lead to the reduction of the inverse Haven ratio in PolyILs, and detailed simulations might provide more insight. Another puzzling observation is that these ion−ion correlations in PolyILs (except PSTF-Li) get stronger upon cooling (Figure 6). This might be explained by a higher mobility and larger fluctuations of structural units at higher temperatures which reduce the proposed ion−ion correlations (Figure 7). We do not know how these ideas can be verified experimentally, but atomistically detailed MD simulations should be able to test the proposed mechanism. In any case, this analysis reveals an interesting result: PolyILs have strong ion−ion correlations suppressing ionic conductivity, in contrast to superionic glasses and crystals where these correlations enhance ionic conductivity. PolyILs lose more than 1 order of magnitude in conductivity due to this effect (Figure 6), thus imposing strong limitations on ionic conductivity in PolyILs. Reducing these correlations might be important for enhancing their single ion conductivity. These ion−ion correlations might also explain the observed anomalous temperature dependence of the amplitude of the conductivity relaxation process in the permittivity spectra (Figure 3). The correlated “backflow” of the mobile ions

nq2 nq2d 2 D= (9) kT kT6τ Here d is the ion jump length. Usually in ionic liquids and PolyILs n will be limited to ∼1−4 × 1021 1/cm3 (see e.g. Table 1). Assuming that an ion jumps d ∼ 1−2 Å with the rate 1/τ ∼ 10−2−10−3 1/s (characteristic structural relaxation rate at Tg), we expect the conductivity at Tg to be σ(Tg) ∼ 10−14−10−15 S/ cm. Higher conductivity at Tg indicates that ionic motion is faster than segmental dynamics, and the difference in some PolyILs studied here exceeds 10 orders of magnitude (Figure 2c). This is consistent with the shorter conductivity relaxation time in comparison to the segmental relaxation time that should be τα ∼ 102−103 s at Tg. The difference in the characteristic time scales of ionic and segmental dynamics again exceeds 10 orders of magnitude for some of the studied σ=

F

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We propose a simple model to describe the observed dependence of the activation energy of mobile ion diffusion on ion size (Figure 9). We assume that Eτ comprises a sum of electrostatic energy for separation of two chargers and elastic energy that we describe following the shoving model of the glass transition.50 The shoving model considers a motion of a structural unit in a frozen environment and suggests that the activation energy for molecular motion is controlled by thermal fluctuations creating a local volume increase. In that case Eel ≈ G∞V, where G∞ is the high-frequency shear modulus and V is comparable to the volume of the moving molecule. In addition, we can assume that frustration in chain packing (free volume) might locally soften the shear modulus. Thus, we arrive at Eel = αG∞Vion, where Vion = 4πRion3/3 (the volume of the mobile ion) and α is a constant for a given polymer with values α < 1. The total activation energy in this case can be written as

PolyILs (Figure 2d), demonstrating an extremely strong decoupling of ion mobility from segmental dynamics in these polymers. Activation Energy for Ion Diffusion below Tg. When ion motions are many orders faster than the segmental dynamics, the ion diffusion can be analyzed using a solid-state approach because ion jumps occur essentially in a frozen segmental environment. To get deeper insight into parameters controlling the ion diffusion in these conditions, we can analyze the activation energy for conductivity relaxation time in the glassy state, i.e., at T < Tg. According to the RBM eq 3, ion diffusion is inversely proportional to τRBM ∼ τσ, and we can neglect minor changes in the Coulombic cage size λ caused by density changes with temperature. By fitting the temperature dependence of τσ(T) at T < Tg to an Arrhenius behavior (Figure 2), τσ = τ0 exp(Eτ/kT), we estimated the activation energy for the ion diffusion in all PolyILs studied here (Table 1 and Figure 9). For consistency check, we also fit the dc

Eσ (T < Tg) =

q2 + αG∞Vion 4πεε0R

(10)

where q is the ion charge and R is the distance between the ion and a counterion, which we assume to be R ∼ 2Rion. We emphasize that representing a large ion such as TFSI by a pointlike charge (eq 10) is a very crude approximation. However, this simplistic model aims to reveal the role of various contributions on a qualitative level and does not pretend to provide accurate quantitative analysis. Yet, surprisingly, this simple model provides a reasonable description of the experimental results (Figure 9) assuming a dielectric constant ε ∼ 6,51 and a shear modulus parameter αG∞ ∼ 1−1.5 GPa.52 Both parameters are reasonable for polymers.51−53 We want to emphasize that the surprising quantitative agreement of this simplistic model with the experimental data may be rather fortuitous. Nonetheless, the model provides very clear qualitative predictions. In particular, it predicts a clear minimum in the dependence of the activation energy on the ion radius (Figure 9). This minimum will shift to lower Rion with increase in both dielectric constant and the shear modulus of the polymer. The analysis based on this model reveals that the elastic force contribution dominates the activation energy for most of the studied ions, and for the same ion it might differ significantly (∼50%) between different polymers. This dependence on the polymer structure could be ascribed to a different frustration in the chain packing (free volume) that reduces the apparent local shear modulus, i.e., the parameter α. Thus, one can enhance the ion decoupling and conductivity by frustrating the polymer chain packing. This works, however, only for relatively large ions and will have rather minor effects for small ions like Li. Decoupling and conductivity of these ions are dominated by Coulombic interactions. These interactions can be strongly reduced by an increase of the dielectric constant. A simple estimate using eq 10 suggests that achieving ε ∼ 50 might reduce the energy barrier for Li diffusion at Tg to Eτ ∼ 25−30 kJ/mol. This is very low activation energy that should enable very high conductivity at ambient temperature. It is not obvious how high the dielectric constant of PolyILs can be. Small molecular electrolytes have very high ε ∼ 50−150.54 Therefore, designing PolyILs with ε ∼ 50 might be feasible. Thus, one of the promising directions in developments of PolyILs with high conductivity is synthesis of polymers with high dielectric constant. Not only will it lead to a lower Tg,13 it

Figure 9. Activation energy for mobile ion diffusion (conductivity relaxation time) at T < Tg vs the ion radius (blue circles). Also literature data from ref 34 (open squares) and from ref 49 (open triangles) are included, where TFO corresponds to mobile anion CF3SO3−. The dashed line presents Coulombic contribution with ε = 6, while the dotted lines present elastic contribution with αG∞ = 1 and 1.5 GPa. The solid lines present the model predictions (eq 10) as a sum of the elastic and Coulombic contributions to the activation energy.

conductivity below Tg by an Arrhenius dependence, σ0 ∝ exp(−Eσ/kT). The estimated activation energy of conductivity below Tg, Eσ, is slightly higher than Eτ in all PolyILs studied here (Table 1). This is expected due to the weak temperature dependence of the inverse Haven ratio that contributes additionally to the temperature dependence of conductivity (Figure 6). In addition we included data from the literature34,49 for the activation energy of ion mobility in PolyILs below Tg (Figure 9). Analysis of Eτ as a function of the mobile ion radius, Rion, reveals nonmonotonous dependence with a clear minimum (Figure 9). Similar nonmonotonous behavior has been observed in recent coarse-grained simulations of PolyILs, in which case the mobile ion diffusion as a function of Rion revealed a maximum.14 The observed nonmonotonous behavior (Figure 9) can be explained by a competition between (i) Coulombic interaction that decreases with increase of ion radius and dominates at very small Rion and (ii) elastic force that increases with the ion size and dominates at larger Rion. G

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Macromolecules ORCID

will also strongly reduce the energy barrier for conductivity of small ions, such as Li and Na. We want to emphasize that the proposed model is also applicable to ion dynamics at temperatures above Tg as long as the ion motion is much faster than the segmental dynamics (e.g., τσ ≪ τα). In that case the ion diffusion still can be considered as a motion within frozen segmental environment. Therefore, the predictions should also be valid above Tg for PolyILs with decoupled ionic conductivity.

Mallory Gobet: 0000-0001-9735-0741 Zaneta Wojnarowska: 0000-0002-7790-2999 Vera Bocharova: 0000-0003-4270-3866 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work was supported by Laboratory Directed Research and Development program of Oak Ridge National Laboratory, managed by UT-Battelle, LLC, for the U.S. Department of Energy. A.P.S. also acknowledges partial financial support for the data analysis by the U.S. Department of Energy, Office of Science, Basic Energy Sciences, Materials Sciences and Engineering Division. The NMR work at Hunter College was supported by the U.S. Office of Naval Research. Z.W. is grateful for the financial support by the National Science Centre within the framework of the Opus 8 project (Grant No. DEC-2014/15/B/ST3/04246). C.P.G. acknowledges financial support from Deutsche Forschungsgemeinschaft under GA2680/1-1 project.

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CONCLUSIONS Analysis of the dielectric and conductivity spectra in several PolyILs with different chemical structure and different size of the mobile ions revealed two critical mechanisms controlling ionic mobility and its decoupling from segmental dynamics. Coulombic (electrostatic) interactions dominate the ion mobility for very small ions (e.g., Li), while elastic forces dominate for larger ions and their contribution increases with the ion size. The activation energy barrier imposed by the elastic forces can be reduced by frustration of chain packing (increased free volume). At the same time, diffusion of small ions has no significant resistance from elastic forces, and a significant increase in the dielectric constant of the polymer should lead to a strong suppression of the energy barrier imposed by Coulombic interactions. Thus, increasing the dielectric constant might significantly enhance ionic conductivity for small ions, such as Li and Na. This suggests one of the promising direction in development of PolyILs with enhanced ionic conductivity: design of PolyILs with high dielectric constant that will not only reduce the glass transition temperature but also strongly reduce the energy barrier for conductivity of small ions, such as Li and Na. The analysis presented here also demonstrates that the previously proposed model21 of conductivity relaxation based on the random barrier model28,29,55 describes well the dielectric spectra of all PolyILs studied here and provides direct estimates of the ion diffusion without any adjustable parameters. The latter has been confirmed by direct comparison to the PFG NMR data. Analysis of the dielectric spectra based on this model also revealed strong ion−ion correlations that reduce ionic conductivity in PolyILs by about one order of magnitude. This effect cannot be ascribed to ion pair motions, and its detailed microscopic mechanism remains unknown. This effect is also manifested in the decrease of the strength of the conductivity relaxation contribution to the permittivity spectra upon cooling, which is behavior opposite to that of classical dipolar relaxation. We speculate that it is related to mobile ion−mobile ion correlations, a kind of backflow effect which leads to no net charge diffusion despite ion diffusion. Atomistically detailed MD simulations should provide a clear test of this idea. We expect that increase in the dielectric constant should lead to an additional enhancement of ionic conductivity through the decrease of the ion−ion correlations. Thus, synthesis of PolyILs with high dielectric constant should strongly enhance ion conductivity of small ions such as Li and Na.

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*E-mail: [email protected] (E.W.S.). *E-mail: [email protected] (C.P.G.). *E-mail: [email protected] (V.B.). H

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