Correlation of the average hopping length to the ion conductivity and ion diffusivity obtained from the space charge polarization in solid polymer electrolytes

Abstract

Herein, a physical model, based on impedance spectroscopy and space charge polarization with the consideration of low frequency capacitance dispersion is presented to evaluate the parameters describing the electrical transport properties of some previously studied polymer electrolytes. Implementing the model, the complete frequency response of complex conductivity within the measured frequency range can be imitated, which enables us to appropriately evaluate the macroscopic DC conductivity, hopping frequency and double layer frequency within the measured temperature range. The temperature dependent mobile ion concentration, free-ion diffusivity and average ion hopping length have also been estimated from the analysis of the frequency dependent real and imaginary parts of the complex permittivity spectra, using the same model. Reasonable agreement of the ion diffusivity obtained from the present analysis to that obtained previously from the pulsed-field gradient (PFG) NMR measurements strongly justifies the applicability of the model to a wide variety of ion conducting systems.

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1. Introduction

The widely accepted mechanism of ion conduction in solid polymer electrolytes (SPE) is the migration of cations, formed by the ion pair dissociation, primarily through the amorphous region of the host polymer matrix by the segmental motion of the polymer chains.^1–3 But it is worthwhile to mention that the electrical properties of these ion conducting electrolytes in the low frequency range are characterized by very high capacitance (>1 μF cm⁻²) and decrease in conductance. These are interpreted^4–6 as in other ion conductors, by the build-up of space charge due to blocking of the ions at the electrodes and an accumulation or depletion of mobile carriers near the electrodes. Diffusion opposes the concentration gradients caused by the applied electric field which results macroscopic polarization. The sample may be considered as a macroscopic dipole which follows the applied alternating electric field with a lag and a space-charge relaxation phenomenon occurs. This low frequency space charge relaxation can be essentially determined by the electrical conductivity and the permittivity, representing the microscopic transport processes and the Poisson–Nernst–Planck (PNP) electrodynamics relations controlling the buildup of charges.^4,7 This accumulation of charged species near an electrode, so-called electrode polarization (EP), is universal in all ionic conductors, including aqueous solutions, polymer electrolytes, ionic liquids, and superionic glasses and is a fundamental process for applications in fuel cells,⁸ supercapacitors⁸ etc.

The fundamental problem inhibiting the understanding of ionic conductors is the incapability to explicitly estimate the free-ion diffusivity and mobile ion concentration from the measured dc conductivity. Moreover, it is also difficult to unambiguously determine the macroscopic dc conductivity of the ionic conductors in presence of the above mentioned EP effect in the experimental ac impedance data. Usually, the dc conductivity has been determined from the narrow frequency-independent region of the real part of the frequency dependent conductivity or from the linear part of the imaginary complex permittivity. But, both the processes can give erroneous results as the contribution of overlapping electrode polarization is generally not considered in these methods. Evaluation of free-ion diffusivity and concentration of mobile ions from the analysis of EP has been modeled by Coelho,⁴ Schutt,⁵ and Trukhan.⁹ Although, all these models do produce reasonable estimates for diffusivity, mobile ion concentration and diffusion coefficients in a few cases, it fails for most of the materials.^10,11 For systems with high ion concentration, the above models overvalues the free-ion diffusivity while undervalues the mobile-ion concentration with respect to that obtained from NMR.^10,11 These may be attributed to the oversimplified consideration of smooth electrode–electrolyte interfaces^5,7 and the method of determination of dc conductivity values used in these models.

In the current study, a physical model is presented that enables us to simulate the complete frequency response of complex conductivity within the measured frequency range to unambiguously determine the macroscopic dc conductivity, mobile-ion diffusivity and the concentration of mobile ions of some lithium bis(trifluoromethanesulfonyl)imide (LiTFSI) in poly(ethylene oxide) (PEO) polymer electrolyte system over a wide range of EO/Li ratio which had been studied earlier by PFG-NMR measurement.¹² The rough electrode–electrolyte interface has taken into account by considering the frequency dependent complex phase element instead of frequency independent double layer capacitance.^10,13 The ion diffusivity evaluated according to the proposed model shows reasonable agreement to the reported PFG-NMR value.¹² The average hopping length of mobile ions, evaluated from the proposed model has been correlated to the ion conductivity and diffusivity of the materials to get better insight to the general understanding of the conduction mechanism within the ion conducting materials.

2. Theory and phenomenology

As a simple model for electrode polarization, we consider the response of a salt dissolved polymer electrolyte, where cations and anions carry the same amount of charges, confined between two blocking electrodes of a parallel plate capacitor, to which a small ac voltage is applied. To analyse the observed frequency, temperature and composition dependence of the conductivity, the process can be described as the frequency dependent bulk conduction process associated with the accumulation of charges at the electrolyte–electrode interface, commonly designated as the double layer capacitive effect (C_dl).^5,7,13

The admittance of the sample can be evaluated from the equivalent electrical circuit shown in Fig. 1a which follows the classical PNP model. Here the bulk conductivity σ_b = (d/A)(1/R_b) (conductivity at much higher frequency than the region where the electrode polarization phenomena dominates i.e. at the frequency independent region of the conductivity spectra), is connected in series with the frequency independent double layer interface capacitance C_dl = Aε₀ε_s/2L_D and this series combination is in parallel with the bulk capacitance C_dl = Aε₀ε_s/d where R_b is the bulk resistance and ε_s the dielectric constant of the electrolyte. Here d and L_D are sample thickness and the Debye length (the length scale of the electrostatic double layer) respectively, satisfying the condition dL_D⁻¹ ≫ 1 and A is the surface area of the blocking electrodes. The total number density of free ions (N_c), which is the sum of the number densities of free cations (n₊) and anions (n₋), can be related the Debye length by the following equation:⁹

here q is the amount of charges carried by an ion and T is the absolute temperature. Considering the circuit shown in Fig. 1a, the effective complex conductivity of the sample within the onset frequency is given by⁵

Fig. 1 Schematic illustration of equivalent circuits to describe the electrode polarization with conduction, (a) with frequency independent double layer capacitance and (b) with frequency dependent CPE.

It is worth evident from the experimental data that the electrode polarization is frequency dependent i.e. show capacitance dispersion with frequency. To explain the observed behaviour, the double layer capacitance must be considered as frequency dependent, which also replaces the simplified assumption of plane electrode–electrolyte interfaces^5,13 in circuit Fig. 1a by the interface with roughness and pores. The model can be demonstrated by the equivalent circuit Fig. 1b, in which the double layer capacitance term has been replaced by the constant phase element (CPE) term with impedance Z_CPE(ω) = C_dl⁻¹τ_∞(iωτ_∞)^−α with 0 < α ≤ 1 and the Maxwell–Wagner relaxation time τ_∞ = R_bC_b = ε₀ε_s/σ_b = ε₀ε_s/N_cqμ to obtain the effective complex conductivity as

Considering eqn (3) with Z_CPE, the expression for the frequency dependent real and imaginary part of effective conductivity with the contribution of electrode polarization become

and

This is the expression for the frequency dependent conductivity up to the high frequency limit as the onset of electrode polarization where σ′_eff ∼ σ_b ∼ σ_dc.

When the applied frequency becomes higher than a characteristic hopping frequency ω_H, the free charge carriers may behave as dipoles with non-Debye-like absorption, the conductivity starts to disperse with frequency, characteristic of the “universal dielectric response”.^11,14 The frequency dispersion of the conductivity occurs at about the frequency ω ≈ ω_H. At higher frequency ω > ω_H both the real and imaginary part of the ac conductivity increase with frequency.¹⁵ Thus, above the onset frequency of electrode polarization, it is illustrative to express the frequency dependence of the real part of the conductivity, σ′(ω) in terms of the dc conductivity, σ_b and hopping frequency ω_H, using the Jonscher power law¹⁵ and Almond–West formalism¹⁶ as

σ′(ω) = σ_b[1 + (ω/ω_H)ⁿ] (6)

Similarly, the linear frequency dependence of the imaginary part of conductivity can be expressed as¹⁵

σ′′(ω) = Aω^s (7)

with both ‘n’ and ‘s’ lies within unity. Replacing σ_b in eqn (4) by eqn (6) and eqn (5) by eqn (7) the complete frequency response of both the components of the conductivity within the measured frequency range can be simulated. The frequency dependent real part of double layer capacitance (C_DR) is expressed as

Considering the equivalent circuit of Fig. 1a the complex dielectric function of electrode polarization is typically demonstrated by a Debye type relaxation as^4,5,7,13

In eqn (9), Δε = ε_s,EP − ε_s is the dielectric strength and ε_s,EP = ε_sd/2L_D = ε_sM is the low frequency dielectric constant in presence of EP. The characteristic double layer relaxation time of electrode polarization is given by

Considering the equivalent circuit in Fig. 1b and expressing Z_CPE in terms of τ_∞ the expression for the complex permittivity becomes⁷

The real part of eqn (11) is

and the corresponding imaginary part is

It is important to notice that the above expression should give the frequency dependence of the complex permittivity up to the onset frequency of electrode polarisation, above which the observed frequency variation should be explained by high frequency non-Debye type dielectric relaxation phenomena⁶ of the electrolytes which are not considered here.

In the case of α = 1, the CPE reduces to a double layer capacitance (eqn (9)) and the expressions for real and imaginary part of the complex permittivity, considering the circuit of Fig. 1a, reduces to respectively.

For the symmetric electrolytes, from the Nernst–Planck expression and from the definition of L_D, the bulk conductivity can be expressed as⁹

σ_b = [ε_s + Δε_EP]ε₀D/L_D² (15)

Following the above equations the ion diffusion coefficient D can be expressed as

From the microscopic description of diffusion and conduction proposed by Einstein and Smoluchowski¹⁷ for the ion conducting systems we get

where γ ∼ 1/6 is a geometrical factor for ion hopping,¹⁴ 〈λ²〉 is the time independent mean square displacement or average hopping length of mobile ions and H_R is Haven's ratio, which is determined as the proportion of the tracer diffusion coefficient to ionic conductivity dependent diffusion coefficient. The Haven ratio is a collective correlation factor that describes inter-particle correlations. For oxide glasses and other ion-conducting solids such as polymer electrolytes the Haven ratio can be seen as unity.^14,18 Using the above equations the average hopping length of mobile ions within the electrolytes can be evaluated as

〈λ²〉 = 6D/ω_H (18)

with the correlation between conductivity, diffusivity, mobile ion concentration and average hopping length of mobile ions, the temperature and composition dependent transport phenomena of the electrolytes can be illustrated.

3. Experimental

3.1. Synthesis of the materials

To establish the applicability of the model to a general understanding of the ion conduction phenomenon within the polymer electrolytes, a series of already well understood lithium ion conducting polymer electrolytes were investigated that had been studied earlier by PFG-NMR experiment.¹²

Polymer electrolytes, comprised of different poly(ethylene oxide) (PEO) (Sigma-Aldrich, M_W ∼ 9 × 10⁵) and lithium trifluoromethanesulfonylimide (LiTFSI) (Alfa-Aesar) were prepared by solution casting method in a vacuum dry box, according to the procedure described in the report.¹² Four polymer electrolytes, corresponding to EO/Li ratios 6, 8, 10 and 20 respectively were prepared by dissolving the polymer and the lithium salt (LiTFSI) in acetonitrile (Sigma-Aldrich) as and subsequently removing the solvent in a vacuum oven at 60 °C for two days.

3.2. Impedance spectroscopy

The complex capacitance and conductance measurements of the free standing electrolyte films were carried out on a Hioki 3532-50 LCR Hi-Tester from 42 Hz to 5 MHz (signal amplitude, 0.5 V). The samples were placed in a sealed stainless steel parallel plate capacitor of 8 mm diameter and 0.3 mm thickness. The measurements were performed within a temperature ranges 363 K to 283 K under dynamic vacuum. The temperature was controlled using a Eurotherm temperature controller and temperature constancy of ±0.2 K was achieved in the entire range of measurements. To examine the arbitrary errors related to the changes in the environmental conditions, pressure contact and thickness of the films etc. the experiments have been repeated with several samples. The deviations or errors involved in the obtained frequency dependent conductance and capacitance values were less than 3% with the identical nature at different temperatures.

4. Results and discussion

4.1. Frequency dependent complex conductivity

Fig. 2a and b show the typical signature in the frequency and temperature dependence of the real part of the complex conductivity σ*(ω) = σ′(ω) + iσ′′(ω) for a polymer electrolyte. The composition dependence of the different components of the complex conductivity at T = 363 K are also shown in Fig. 2c and d respectively.

Fig. 2 (a and b) Temperature and (c and (d) compositional variation of frequency dependent real and imaginary components of the complex conductivity respectively. Different regions corresponding to the electrical response due to electrode polarization and conductivity relaxation are indicated for the guide of eye. Solid lines are best fits to the combination of eqn (4) with eqn (6) and eqn (5) with eqn (7) respectively.

It is observed from the figures that the frequency dependence of ion conduction within the materials can be characterized by three mutually correlated phenomena viz. low frequency electrode polarization, intermediate frequency independent macroscopic charge conduction and high frequency conductivity dispersion. The most striking features of electrode polarization are manifested in the frequency variation of complex conductivity are emphasized by the arrows in Fig. 2b.

The phenomena come into play at a certain onset frequency (ω_on), where a minimum in σ′′(ω) can be observed first. Below the onset frequency (ω_on), σ′(ω) slowly decreases with lowering the frequency. At lower frequencies with the full development of the electrode polarization near the frequency ω_max, a peak in σ′′(ω) starts to develop (Fig. 2c and d) with a simultaneous sharp decrease in σ′(ω) (Fig. 2a and b). Above the onset frequency of electrode polarizations σ′(ω) remains frequency independent up-to a certain frequency, which, in accordance with other reports^19–21 can be ascribed to the macroscopic dc conductivity of the materials. With further increase of frequency the slope of the conductivities change in frequency and σ′(ω) approaches power law behaviour. The crossover from the frequency independent conductivity to the conductivity dispersion indicates the commencement of the conductivity relaxation phenomena.^19,20

It is observed from Fig. 2 that as the temperature decreases, the overall conductivity decreases and the onset of electrode polarization (ω_on) shifts toward lower frequencies with a reduction in the frequency range of electrode polarization. Simultaneously, the onset of the conductivity dispersion from frequency independent region also shifts towards lower frequency. This indicates that both the phenomena are strongly correlated with each other. It is interesting to observe from the real and imaginary components of the frequency dependent complex conductivity have been analysed by the combination of eqn (4) with eqn (6) and eqn (5) with eqn (7) respectively. The absolute agreement of the simulated data by the combination of the above equations with the experimental data justifies the applicability of the model to analyse the frequency dependence of the total conductivity within the measured frequency and temperature range. In this fitting, the value of ‘s’ remains close to 1 for all the measured temperatures and the frequency exponent ‘n’ remains almost temperature independent.

The temperature dependence C_DR for an electrolyte is shown in Fig. 3a. It is worthy to observe that for each of the temperatures in contrast to the frequency independent double layer capacitance C_dl considered in the PNP model (Fig. 1a), C_DR gradually decreases with frequency and attains a frequency independent constant value after the nearby frequency of ω_on. Also the magnitude of C_DR at the lower frequency gradually decreases with the decrease of temperature.

Fig. 3 (a) Temperature and (b) composition dependency of the real part of constant phase element (C_DR).

The temperature dependence of the Maxwell–Wagner relaxation time τ_∞ obtained from the above fitting are depicted in Fig. 4a. The decreasing value of τ_∞ with increasing temperature clearly demonstrates that the electrode effect dominates at higher temperature. Furthermore.

Fig. 4 Temperature variation of the Maxwell–Wagner relaxation time τ_∞ for different electrolytes.

The temperature dependence of the macroscopic dc conductivity σ_dc and hopping frequency ω_H obtained from the above fitting are shown in Fig. 5a and b respectively. It is worthwhile to mention that the variation of the conductivity with respect to the EO/Li ration show similar trend to that observed previously.^12,22,23 It is observed that all the samples exhibit a non-Arrhenius temperature dependence of the dc conductivity within the measured temperature range, similar to the previously studied identical systems.²⁴

Fig. 5 Temperature variations of (a) dc conductivity and (b) hopping frequency for different electrolytes. Solid lines are best fits to VTF equations (eqn (18)).

This non-Arrhenius temperature dependence of the conductivity has been fitted by the modified Vogel–Tammann–Fulcher (VTF) equation expressed as^20,25,26

σ_dc = σ₀ exp(−BT₀/(T − T₀)) (19)

where σ₀ is the pre-factor, B is a parameter quantifying the divergence from Arrhenius temperature dependence and also determines the fragility of the system. ‘B’ can be seen to contain, in addition to constants, two parameters, as ‘B = (constant)(Δμ/K)’, where K can be related to the density of minima on the potential energy surface, while, Δμ, determines the barrier heights separating the minima.²⁵ Higher ‘B’ corresponds to less fragile, or more Arrhenius-like, behavior.²⁰ The characteristic Vogel temperature T₀ is considered as the equilibrium glass transition temperature where molecular motions cease. The solid lines in Fig. 4a and b show the results of the non-linear least square fitting of the conductivity data to eqn (19). The best fitted parameters, obtained from the above fittings are listed in Table 1.

Table 1 Different parameters obtained from the fitting of non-linear VFT equation to the temperature dependent dc conductivity, hopping frequency and ion diffusivity

Samples	Dc conductivity			Hopping frequency			Ion diffusivity
	log σ0	T0 (K)	B	log ωH0	T0 (K)	B	log D0	T0 (K)	B
(PEO)6 (LiTFSI)	0.27	177	6.34	10.23	176	6.73	−4.22	171	6.63
(PEO)8 (LiTFSI)	0.37	186	5.70	10.57	189	5.87	−4.39	181	6.07
(PEO)10 (LiTFSI)	0.43	162	5.28	10.51	164	5.20	−4.54	186	4.51
(PEO)20 (LiTFSI)	0.38	177	6.00	10.45	175	5.99	−4.13	169	5.87

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It can as well be mentioned that the hopping frequency ω_H for all the electrolytes follows VFT temperature dependence, and the Vogel temperature and strength parameter for ω_H closely follow those of the dc conductivity process (Table 1) for the electrolytes with different EO/Li ratio, confirming that for all the electrolytes there is a strong coupling between the macroscopic ion transport, ion pair dissociation and the hopping frequency.

The observed temperature independence of the power law exponents for these electrolyte systems prompts us to analyse the scaling behaviour of the conductivity spectra of the systems, considering the hopping frequency and dc conductivity as the scaling parameters as reported earlier.²⁷ Fig. 6 depicts the scaling of the real and imaginary part of the conductivity spectra for an electrolyte at different temperatures respectively. The perfect overlap of both the conductivity spectra for different temperatures justifies the selection of the parameters. Similar nature of scaling for all the electrolytes with different EO/Li ratio clearly indicates that the physical mechanism involving the ion dynamics within the electrolytes is temperature independent.

Fig. 6 Temperature scaling of the real and imaginary part of the conductivity spectra for an electrolyte.

4.2. Determination of the ion diffusivity, mobile ion concentration and hopping length

To attain better insight to the charge transport within the electrolytes and to evaluate the contributions of mobile ion concentration N_c and ion diffusivity ‘D’ to the macroscopic dc conductivity, the temperature and composition dependent real and imaginary components of the complex permittivity data of the electrolytes have been analysed by eqn (12) and (13). Fig. 7a and b show the fitting of the frequency and temperature dependence of the real part of the complex conductivity ε*(ω) = ε′(ω) − iε′′(ω) for a polymer electrolyte. The composition dependence of the different components at T = 363 K as fitted by the above equations are also shown in Fig. 7c and d respectively. In these fitting the values of the temperature dependent Maxwell–Wagner relaxation time τ_∞ obtained from the fitting of the imaginary part of ac conductivity, as indicated in Fig. 4, have been used. Different parameters obtained from the fittings are almost same. It is interesting to observe that both eqn (12) and (13) can absolutely simulate the frequency profile of the experimental data over almost entire frequency range, in contrast to the fitting observed by the eqn (14) which could not depict the shape of the curve at lower frequency (ω < 1/τ_EP) as were reported earlier.^13,26

Fig. 7 Frequency dependence of (a) and (c) real and (b) and (d) imaginary part of the permittivity at different temperatures for an electrolyte and at a particular temperature for different electrolytes. Solid lines are best fits to eqn (12) and (13) respectively.

However, the fitting of both the components start to deviate at higher frequency and at lower temperature, where the dielectric relaxations of the Li ions coordinated polymer segments become prominent and the above equations cannot explain the same. Typically the empirical equation of Havriliak and Negami (HN) may be well fitted to experimental data for the polymer relaxations at the high frequency region⁶ which beyond the scope of this paper.

Plots of temperature dependent relaxation times for electrode polarization τ_EP obtained from the above fittings are presented in Fig. 8. As observed, τ_EP shows an identical temperature dependency to that of τ_∞ for each of the electrolytes, but an opposite composition dependency. Moreover, the value of τ_EP is found to be much higher than that of τ_∞ at each temperature for all the electrolytes.

Fig. 8 Temperature variation of relaxation times for electrode polarization τ_EP for different electrolytes.

The ion diffusivity (D) at different temperatures for different electrolytes have been estimated from eqn (16) and are shown in Fig. 9 in an Arrhenius fashion. The average of the absolute translational diffusion coefficients D_avg = D_Li⁷ + D_F¹⁹/2 estimated from the PFG-NMR measurement by Gorecki et al.¹² has been incorporated in the figure for comparison. The diffusion coefficient from EP analysis is found to be slightly higher than the values obtained from NMR experiments, but shows similar trend with respect to EO/Li ratio. Also, the ‘D’ value obtained from the current analysis is much lower than that was estimated using Debye model for identical electrolyte system by Wang et al.¹⁰ Nevertheless, it is worthy to mention that the PFG-NMR measurement evaluates the average diffusivities of both free ions and ion pairs, whereas the EP phenomenon in theory only reflects the diffusivity of free ions. Also there is a difference in the experimental and other allied conditions. Considering the above aspects, the estimated value of ‘D’ from the current model is quite reasonable as compared to the PFG-NMR value.¹² The observed non-linearity with temperature in contrast to the PFG-NMR data may be attributed to the difference in the measured temperature range.

Fig. 9 Temperature variation of ion diffusivity for different electrolytes determined from electrode polarization and PFG-NMR.¹² Solid lines are best fits to VTF equation.

Considering the relation between the conductivity (σ_b), mobile ion concentration N_c and the free ion diffusivity the temperature dependent mobile ion concentration of different electrolytes has been evaluated as shown in Fig. 10a. It is observed that with the increase of temperature the mobile ion concentration linearly increases for all the electrolytes.

Fig. 10 (a) Arrhenius temperature dependence of mobile ion concentration and (b) temperature variation of average hopping length for all the electrolyte samples. Solid lines in (a) are linear best fits.

The nonlinear variation of the data are well described by similar VFT equation as that of eqn (19). The VFT temperature dependence of diffusivity reflects the coupling of segmental motion and ion transport. Fitting parameters are included in Table 1. The estimated value of the free ion concentration is in agreement with the general expectations for polymer electrolytes, as reported earlier.¹⁰

Yet, for the electrolyte with EO/Li ratio 20, the value of N_c shows minimum suggesting incomplete dissociation of the salt within the polymer matrix. The temperature dependence of N_c is well described by an Arrhenius equation N_c = N_tot exp(−E_a/k_BT) (solid lines) with N_tot is the total ion concentration at full salt dissociation and E_a is the activation energy of the free ion concentration, signifies the energy needed to separate an ion pair (binding energy). The Arrhenius temperature dependence of ion concentration implies that the ion becomes unpaired (truly free) through a thermally activated process of fully overcoming electrostatic attraction. The concentration of unpaired ions will therefore increase with increasing temperature and the ion association seems therefore to be ridden by the electrostatic attraction between anion and cation. The activation energy for the electrolytes with EO/Li ratio 6, 8, 10 and 20 are found to be 0.1, 0.07, 0.05 and 0.13 eV respectively. Thus, for the electrolyte with EO/Li ratio 20 the ion dissociation requires much larger energy than that of the electrolyte with EO/Li ratio 10 (0.05 eV).

The above features of ion concentration are in accordance with earlier investigations for different electrolyte samples based upon the analysis of electrode polarization.^1,26,28

In this regard, it is worthwhile to mention that a correction procedure in the estimation of diffusivity and mobile ion concentration has also been proposed by different researchers^10,11,26 to accommodate the electrode roughness into the electrode polarization analysis. However, the applicability of the above correction procedure is under debate¹⁰ as that approach cannot extend to the reasonable estimation of the electrical parameters for polymer electrolytes where, the dissociation energy would change significantly with salt concentration.¹⁰ The current model pre-considers all the concerns related to the electrode roughness and hence the values of parameters estimated from the analysis are physically reasonable and far more accurate.

From eqn (18), the average hopping length of mobile ions within the electrolytes has been evaluated as depicted in Fig. 10b with temperature. It is evidenced from Fig. 10b that 〈λ²〉 sharply decreases with increase of temperature for each of the electrolytes i.e. it follows reciprocal trend to that of N_c. For the electrolyte with EO/Li ratio 20, is much higher than that of the electrolyte with EO/Li ratio 10. It has been reported earlier²⁶ that the interactions between the ions and polymer backbones transforms the crystalline regions of PEO into amorphous regions by distorting or stretching its helical conformation and local chain organization. Furthermore, the phase diagram of the (PEO)_m–(LiTfSi) system studied earlier showed that the electrolytes with m < 10 are more amorphous that that of the electrolytes with m > 10.²³ Thus, the observed difference in the crystallinity for the above electrolytes strongly affects to the ion dissociation and formation of hopping cites within the electrolytes.^12,23,26

4.3. Correlation of hopping frequency with polymer segmental motion

The free ion diffusivity significantly depends upon the hopping motion of mobile ions transferring from one ion pair to a neighbouring one associated with the segmental motions of polymer chains.

As the amorphous region progressively increases at higher temperature; the polymer chains obtain quicker internal modes in which bond rotations enhance segmental motion. As a consequence, the inter-chain and intra-chain hopping movement of ions is favoured and the ions can hop with higher frequency to avail higher conductivity with shorter average hopping length. To examine this correlation of segmental motion and hopping frequency directly we plot both dc conductivity and free ion diffusivity against the frequency of segmental motion i.e. the characteristic hopping frequency ω_H as shown in Fig. 11a and b respectively. For ion motion strongly coupled to polymer segmental motion, conductivity is expected to obey the Debye–Stokes–Einstein (DSE) equation²⁶ σ_b ∝ ω_Hⁿ. The conductivity for all the electrolytes behaves according to the ideal DSE equation with a slope nearly equal to 1 in the log–log plot (Fig. 11a, implying that the conductivity is strongly regulated by the segmental motion, i.e. the hopping frequency of the ionic clusters.

Fig. 11 (a) Dc conductivity and (b) free ion diffusivity against hopping frequency. Solid lines are best fits to DSE equations.

Similar relation may be observed for the diffusivity also²⁶ as shown in Fig. 11b. However, the ion diffusivity follows a fractional DSE type behavior with n < 1 as shown in Fig. 11b. The deviation from ideal DSE behavior for the ion diffusivity than for the conductivity can be attributed to the considerable temperature dependency of the mobile ion content, which is more significant for the electrolyte with EO/Li = 6 and 20 respectively.

5. Conclusions

The physical model, in relation to the frequency dependent conductivity and dielectric permittivity of the electrode polarization, implemented in this research, is found to be successful to explicitly evaluate different parameters quantifying the ion transport behaviour within different concentration of LiTFSi doped PEO polymer electrolytes. It has been observed that this method could yield free-ion number density and diffusivity that are in reasonable agreement with PFG-NMR measurements of the same system as reported earlier. The dc conductivity is strongly coupled with segmental motion over the entire range of ion content studied following the ideal DSE relation. However, a fractional DSE relation is found to be appropriate for the free ion diffusivity and segmental relaxation. Temperature dependence of mobile ion concentration and the reciprocal temperature dependence of the average hopping length of the charge carriers for all the electrolytes also bears the same conclusion.

Acknowledgements

The authors thankfully acknowledge the DST-FIST scheme of the Department of Physics, University of Kalyani and DST-PURSE scheme, University of Kalyani for providing the instrumental facilities.

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