Mechanism of Conductivity Relaxation in Liquid and Polymeric

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Mechanism of Conductivity Relaxation in Liquid and Polymeric Electrolytes: Direct Link between Conductivity and Diffusivity Catalin P. Gainaru, Eric W. Stacy, Vera Bocharova, Mallory Gobet, Adam P. Holt, Tomonori Saito, Steve Greenbaum, and Alexei P. Sokolov J. Phys. Chem. B, Just Accepted Manuscript • DOI: 10.1021/acs.jpcb.6b08567 • Publication Date (Web): 28 Sep 2016 Downloaded from http://pubs.acs.org on October 4, 2016

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The Journal of Physical Chemistry

Mechanism of Conductivity Relaxation in Liquid and Polymeric Electrolytes: Direct Link between Conductivity and Diffusivity

C. Gainaru,1,2* E. W. Stacy,3 V. Bocharova,4 M. Gobet,5 A. Holt,3 T. Saito,4 S. Greenbaum,5 and A. P. Sokolov1,4*

1

Department of Chemistry, University of Tennessee, Knoxville, Tennessee 37996, U.S. 2

Fakultät Physik, Technische Universität Dortmund, D-44221 Dortmund, Germany

3

Department of Physics and Astronomy, University of Tennessee, Knoxville, Tennessee 37996, U.S.

4

Chemical Sciences Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, U.S.

5

Department of Physics & Astronomy, Hunter College of The City University of New York, New York, NY 10065, U.S.

1

ACS Paragon Plus Environment


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ABSTRACT: Combining broadband impedance spectroscopy, differential scanning calorimetry, and nuclear magnetic resonance we analyzed charge and mass transport in two polymerized ionic liquids and one of their monomeric precursors. In order to establish a general procedure for extracting single-particle diffusivity from their conductivity spectra we critically assessed several approaches previously employed to describe the onset of diffusive charge dynamics and of the electrode polarization in ion conducting materials. Based on the analysis of the permittivity spectra we demonstrate that the conductivity relaxation process provides information on ion diffusion and the magnitude of crosscorrelation effects in ionic motions. A new approach is introduced which is able to estimate ionic diffusivities from the characteristic times of conductivity relaxation and ion concentration without any adjustable parameters. This opens the venue for a deeper understanding of charge transport in concentrated and diluted electrolyte solutions.

2


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I. INTRODUCTION

Fundamental understanding of mechanisms controlling ion transport in liquid and polymeric electrolytes is crucial in design of advanced materials for application in energy related technologies. The classical mechanism usually assumes a strong coupling of ion diffusion to structural relaxation (viscosity) of the liquid electrolytes.1 The same mechanism is also proposed for polymer electrolytes with segmental dynamics controlling ion diffusion.2,3 However, superionic glasses and crystals exhibit rather fast ion motions even when structural relaxation is completely frozen.4,5 Similar decoupling of ion transport from segmental dynamics has been clearly demonstrated for many polymeric materials including polymerized ionic liquids (PolyILs).6,7,8,9,10,11,12,13 Moreover, it has been suggested that the decoupling of ion transport from segmental dynamics is the most promising way to achieve desired ion conductivity in polymers at ambient conditions.11,14 Ion conductivity in electrolytes is often studied using Broadband Dielectric Spectroscopy (BDS) that provides accurate measurements in an extremely broad frequency range.15 However, the direct measure of conductivity provides a convoluted measure of the number of ions involved and their diffusivity/mobility.16 This always complicates the BDS data analysis and does not allow direct measure of ion diffusivity. The latter can be obtained from pulsed field gradient nuclear magnetic resonance (PFG NMR) measurements.17 However, this technique covers high diffusivities only.18 There were several attempts to estimate ion diffusivity in ionic systems from analysis of the BDS data alone.16,19 These studies employed Electrode Polarization (EP) effect11,20 and conductivity spectral shape

σ(ν)21 to gain information on ion diffusivity. However, analysis of EP revealed significant quantitative disagreements with NMR data,22 while analysis of the conductivity spectra required assumptions of large ion jump distances to get an agreement with the NMR data.16 BDS spectra of ionic conductors has another important contribution, the so-called conductivity relaxa-

3



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tion process.23,24,25 It is usually analyzed in dielectric modulus presentations,15 although it is also obvious in permittivity spectra. The mechanism of this relaxation process remains unclear, and the Random Barrier Model (RBM) is often employed to analyze the conductivity spectra in this frequency range.26,27 However, in its original version it does not reproduce the permittivity relaxation spectra well12,28,29 and

Table1. Parameters Characterizing the Systems Investigated in This Work

material monomer LTg-PolyIL HTg-PolyIL

Tg (K) 18133 261 344

ρ (g/cm3)

σ∞ (S/cm)

1.43 1.48 1.61

1.7 0.6 1.9

B (K) 895 1620 1890

T0 (K) 152 199 238

ε∞ 3.4 2.5 5.2

np (m-3) 2.05×1027 1.72×1027 2.42×1027

ri (nm) 0.49 0.52 0.46

λ (nm) 0.24 0.46 0.52

only provides estimates of the characteristic frequency of the process, leaving open the question of its characteristic length scale. In the present contribution we analyze diffusivity of ions in several liquid and polymeric electrolytes employing dielectric spectroscopy combined with PFG NMR and differential scanning calorimetry (DSC) measurements. Using these data we analyzed several model approaches and demonstrated their deficiencies. We propose a modified model approach that provides direct estimate of the ion diffusivity from the conductivity relaxation spectra without any adjustable parameters. This model opens exciting opportunity for direct measurements of ion diffusion in a broad range of diffusivities based on BDS spectra alone, and might be instrumental in detailed analysis of mechanisms of ionic conductivities in liquid and solid ionic conductors and electrolytes.

II. EXPERIMENTAL DETAILS

For our model studies we chose a classical ionic liquid (IL), and two polymerized ILs. The latter have the advantage of being single ion conductors that simplifies the model analysis. 1-Butyl-3methylimidazolium bis(trifluoromethylsulfonyl)imide (BmimTFSi) (Fig.1) was purchased from IoLiTech (purity 99%) and measured as received. Two TFSi--based PolyILs (Fig.1) were synthesized in 4


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our laboratory at ORNL. All three systems have the same anion, TFSi-, and the imidazolium-based cation. The only difference between the two PolyILs is the structure of their cationic side groups. Special attention has been paid to the careful drying of the samples before any measurements.

Fig. 1 Molecular structure of the systems investigated in this work.

The BDS measurements were performed in a frequency range 10-2 Hz to 107 Hz using an Alpha-A analyzer from Novocontrol and voltages amplitude of 0.1 V. Prior to each spectrum accumulation the temperature was stabilized within 0.2 K by a Quattro temperature controller also provided by Novocontrol. The IL was placed between two parallel plates made of brass and separated by a Teflon spacer ring with a thickness L = 125 µm. The two PolyILs were pressed as uniform thin sheets which were transferred into a spacer-free sapphire-invar cell30 with the separation between electrodes L = 47 µm. For DSC investigations PolyILs were hermetically sealed in aluminum pans and transferred to a Q1000 analyzer from TA Instruments. An empty pan was used as reference. The values of the glass transition temperature Tg (Table 1) were determined at the midpoint of the endothermic step31 recorded at a constant heating rate of 10 K/min. No signs of crystallization were noticed in the investigated Trange extending from 200 °C down to temperatures well below Tg. Also first indications of irreversible degradations occurred for temperatures well beyond the range of the present study. 5



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It is interesting that the increase in side group chain length in PolyILs (Fig.1) leads to a significant decrease in Tg (Table 1). This behavior is consistent with the earlier reported decrease in PolyIL’s Tg with an increase of molecular volume of the monomer+anion pair,13 and might be explained by the high flexibility of the side group which acts as a plasticizer. For simplicity, the two PolyILs will be identified as “HTg-PolyIL” for the system with higher Tg and “LTg-PolyIL” for the one with lower Tg, respectively, while the IL will be called “monomer”. For PFG NMR diffusivity measurements HTg-PolyIL was smashed to a rough powder and placed in sealed NMR glass tubes. LTg-PolyIL was initially dissolved in a small quantity of dried acetone. The solution was then transferred in unsealed NMR tubes which were placed overnight under 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 VarianS Direct Drive Wide Bore spectrometer equipped with a z-gradient DOTY Scientific, Inc. probe and using a stimulated-echo PFG sequence.32 Sixteen field gradient values linearly increased from 2 and up to 1000 G/cm as needed were used. Field gradient pulse durations δ were 1-3 ms and diffusion delays ∆ 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 were recorded. For the monomer (IL) the corresponding values for the glass transition temperature Tg (Table 1) and for the temperature-dependent self-diffusion coefficients D were taken from literature.18,33 For the two polymeric materials the room-temperature mass densities ρ were determined using pycnometry and the corresponding results are included in Table 1.

III. RESULTS AND ANALYSES

A. The dc Conductivity Regime. The conductivity spectra obtained at several selected temperatures 6


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for the monomer and the two polymerized ILs are displayed in Fig. 2. The response of both PolyILs is similar to the one usually exhibited by any other ionic conductors including ILs.15,16,13,34 For all three systems the spectra in Fig. 2 are dominated at high temperatures by the dc conductivity plateau followed by a decrease in amplitude at lower frequencies signaling the onset of electrode polarization effects. As temperature decreases both the dc level and the frequency marking the crossover to EP shift monotonously to lower values, as the overall dynamics progressively slows down. Below certain temperature an upturn in conductivity values becomes visible at high frequencies, and by further cooling the latter contribution extends to lower and lower frequencies as both EP and DC regimes shift out from the investigated dynamic range. 10

-2

10

-5

270

monomer

240

220 205

10

-8

195

-11

10

conductivity σ' (S/cm)

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180

188

(a) -14

10

10

-5

380

LTg-PolyIL

320

300

10

-8

285 275

-11

10

265

245

(b)

-14

10 -3 10 10

-5

10

-7

10

-9

473

HTg-PolyIL 423

393

363 343 323 303

(c)

-11

10

10

0

10

2

10

4

frequency ν (Hz)

7


10

6


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Fig. 2 Conductivity spectra of (a) monomer, (b) LTg-PolyIL, and (c) HTg-PolyIL at few selected temperatures indicated by the corresponding numbers in K. Solid lines are fits to the eq. (7), see text for details.

The values of dc (or steady-state) conductivity σ0 can be directly read out as the amplitudes of the intermediate σ’(ν) plateaus (Fig. 2). The σ0 results for the three ionic conductors are plotted as functions of inverse temperature in Fig. 3. The temperature evolution of σ0 for the monomer can be well described in the entire T-range by the Vogel-Fulcher-Tammann (VFT) expression: 

B .   T − T0 

(1)

σ 0 = σ ∞ exp  −

The VFT parameters σ∞, B, and T0 obtained from the fits (Fig. 3) are included in Table 1. Fig. 3 also demonstrates much lower conductivity in the two PolyILs than in the IL. This is caused by the lower segmental mobility in PolyILs that is also reflected in their higher Tg, as demonstrated by the DSC measurements (Table 1). However, the conductivity in PolyILs exhibits a clear crossover from a VFT-like temperature dependence at T > Tg to an Arrhenius-like behavior σ0 ∝ exp(-E/T) at T < Tg . This crossover reflects the change in the transport mechanism of the anions which are diffusing through a viscous melt above Tg, and through a frozen glassy matrix below Tg, respectively. The value of conductivity at Tg characterizes the degree of decoupling of ion transport from structural (segmental) dynamics.35 It should be σ0(Tg)~10-14-10-15 S/cm for systems with ion motion strongly coupled to structural relaxation.36 In particular case of HTg-PolyIL σ0(Tg)~10-8 S/cm (Fig.3) suggests that anion diffusion rate is ~106-107 faster than the rate of segmental and cation dynamics.

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-2

10

dc conductivity σ0 (S/cm)

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-4

10

monomer

-6

LTg-PolyIL

10

-8

10

-10

10

HTg-PolyIL

1/Tg

-12

10

-14

10 0.002

0.003

0.004

0.005 -1

inverse temperature 1/T (K )

Fig. 3 Arrhenius plot of dc conductivity for the three systems considered in this work. Solid lines are VFT fits (eq. 1) and dotted lines correspond to Arrhenius laws. Vertical dashed lines correspond to reciprocal values of the glass transition temperatures as determined from DSC measurements.

For establishing a connection between the macroscopic conductivity σ0 and the single-particle diffusion constant D, one may proceed with the definitions of the two coefficients as provided by the statistics of stochastic processes. According to the Green-Kubo formalism,37,38 σ0 =

1 3Vk BT

∞

∫

r r J(0) J(t ) dt

(2)

0

and ∞

D=

r 1 r vi (0)vi (t ) dt , ∫ 30

(3) r

r

r

r

where V represents the sample volume, kB Boltzmann constant, while J(0) J(t ) and vi (0)vi (t ) are the current-current correlation function and the single-particle velocity autocorrelation function, respectiver

N

ly. Since by definition J = q∑ sgn(qi )vri , where q is the charge, sgn is the sign function and N the total i =1

number of charge carriers in volume V, the two correlation functions are connected via: 9



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r r J(0) J(t ) = q 2

N

r

N

r

∑ [sgn(q )v (0)]∑ sgn(q )v (t) i

i

i =1

j

j

=

j =1

.

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(4)

r r = Nq 2 H vi (0)vi (t )

The proportionality factor in eq. (4) N

N

r

i

H = 1+

r

∑∑ [sgn(q )v (0)] sgn(q )v (t) i

i =1 j ≠i

N vi (0)vi (t)

j

j

accounts for the contribution of the cross-correlated velocity

term relative to the contribution of the self-correlation velocity term, and in the present context is the inverse of what is generally known in literature as Haven ratio.16,39 Combining eqs. (2), (3), and (4), one obtains the generalized Nernst-Einstein relation: σ0 =

n * q2 nq 2 D= Dσ , k BT k BT

(5)

where n=N/V is charge carriers density, while n* = n×H and Dσ = D×H are regarded as effective number density and effective diffusivity constant, respectively, for the charges contributing to conductivity. Theoretical estimates of H in non-crystalline materials are by far nontrivial,39,40 but the amplitude of cross-correlation terms can be estimated experimentally by combining results for the self-diffusion coefficient D provided by NMR and for Dσ obtained via eq. (5). H equals unity is a crude approximation for most of conductors and implies that the species involved in diffusion are also the ones transporting charge in an uncorrelated manner. This situation may occur for highly diluted electrolytes41,42 in which the mean separation between carriers far exceeds their Bjerrum length.43 For highly and moderately concentrated electrolytes, such as polymer electrolytes or ILs, the cross-correlation terms are not negligible, hence, H < 1 generally holds. This situation is usually explained by considering migration of ionic pairs in addition to “free” charges,44 as the oppositely charged ions [sgn(qi)sgn(qj)=−1] moving together (vi=vj) contribute to diffusion but not to conductivity. We want to note here that the idea of “free” ions for so concentrated ionic liquids might be conceptually incorrect, because each ion is always surrounded

10


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by several counterions. B.

The Ionic Relaxation Regime. The increase in conductivity σ(ν) at frequencies above the dc

plateau (Fig.2) is generally attributed to the onset of the so called hopping regime.16 As nicely revealed by the time dependence of mean square displacement,45 at short times (which correspond to high frequencies) the particles probe the energy landscape of the disordered environment in a sub-diffusive manner. Only at longer times (lower frequencies) their dynamics become fully diffusive and give rise to the dc conductivity plateau.42 In the conductivity spectra the transition between the two regimes is rather smooth (Figs. 2 and 4a). The response of conducting materials is usually discussed in terms of conductivity spectra,15 and much less attention is paid to the analysis of permittivity data which may reveal some spectral features not obvious in σ(ν) spectra. For example, Fig. 4b reveals that the upturn in conductivity corresponds to a relaxation process. In this respect PolyILs share high similarities with other amorphous23,24and crystalline ion conductors.25 Starting from highest frequencies where ε’ approaches values corresponding to ultrafast resonance processes (ε∞), the decrease in ν leads to a small but significant increase of ε’ before the latter rises strongly due to electrode polarization. In this intermediate frequency range ε’(ν) displays a sigmoidal shape resembling the behavior of reorienting dipoles and its inflexion point, indicated by the vertical dashed line in Fig. 4, occurs at a frequency at which the conductivity starts to become dispersive (Fig. 4). In order to reveal the contribution of this relaxation mode in the dielectric loss spectrum one may either subtract from the raw ε” data the conductivity term σ0/(ωε0), or employ the so-called “conductionfree” approximation46 ε” ~ -(π/2)∂ε’/∂ln(ω). Applied to our systems both procedures render similar results (not shown), and we included as crosses in Fig. 4(c) only those obtained with the derivative method. The treated data reveal the presence of a submerged relaxation peak with a characteristic frequency ν i

or time τ i=1/(2πνi). 11



The characteristic frequency marking the crossover between the diffusive and the sub-diffusive regimes has been used in many publications to estimate the diffusion coefficient and/or the effective number of carriers in conductive materials. According to the formalism introduced by Almond and West,47 the crossover frequency represents the hopping (or jump) rate of the charge carriers performing an ele-

-9

σ'/10 (S/cm)

mentary step in the diffusion process. 15

(a)

HTg-PolyIL

12

T = 333 K

9 6

(b)

15

ε' 10

ε∞

5 2 10

(c)

ε" 101 0

10

tan δ

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(d) 0.06 M"

1

10

0.03 0

10

0

10

1

10

2

10

3

10

4

10

5

10

0.00 6 10

frequency ν [Hz]

Fig. 4 Different representations of the impedance data recorded for HTg-PolyIL at 333 K, namely conductivity (a, black circles), dielectric storage (b, black open squares), dielectric loss (c, black open stars), tangent of loss angle (d, left axis, black dots), and modulus loss (d, right axis, green open triangles). In (a) the black dashed line is a fit to the eq. (6), and the vertical black dotted line corresponds to ν = ν*. The red solid lines are in (a), (c) and (d) calculated using the parameters obtained from the fit of the spectrum in (a) to the eq. (9). The black solid line in (b) is a fit based on eq. 9 with two additional storage contributions: a power-law accounting for EP at lowest frequencies and a constant term for ε∞, the 12


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latter being represented in this frame by the horizontal black dashed line. In (c) the crosses are obtained using the derivative of ε’ spectrum (see text for details), revealing the presence of a relaxation peak and the onset of EP at frequencies below 100 Hz. In (b) and (c) the blue dotted lines are calculations based on RBM using eq. 7, and the red dashed lines are the predictions of RBM based on eq. (9). The vertical dashed line in (a)-(d) indicates the position of the inflexion point displayed by ε’(ν) corresponding to ν =

νi. Similar spectra are obtained for all other studied materials (not shown).

In the original version of this approach47 the conductivity spectrum is fitted with the popular Jonscher’s expression:48 σ '(ν ) = σ o 1 + (ν / ν *) a  .

(6)

The critical frequency ν* is related to the ion hopping time extracted as τI = 1/(2πν*). Lacking theoretical foundations, expression (6) containing three variables provides an excellent fit of the dc-ac conductivity regime for HTg-PolyIL (black dashed line in Fig.4a). Similar fit quality is obtained also for LTgPolyIL and for the monomer (not shown). A theoretical alternative to describe the conductivity relaxation spectra was proposed by the Random Barrier Model (RBM).26,27 RBM considers non-interacting charge carriers performing hopping on a simple cubic lattice. A broad distribution of energy barriers governs the charge transport in amorphous conductors and that the carriers must overcome a certain ‘percolation’ barrier in order to exhibit random diffusion.26,27 The hopping rate corresponding to this threshold barrier determines the characteristic frequency marking the onset of dc conduction towards lower frequencies. Derived within continuous time random walk approximation, the original simplified solution of the RBM model26 for the complex conductivity reads: 

, i 2πντ RBM  ln(1 + i 2 ) πντ  RBM 

(7)

σ * (ν ) = σ 0 

13



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with σ0 the dc conductivity and τRBM the percolation time. Using only two variables, eq. (7) describes well the conductivity spectra of the monomer and the two PolyILs (Fig. 2). The results for σ0 obtained from the fit are practically indistinguishable from the values obtained via the model-independent evaluations of the amplitude of the dc plateau (Fig. 3).

C.

The Electrode Polarization Regime. Blocking effects occurring in impedance measurements at

the interface of the probing electrodes49 have been considered for a long time as artifacts. Consequently, their strong contributions at low frequencies are often masked in conductivity spectra since their reproducibility depends not only on the geometric factors of experiment, but also on the amplitude of the applied electrical field and the electrode’s material.21 Several approaches were employed in the past to describe the salient features of EP,50,51 however only few52,53,54 were able to provide access to information regarding the dynamics of charges in the bulk. As was recently demonstrated, the range of applicability of these models is restricted to highly diluted electrolytes,20 as they fail badly in providing reasonable diffusivities or effective number of charges for materials with high ion concentrations.22 According to the most recent version of Macdonald-Trukhan approach22,54 the diffusivity constant of charge carriers in the bulk is related to the distance between the probing electrodes L and position νδ and amplitude tanδmax of the low-frequency peak arising in the spectra of tanδ = ε”/ε (black dots in Fig. 4d). Dδ =

2πν δ L2 . 3 32 tan δ max

(8)

Using the tanδ spectra (see, e.g. Fig. 4d) and eq. (8) we estimated Dδ for the three systems studied here (Fig. 6a). Clearly, the EP model significantly overestimates the values of diffusivity for both PolyILs and the IL, as has been already discussed in several earlier publications (see, e.g. [22]).

14


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IV. DISCUSSION

A.

Applicability of RBM to Permittivity Spectra. Although original solution of RBM describes

reasonably well conductivity spectra, previously it was found that eq. (7) does not captures all spectral features of the relaxation process which is not recognizable in conductivity representation, but is obvious in permittivity spectra (Fig. 4b). Using the parameters σ0 = 5.7×10-9 S/cm and τRBM = 0.1 ms obtained from the fit of the conductivity spectrum of HTg-PolyIL at 333 K (blue dotted line in Fig.4a), we calculated the ionic contribution to the permittivity spectrum according to ε*(ν)=σ*(ν)/(i2πνε0). Adding the instantaneous permittivity (unrelated to ionic motion) ε∞, the theoretically predicted dielectric storage and loss spectra are estimated and plotted in Fig. 4b and 4c, respectively. As revealed here, eq. (7) clearly fails to describe the permittivity spectra (Figs. 4b, 4c). Similar deviations have been recently reported in other studies testing the applicability of eq. (10) for ILs28,29 and PolyILs,12 and adding processes of unspecified origin to the permittivity spectra have been proposed.28,12 However, in 2008 Schrøder and Dyre [55] provided more accurate solution for the RBM by including in the theoretical description energy barriers which are slightly above the percolation threshold. Within this new approach the eq. (7) is replaced by the following expression: ln

i 2πντ RBM σ 0  σ * (ν ) i 2πντ RBM σ 0  = 1 + 2.66  * σ0 σ (ν )  σ * (ν ) 

−1/3

.

(9)

Our analysis shows that the numerical solution of eq. (9) describes perfectly the conductivity spectrum of HTg-PolyIL (red solid line in Fig. 4a) with the new parameters σ0 =5.3×10-9 S/cm and τRBM=0.08 ms not so different from those obtained from the fit to the eq. (7). More important, the modified solution of RBM provides also a good description for the conductivity relaxation process in permittivity spectra (Figs. 4b, 4c) with a characteristic time τi ≈ τRBM, as well as the dielectric loss modulus spectra M” =

ε”/(ε’2 +ε”2) (Fig. 4d). Thus the revised solution of RBM (eq. 9) describes well the experimental spectra

15



in any chosen presentation. Fig. 5 demonstrates that this approach describes well the dielectric spectra of all three studied here systems. dielectric storage ε' (arb. units)

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Page 16 of 31

30

HTg-PolyIL 25

LTg-PolyIL

333 K

monomer

20 15 285 K 10 195 K

5 0 0

10

1

10

10

2

3

10

10

4

5

10

6

10

7

10

frequency ν (Hz)

Fig. 5 Spectra of the real part of the dielectric permittivity for the three systems studied in this work at temperatures when the ionic relaxation frequency occurs between 102 and 103 Hz. For clarity reasons the raw spectra were shifted vertically by arbitrary factors. The dashed and the solid lines are calculated using the parameters ε∞ included in Table 1 and σ0 and τRBM obtained from the fit of the corresponding conductivity spectrum to the eqs. (7) and (9), respectively.

B.

Determination of Single-Particle Diffusivity. Let us focus now on various ways to estimate ion

diffusivity from the dielectric spectra. The simplest approach for accessing diffusion constants from conductivity measurements is provided by the Nernst-Einstein equation (eq. 5). Having at hand the values for σ0 and the number density of ions one may determine the diffusivity assuming that all mobile charges contribute to conductivity. In Table 1 we included the number densities np of anion-cation pairs estimated from mass densities and molecular weights for the three systems considered in this work. For the two PolyILs np directly reflects the number of anions which are governing dc conductivity in this case. For monomer both species have similar mobilities, therefore the number density of mobile charges is considered twice larger than the np value listed in Table 1. Under such considerations we estimated the diffusivities Dσ which are plotted as filled symbols in Fig. 6(a). Here we included for comparison as 16


Page 17 of 31

open symbols the D values for TFSi- diffusivities probed directly via NMR. For monomer the crosses represent the diffusivities of BMIM cations also reported by NMR.18 Although the differences between NMR and BDS data sets do not appear large on logarithmic scales, the estimated diffusivities Dσ are systematically smaller than the ones measured by NMR, especially in the case of PolyILs (Fig. 6a). Accordingly, H < 1 holds for these two systems at least in the high temperature range covered by both techniques.

-10

10

-14

10

-18

10

-22

(a)

10 -10 10

approach I

-14

2

diffusivity D (m /s)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


10

0.17 nm

-18

10

-22

10 -10 10

(b)

0.075 nm

0.14 nm

approach II 0.24 nm

-14

10

-18

10

-22

0.52 nm (c)

10 -10 10

0.46 nm

approach III monomer

-14

10

-18

10

-22

(d) HTg-PolyIL

10 0.002

0.003

LTg-PolyIL 0.004

0.005 -1

inverse temperature 1/T (K )

Fig. 6 Temperature-dependent self-diffusion coefficients D determined from NMR experiments (open symbols, crosses) are compared with those estimated from the various approaches discussed in this work. In (a) filled symbols and solid lines correspond to Dσ and Dδ calculated via eqs. 5 and 8, respectively. In (b) filled symbols are estimations for DI obtained from eq. (10) using dP = 0.77 nm (see text for 17



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Page 18 of 31

details). Here dashed lines are predictions of the approach I using the indicated jump lengths λ, as a free parameter. In (c) dashed lines are DII estimations based on eq. (11) for the indicated jump lengths λ. In (d) filled symbols are predictions of approach III based on eq. (13). For monomer the solid line is obtained by dividing DIII by a factor of 4.

Several other models relate ion diffusion to characteristic frequency of the conductivity spectra regarded as the rate of ionic “jumps” over a characteristic distance. According to the formalism introduced by Almond and West,47 called in the following approach I, the diffusion coefficient DI is estimated considering the jump lengths to be as large as the “Pauling’s diameter” dP of ions:56 DI = d P 2 / (6τ I ) .

(10)

In order to test the predictions of the approach I we first fitted the conductivity spectra of our systems to the eq. (6) to obtain ν*. For PolyILs we considered dP as the diameter of anions while for the monomer as the average diameter of both ions dP = 0.77 nm (for TFSi- dP is 0.87 nm57 while for BMIM+ 0.66 nm, according to estimations based on Ref. 58) to estimate the DI values using eq. (10). This approach obviously overestimates diffusivity for all 3 liquids (filled symbols in Fig. 6b). Substituting dP with a free jump distance parameter λ, a good agreement with the NMR results (for monomer the agreement with the mean value of the two NMR diffusivities was imposed) is obtained with λ values indicated in Fig. 6(b). These values are not only smaller than the diameter of migrating ions, but are also system dependent. So, this approach obviously requires adjustable parameters instead of the ion diameter which is the same for the two PolyILs. More recently, it was proposed19 by Sangoro et al. that the ionic hopping time can be considered the time constant provided by the RBM, τRBM = 1/(2πνRBM). Initially it was demonstrated19 that this approach (denoted as II in the following) provides a good agreement between estimated and measured diffusivities for various ILs if (i) the ‘jump’ length λ is chosen as “comparable to the Pauling diameter”, 18


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and (ii) expression (10) is modified according to DII = λ 2 / (2τ RBM ) . Later on21 Sangoro et al. reconsidered the case of three-dimensional random walk: DII = λ 2 / (6τ RBM ) .

(11)

Direct comparison between DII and the NMR results obtained for a series of imidazolium-based ILs yielded a rather surprising result: The larger is the ion size the larger is its associated ‘jump’ length λ.21 Moreover, the results for λ reported by Sangoro et al. for ILs varied between 0.24 and 0.31 nm. These values are quite large for a single ionic jump, raising doubts of whether τRBM should be identified with a hopping time.16 Following approach II considerations, we estimated DII for our systems using eq. (11) with τRBM obtained from the fit of the conductivity spectra, and the hopping length λ as a free parameter to achieve an agreement with the NMR data (Fig. 6c). The values for λ obtained as such are included in Table 1. While for the monomer this value is close to 0.2 nm considered for other ILs,21 for the two PolyILs λ values are obviously too large to be considered as elementary jump lengths. To illustrate this problem even stronger we performed similar analysis using earlier obtained data of poly(propylene glycol) (PPG) with relatively low content 11wt% of LiTFSi salt.22 In Fig. 7(a) we present the results obtained for this system, together with the fits to the RBM. This procedure provides the temperature dependent time constants of the ionic relaxation process. In Fig. 7(b) we included the NMR diffusivities associated with the two counterions. At low temperatures both reach similar values. Using the estimated time constants and following the considerations of approach II, a good agreement with NMR data is obtained (in the commonly investigated T-range) for λ=0.9 nm, a value which obviously cannot be attributed to ion hopping.

19



frequency ν (Hz) 10

σ (S/cm)

-1

1

10

-5

(a)

-7

260 248 239

10

10

10

3

5

7

10

10

293 281 272

-9

10

230

224

11% LiTFSi in PPG

-11

10 -7

2

log10 D (cm /s)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 20 of 31

-8

(b)

-9 -10

DF, PFG-NMR

-11

DLi, PFG-NMR

-12

approach II, λ = 0.9 nm approach III

-13 3.0 3.2 3.4 3.6 3.8 4.0 4.2 4.4 -1

1000 / T (K )

Fig. 7 (a) Conductivity spectra of 11 wt% LiTFSi in poly(propylene glycol), data from [9]. The numbers indicate temperatures in Kelvin units. The red solid lines are fits to the eq. (9). (b) For the same material the NMR diffusivities measured for the anions (open triangles) and cations (open star) are compared with estimates based on approach II (dashed line) and approach III (green dots).

An alternative would be to consider that a single ion jump is not equivalent with the elementary step length in the Random Barrier Model. In systems with high ion concentration, at short times the ions are trapped in local cages formed by neighboring ions mainly carrying opposite charge. An external electric field induces a dipole moment for the ion-cage system, dipole which builds up via microscopic displacements (jumps) of the trapped ion. The external energy is stored in this polarization process, and hence permittivity exhibits a relaxation profile with characteristic time defined by the escape of the ion from the cage. Within this approach (we will call it approach III) each ion performs as many local jumps as necessary until it is able to overcome the threshold (percolation) energy barrier which corresponds to 20


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local coulombic interactions, so it can relocate in an adjacent cage and restore the overall charge equilibrium. In another words, we relate the conductivity relaxation process to the reorientation of ion pairs. For PolyILs the large difference between the temperature at which this relaxation process freezes and the glass transition temperature revealed by DSC demonstrates that this process does not reflect structural fluctuations, as was previously suggested for aprotic ILs.59 To get deeper insight into the nature of the conductivity relaxation process let’s analyze its strength ∆ε. A close inspection of Fig. 5 reveals that ∆ε has a similar value for both PolyILs and is about twice larger for the monomer. This suggests that the amplitude of the ionic relaxation might reflect the number density of mobile charges. The strength and the time constant of the relaxation process are generally considered to be connected with each other via the empirical Barton-Nakajima-Namikawa (BNN) relation σ 0τ i = pε 0 ∆ε , with p a proportionality constant close to 1. According to this relation and assuming p = 1,

∆ε for HTg-PolyIL at 333 K can be estimated using the set of parameters provided by the fit of the conductivity spectrum to eq. (9), and is ∆ε ~ 4.8. This value matches very well with the experimental observations (Figs. 4b and 5). Assuming that the conductivity relaxation process indeed reflects a dipolar reorientation of ion pairs and follows the Curie law, and using the Nernst-Einstein relation for conductivity, the BNN expression can be rewritten in different form (assuming p=1): nd µ 2 n * q2 τ i . = D 3ε 0 k BT k BT ε0

(12)

with nd the number of dipoles with dipole moment µ. Further considering that each ionic pair forms a local dipole,60 nd = n*/2 and µ = qd, where d is the interionic distance, Eq. (12) reduces to the wellknown relation for isotropic random walk d2 = 6Dτi. Although based on rough approximations, this simple exercise suggests that the relaxation strength of the ionic process indeed reflects the effective number

21



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Page 22 of 31

of charges performing conduction and in addition, that random diffusion establishes at length scales which are equal or smaller than interionic distances. Accordingly, we propose that the elementary step for the conductivity relaxation process is the distance an ion needs to move to break out of the Coulombic cage formed by neighboring ions. This radius ri can be estimated based on the chemistry of each compound as ri = [3/(4πnp)]1/3 where np is the number density of charge pairs. The ri values obtained for our systems are included in Table 1. The time scale of ions escaping their cages is the characteristic time of the ionic relaxation process τi which can be extracted model-independently from the position of the inflexion point in ε’(ν). In that case, the approach III provides estimate of the ion diffusivity without any adjustable parameters: DIII = [3 / (4π n p )]2/3 / (6τ i ) .

(13)

The diffusion coefficients estimated via approach III DIII are in good agreement with NMR results for the PolyILs, while they seems to be factor of ~4 off for the monomer (Fig.7d). This factor may be explained by taking into consideration that during τi both anions and cations have similar mobility and explore their surroundings, reducing by half the distance required for each ion to escape its transient cage. Indeed the estimated free parameter λ is ~ri/2 for the analyzed here IL (Table 1). In order to test the applicability of the eq. (13) for systems in which the interionic distances are much larger than ionic sizes we consider once again the case or PPG with 11 wt% LiTFSi. Using np = 2.3×1026 1/m3 as reported in Ref. [22] we estimated ri to be 1 nm, a value which is close to the adjusted λ parameter of approach II (Fig. 7b). In Fig. 7(b) we added the DIII values estimated for this polymer electrolyte. They are in good agreement with NMR results, indicating that the applicability of approach III could be extended to both concentrated and semi-dilute electrolytes.

C.

Inverse Haven Ratio in PolyILs. Approach III provides a good, although not perfect overlap

between estimated and experimental diffusivities (Fig. 6d). Let us assume in the following that at low 22


Page 23 of 31

temperatures D obeys the temperature dependence as presented in Fig. 6(c), where BDS data are adjusted to achieve best agreement with NMR. Using the Dσ values plotted in Fig. 7(a) we estimated the temperature dependent inverse Haven ratio H = Dσ/D for the three conductors and collected the corresponding results in Fig. 8. 100

H = Dσ/D

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


monomer LTg-PolyIL

10-1

0.002

HTg-PolyIL

0.003

0.004

0.005

inverse temperature 1/T (K)

-1

Fig. 8 Inverse Haven ratio for the three systems investigated in this work. The horizontal dashed line corresponds to H = 0.5. The open symbols are obtained using NMR data and the filled symbols are estimations based on BDS data only.

Here one may clearly observe that for all three systems H appears to be smaller than 1 in the entire Trange and strongly temperature dependent. Several points need to be mentioned here: (i) At highest temperatures H is obtained model-independently using experimental results for NMR diffusivity and dc conductivity (open symbols in Fig. 8). Disregarding the weak temperature variation in the number density of charges, the main approximations used here are that for monomer both types of ions contribute to conductivity while for the PolyILs only the anions. (ii) The strong T-dependence observed in Fig. 8 for H at lower temperatures (filled symbols) is not biased by a particular choice between approach I, II, or III, since in three cases the estimated D display similar temperature variation. What could have some impact here is that the spatial parameter considered by these approaches (dP, λ, or ri) may display some 23



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Page 24 of 31

weak T-dependence. For other ILs H has been reported to vary between 0.15 at low temperatures and 0.35 at higher T.44 However, our results for monomer reveal that H may reach even higher values of about 0.5. For ILs H is mainly interpreted as the ratio between the effective number density of charges contributing to conductivity and the total number density of charges present in the sample. The difference between these two quantities is considered to be governed by the number of ions migrating as pairs, since it is hard to conceive the existence of immobile ions in the liquid state. Usually the temperature dependence of H is used for the extraction of activation (or dissociation) energy associated to pair formation.19 Fig. 8 demonstrates that such a thermally activated approach can be taken into consideration only if data are analyzed in a restricted temperature range. Moreover, our H values saturate at high temperatures, hence a complete “dissociation in the infinite temperature limit” as recently proposed22 does not seem to be generally valid. Finally, the present results obtained for the two PolyILs also demonstrate that small H values cannot be entirely attributed to ion pairing. The large difference between the mobilities of anions and of polymeric segments containing the cations precludes their long range displacements as pairs. One possibility is that in concentrated ionic conductors the local field created by the migration of an ion will bias the motion of neighboring ions, leading to significant correlations in the dynamics of charges rather than just motion of ion “pairs”. In another words, velocity-velocity correlations of different ions is not negligible in the case of concentrated ionic systems. Further experimental and theoretical efforts are essential in solving the microscopic origin of the ionic cross-correlation terms in materials with high charge density.

V. CONCLUSIONS

In the present contribution we employed impedance spectroscopy, calorimetry, and NMR diffusivity to

24


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investigate ionic dynamics in two polymerized and one monomeric ionic liquids, all having the same anions and slightly different cations. Presented analyses demonstrate that the revised solution of Random Barriers Model presents the best description of both conductivity and permittivity spectra. Most important, we ascribe conductivity relaxation process to the dipolar reorientation mechanism caused by ions escaping from the cage formed by surrounding counterions. This model provides direct estimate of the ion diffusion from the characteristic frequency of the conductivity relaxation spectrum and knowledge of ion concentration without any adjustable parameters. This provides a simple, yet powerful tool for analysis of ion diffusivity in both concentrated and diluted ionic systems in an extremely broad temperature range not accessible to NMR technique. Our results also suggest that the large crosscorrelation effects in ionic motions might be the main reason for strong reduction of conductivity in highly concentrated ionic systems. This decrease cannot be attributed simply to migration of ion pairs, because there is no long range ion pair diffusion in the case of polymerized ILs.

*Corresponding authors. [email protected]; +49 231-755-3515 [email protected]; +1 865-974-3852

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. UT Knoxville team also acknowledges partial financial support by NSF Polymer program (award DMR-1408811). 25



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Mechanism of Conductivity Relaxation in Liquid and Polymeric

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