Plasticizer ethylene carbonate facilitates new ion coordination site in blend polymer electrolyte: Dielectric relaxation through two-parameter Mittag-Leffler function Open Access

The solid state batteries have drawn much attention for new generation sustainable energy storage devices due to their high energy density capacity,¹ leading to high performance and safety. The performance of large-scale energy storage devices depends on the proper choice of the electrolyte. The solid polymer electrolyte (SPE) possesses many advantages due to their proper shape, favorable mechanical strength, better thermal stability, good electrode–electrolyte contact, and long life cycle.² Therefore, SPE is an emerging option for the fabrication of next-generation charge storage devices promising low cost³ and overcoming the problem of leakage and corrosion at the electrode. The practical applications of SPE in the charge storage device have impeded due to their low conductivity at room temperature.^4,5 Therefore, one of the objectives of the present study is to improve the ionic conductivity of SPE by making a composite film containing a suitable salt and a plasticizer.

In the present paper, we have used the blend polymer composite of polyvinylidene fluoride (PVDF)–poly(ethylene oxide) (PEO) (1:4) along with the ammonium iodide (NH₄I) salt and ethylene carbonate (EC) as a plasticizer. The polymer composite PVDF–PEO has been widely used for macro-porous solid electrolytes (MPEs) as a backbone of Li⁺ ion batteries. PVDF is considered a good candidate for MPEs as it provides good flexibility and corrosion resistance for the safety of Li batteries. Moreover, it facilitates the dissolution of lithium salt due to its strong electron-withdrawing groups (C–F) and high dielectric constant (8.4).⁶ The presence of PEO in the PDVF backbone creates pores in the membrane, facilitating the ion transport, which eventually leads to the enhancement of ionic conductivity.⁷ It was also reported that the ionic conductivity of the PVDF–PEO composite is higher than that of a single host (PEO).^5,8

The ionic conductivity of the polymer composite system is mostly discussed in terms of transport of alkali metal ions, such as Li⁺ and Na⁺. However, proton (H⁺) ion conductors have immense technological applications, particularly in the fuel cell. Therefore, a large number of proton conducting polymer electrolytes have been studied in the literature.⁹ A proton conducting polymer electrolyte based on PEO and salt NH₄I was reported for the first time by Maurya et al.¹⁰ The complexation of PEO with NH₄I increases the flexibility and hence decreases the crystallinity, which eventually facilitates the ion conduction. Although the conductivity of the PEO–NH₄I complex increases 10⁴ times the conductivity of pure PEO (∼10⁻⁷ S/cm), the system is not very stable mechanically. Therefore, the use of PDVF makes the system more robust and stable.¹¹ We have previously studied the PEO–PDVF based blend polymer composite and found the maximum enhancement of conductivity (∼10⁻³ S/cm) at 35% NH₄I concentration. The transport mechanisms of this polymer composite were envisaged via proton (H⁺) hopping from one coordination site of a tetrahedron structure of $NH4+$ cations to another. This mechanism was also explained for the polymer composite made from chitosan-NH₄I and ethylene carbonate (EC).¹² 

It has been known that one of the four hydrogen atoms in $NH4+$ cations is most weakly bound and can dissociate easily under the influence of the electric field. We intend to increase further the coordination sites of H⁺ ions in the polymer composite matrix, which will eventually enhance the ionic conductivity. Furthermore, it has been established that the EC is able to dissociate more NH₄I due to an enhancement of the dielectric constant of Chitosan-NH₄I containing EC. Therefore, it is conceivable that the polymer complex containing EC might increase the number density of mobile ions resulting in an increase of ionic conductivity. With this view, we have used EC to enhance the ionic conductivity at room temperature.¹³ Nevertheless, EC has higher dielectric constant (89.1) with higher donor number (16.4) and higher boiling temperature (248 °C),^14,15 as compared to other plasticizers, such as polyethylene glycol (PEG) and polycarbonate (PC).

The crystallinity of the composite plays an important role in ion mobility and hence in conductivity. The increase in amorphousity facilitates the segmental motion as well as ion transport through the polymer matrix and hence enhances the conductivity.¹⁶ We have reported in our previous study that the blending of PEO and PVDF reduces the overall crystallinity and also enhances the conductivity.¹¹ We have also found the optimum conductivity at a PEO–PVDF ratio of 4:1 with 35 wt. % of NH₄I. In the present study, we have used this composition with varying amounts of EC. Although the polymer composites have been extensively studied, the understanding of the mechanism of charge carrier transport and relaxation behavior still remains an open question. In the present study, we have explained the efficient proton conduction mechanism via EC. The experimental decay function has been derived by Laplace transforming the modulus function. Several models of dielectric relaxation were employed to fit the experimental decay function over a particular time scale.^17–19 The Kohlrausch–Williams–Watts (KWW) function is one such widely used function to fit with the experimental electric decay function^17,20,21 A novel approach using a fractional calculus has been described to generate a relaxation function, namely, Mittag-Leffler function (MLF). The Mittag-Leffler function is a generalized form of the exponential function.²² The histogram function of the Mittag-Leffler function is a resultant sum of the Debye and non-Debye decay functions with different decay rates.²² The MLF has been found to best describe the relaxation behavior compared to the earlier used KWW function.

PVDF [molecular weight (M_w) = 5.34 × 10⁵], PEO (M_w = 10⁵), and EC (M_w = 88.06) were obtained from Sigma-Aldrich Pvt. Ltd. The ammonium iodide (NH₄I) salt and the N,N-dimethylformamide (DMF, 99.5% purity) solvent were procured from Merck.

A solvent free self-standing film is prepared by the solution casting technique. A required amount of PEO, PVDF, and salt (NH₄I) are dissolved in the DMF solvent and magnetically stirred at 70 °C temperature. Then, an appropriate amount of EC is mixed with this solution. The total solution is stirred at 70 °C for 3 h and continued up to 12 h at room temperature to get a homogeneous solution. Then, the homogeneous solution is deposited on a glass Petri dish. The Petri dish is kept in a vacuum oven at 100 °C for 24 h to remove the solvent. This procedure results in a formation of the free self-standing film, which can be preserved in the vacuum desiccator for, further, characterization. The final composition of the film is 80 wt. % PVDF, 20 wt. % PEO, and 15 wt. % NH₄I in which the concentration of EC is varied from 0 wt. % to 90 wt. % of the total polymer of blend composite. Four different EC concentrations have been presented here to prepare the composites. The sample nomenclature is given in Table I.

The microstructural property of SPE can be analyzed by FTIR spectroscopy. IR spectroscopy is done by Bruker, Tensor II in the range of wave number from 550 cm⁻¹ to 4000 cm⁻¹. This experiment is performed in the attenuated total reflection (ATR) mode to collect the output spectra. Background correction has been done under air atmosphere. The significance of the individual band is analyzed by using the previous literature.¹¹ The deconvolution method is used by using the Origin software to identify the individual contribution of each band when two peaks overlap each other. We identify the position, new peak appearance, and variation in the normalized intensity of the infrared spectrum in order to understand the miscibility and the interaction among the component polymers in the presence and absence of EC.

The dielectric property of the SPE is analyzed by the impedance spectroscopy (IS) method. Impedance spectroscopy is carried out by using the Agilent 4294A-precession impedance analyzer with a frequency range of 40 Hz to 20 MHz. The self-standing film has been sandwiched between two copper electrodes for IS measurements. The DC conductivity of SPE is calculated by using the Cole–Cole plot (Z′ vs Z″), where Z′ and Z″ are the real part and imaginary part of complex impedance, respectively. The dielectric relaxation property has been discussed from the result of impedance spectroscopy.

Dielectric relaxation refers to the reorientation of electric dipoles or displacement vectors when the time dependent electric field is applied. Such an adjustment of dipoles is not instantaneous but rather occurs over a period of time, known as relaxation time. If the electric field is suddenly removed or changes its polarity, the polarization starts to decay due to the thermal motion and is governed by a relaxation function or decay function ϕ(t), defined as P(t) = P(0)ϕ(t), where P(t) is the time dependent polarization vector.²⁴ The complex electric modulus |M|, defined as the inverse of complex relative permittivity ϵ^* (=ϵ′ + iϵ″), characterizes the dielectric behavior of polymeric materials. ϵ′ and ϵ″ are the real and imaginary parts of complex permittivity. A detailed theory for obtaining the complex electric modulus from the impedance spectroscopy measurement has been discussed in our previous paper.¹¹ The electric modulus function can be represented as

where M′ and M″ can be written as

The complex dielectric modulus function M^*(s) is connected with the relaxation function by a very simple relationship,^23,24

where M_∞ is the electric modulus at a very high frequency limit. Equation (4) can be rewritten using the Laplace transform as

where $L$ represents the Laplace transform,

ϕ(t) can be obtained from the inverse Laplace transformation of Eq. (5),

Let

where s = σ + iω.

We choose Re[s] = 0, i.e., σ = 0 to avoid any singularity (Pol) of the function M^*(s). M^*(s) = Re[M(s)] + iIm[M(s)], so Eq. (7) can be written as

We can write

According to the Berberan–Santos method, the inverse Laplace transform of X(s) takes the following form:^25–27 

It is important to mention that the response function ɛ^*(ω) = Re[ɛ(ω)] + iIm[ɛ(ω)], M^*(ω) = Re[M(ω)] + iIm[M(ω)] follows the causality principle, and their real and imaginary parts follow the Hilbert transformation to each other.²⁸ This gives Eq. (10) in the following form:

or

Both Eqs. (11) and (12) are the experimental decay function in the time domain. The nature of the decay function determines the mode of relaxation of charge carriers. In the present context, we are interested to establish the non-Debye relaxation function that will be appropriate for the experimental decay function. Here, we shall now discuss the two-parameter Mittag-Leffler function, which has not been used to fit with the experimental decay function in any of the previous literature. The two-parameter Mittag-Leffler function can be represented as^22,28

where α and β are the two parameters.

Similarly, we can write

where k is a constant and the γ value lies between 0 and 1.

EC induced microstructural variation has been studied using FTIR spectroscopy. IR bands are normalized with respect to the 876 cm⁻¹ band²⁹ for comparison. The FTIR spectrum of SPE without a plasticizer (EC0) has already been reported in our previous work.¹¹ Significant changes in the FTIR spectrum in the presence of EC are observed, as depicted in Fig. 1. In the presence of EC, there is a clear signature of three new bands at 717, 1774, and 1801 cm⁻¹ [Figs. 1(b) and 1(c)], corresponding to the C=O bending¹³ and stretching of EC,¹⁴ respectively. In comparison with pure EC, the C=O stretching bands were reported at the positions 1770 cm⁻¹ and 1798 cm⁻¹, whereas the position of the C=O bending band was found to be unaltered.¹⁴ Interestingly, the intensity variation of these three bands follows a definite trend. The intensities of these three bands increase with increasing EC concentration up to 80 wt. %, and then the intensity seems to decrease at 90 wt. %. Such an intensity variation is clearly shown in the inset of Fig. 1(b). This result suggests that the plasticizer renders the optimum effect at an EC concentration of 80 wt. %. The shift in positions as well as the variation of intensities of these two stretching bands (C=O) confirms that the cation (⁠$NH4+$⁠) interacts with the carbonyl group of EC through the hydrogen bonding network. Intensities of all other bands up to the wave number 2000 cm⁻¹ in FTIR spectra of EC containing a polymer matrix show a higher value compared to EC0, indicating a significant micro-structural change in the polymer matrix containing EC.

Unlike the bands at lower wave numbers discussed earlier, the normalized intensities of two bands at wave numbers 3176 and 3450 cm⁻¹ reduce significantly in the presence of EC [Fig. 1(a)]. These bands correspond to the stretching of the hydroxyl group of PEO.³⁰ The –OH group interacts with the cation $NH4+$ through the hydrogen bond. The incorporation of EC introduces a new coordination site with the cations, leading to weakening the strength of the hydrogen bonding with the –OH group. This might be the cause of considerable decrease in the intensities. It is important to mention that the band at 3176 cm⁻¹ and 3450 cm⁻¹ is previously identified as the presence of free ion and contact ion, respectively.¹¹ The strength of the –OH vibration and, hence, the area of the band profile provide the relative proportion of free and contact ions. The percentage of free and contact ions is reckoned from the percentage of area under the individual band, as shown in the inset of Fig. 1(a). It is clearly evident from Fig. 1(a) that the area under the band (3450 cm⁻¹) corresponding to contact ions increases with increasing EC concentration to a maximum value at an EC concentration of 80 wt. %, whereas the band corresponding to contact ions follows the opposite trends. In this situation, most of the cations are attached to the carbonyl group of EC, leading to the increase of contact ions.

The ion transport phenomenon usually occurs through mobile or free ions within the polymer matrix. Furthermore, the amorphous region of the polymer matrix facilitates the movement of mobile ions.¹¹ The blend polymer containing EC shows a substantial increase in the amorphous region, as evidenced from the broad and low intensity XRD profile (see Fig. 1S of the supplementary material). It is expected that the concentration of free ions would increase, as the increase in conductivity is observed with increasing EC concentration. Therefore, the increase in contact ions with increasing EC concentration, as obtained from Fig. 1(a), is thus intriguing. Therefore, we have proposed a different ion–polymer interaction picture, leading to an enhancement of ionic conductivity. A schematic diagram describing the proposed mechanism is shown in Fig. 2. As discussed before, cations (⁠$NH4+$⁠) find a new coordination site via hydrogen bonding of the carbonyl group of EC.¹³ Therefore, such a hydrogen bonding network acts as the bridge for the cations to hop from one site to another. In other words, ion hopping takes place via EC. In this system, the maximum number of coordination sites forms between EC and cations at an EC concentration of 80 wt. %. This leads to an increase in the conductivity up to a maximum value at 80 wt. % of EC. Interestingly, above 80 wt. % of EC (90 wt. %), the DC conductivity starts to decrease. We believe that, at higher EC concentration, cations recrystallize, resulting in a decrease in the ion movement within the polymer matrix.³¹ Such recrystallization at 90 wt. % of EC has been further supported by the results on XRD and differential scanning calorimetry (DSC) (Figs. 1S and 2S of the supplementary material). Now we intend to discuss the effect of microstructural change on the relaxation properties of the system.

Figure 3 shows the variation of the room temperature DC conductivity (σ_dc) with EC concentration. The conductivity increases with increasing EC concentration and reaches its maximum value (1.2 × 10⁻⁴ S/cm) at 80 wt. % that is two order of magnitude higher than the value obtained from EC0. Above 80 wt. % of EC concentration, the conductivity tends to decrease. The hydrogen bonding network between the carbonyl group of EC and the cations facilitates the ion transport via hopping mechanisms. This efficient hopping of cations via EC throughout the polymer matrix eventually enhances the conductivity. The lowest percentage of crystallinity obtained at 80 wt. % of EC from the DSC study (supplementary material) is in consistent with the conductivity enhancement discussed before. Having said the ion transport mechanism and conductivity enhancement of the composite polymer matrix, it is important to discuss the behavior of dielectric relaxation. The relaxation phenomenon is described by analyzing the results of impedance spectroscopy.

The dispersion relation of the real part of the dielectric constant (ɛ′) gives us the net polarization of the bulk material [Fig. 4(a)]. At an EC concentration of 80 wt. %, we find a significant decrease in ɛ′ from its maximum value 2.6 × 10⁵. At other EC concentrations, ɛ′ do not show any considerable decrease with frequency, as compared to 80 wt. % of EC. The loss tangent (tan δ) spectrum [Fig. 4(b)] provides us the dielectric relaxation behavior of the polymer matrix. Figure 4(b) shows two relaxation peaks at two different frequency regimes. The sharp peak at the higher frequency regime corresponds to the amorphous region, and the broad peak at the lower frequency regime is due to the crystalline region.¹¹ 

In system EC80, it is evident from Fig. 4 that the peak at the low frequency region is shifted toward higher frequency, indicating the increase in the amorphousity of this system, which is consistent with the results obtained from XRD (Fig. S1). Such an increase in amorphousity facilitates the ion transport, which has already been discussed. Frequency dependent tan δ clearly indicates the non-Debye relaxation process in this system.¹¹ We obtained the electric modulus function spectrum using impedance spectroscopy in order to get the relaxation behavior of the composite. As discussed in the theory, the decay function was determined from the inverse Laplace transformation of the electric modulus function in the frequency domain. The behavior of the decay curves in Fig. 5 clearly rules out the possibility of simple single exponential Debye relaxation. Now, we tried to fit this curve with the standard non-Debye Kohlrausch–Williams–Watts (KWW) function available in the literature. None of the conventional non-Debye functions are fitted well with the experimental decay function. Our previous report on a similar polymer composite with the nanofiller has shown that the one-parameter Mittag-Leffler function fits well compared to the conventional KWW function.²⁹ Therefore, we have used both the one- (α) and two-(α, β) parameter Mittag-Leffler function [Eq. (14)] over the entire time scale region (Fig. 5) to fit the decay curve in the present system. However, the fitting of the experimental decay curve is best described by the two-parameter Mittag-Leffler function, compared to the one-parameter Mittag-Leffler function over the whole time scale region (Fig. 5). Figure 5 shows the fit using the two-parameter M–L function. In the M–L function, there are three unknowns, α, β, and γ. In the fit, we fixed the value of α = 0.85 obtained from the one-parameter fit and then other two parameters are allowed to vary. The best fitted value of γ and β is found to be 0.39 and 0.77, respectively.

The shape of the Mittag-Leffler function [Eq. (13)] depends on the value of two exponents (α and β). For example, for α = 0, the decay function behaves like a hyperbola and for α = 1, it corresponds to pure exponential behavior, suggesting Debye relaxation.²⁸ If the α value lies between 0 and 1, it corresponds to non-Debye relaxation.²² Figure 6 shows the variation of β and γ values for different EC concentrations. All these parameters are positive and less than one, suggesting the non-Debye relaxation behavior of the system. The non-Debye behavior suggests that the system is not purely resistive or capacitive in nature but rather a combination of both with the presence of a leaky capacitor in the system.

A systematic investigation has been presented here to show the effect of the high dielectric plasticizer (EC) on the microstructures as well as on the electrical properties of PVDF rich blend SPE. In this study, we have proposed a novel mechanism of ion transport through the polymer composite matrix containing EC. The cations form new coordination sites via the hydrogen bonding network of the carbonyl group of EC. Furthermore, cations also form hydrogen bonds with the hydroxyl group of PEO. Such hydrogen bonding networks were inferred from the FTIR study. The hydrogen bonding network acts as the bridge for the cations to hop from one vacant site to another within the polymer matrix. This eventually leads to the enhancement of conductivity up to the maximum value (1.2 × 10⁻⁴ S/cm) at 80 wt. % of EC. As expected, the amorphousity of the polymer matrix increases with increasing EC concentration and attains its maximum at 80% EC. We have estimated the relative proportion of free ions and contact ions from the area under the FTIR bands at 3176 cm⁻¹ and 3450 cm⁻¹, respectively. We have analyzed our results of impedance spectroscopy in order to gain insights into the dielectric relaxation behavior of the polymer composites. The existence of two peaks in the tan δ spectrum suggests that two types of relaxation phenomena, such as short range and long range, exist in this system. The experimental decay function has been derived from the inverse Laplace transformation of the complex modulus function. The decay curve for all EC concentrations indicates the non-Debye relaxation behavior, which led us to look for an appropriate non-Debye function. We have used the two-parameter Mittag-Leffler function, which was originally used for the visco-elastic system to fit the experimental decay curve. It has been found that the experimental decay curve is well described by the Mittag-Leffler function compared to the previously used Kohlrausch–Williams–Watts function. Exponents of the Mittag-Leffler function were found to be positive and less than one, suggesting non-Debye relaxation in this system. Our results suggest that the polymer composite is a highly conducting as well as high dielectric material, leading to the possibility of efficient energy storage devices.

See the supplementary material for experimental details of XRD, thermogravimetric analysis (TGA), differential thermal analysis (DTA), and DSC, the effect of EC on the structure of the polymer matrix in the light of XRD results, and the thermal stability of the SPE delineated from the TGA, DTA and DSC results.

S.K.P. would like to acknowledge the Department of Higher Education, Government of West Bengal for providing the research fellowship. The authors would also like to acknowledge Dr. Arup Gayen and Mr. Kalyan Ghorai, Department of Chemistry, Jadavpur University for providing the XRD facility. S.K.P. would also like to thank Dr. Dipankar Mandal and Mr. Kuntal Maity, Department of Physics, Jadavpur University for providing the FTIR facility.

The data that support the findings of this study are available from the corresponding author upon reasonable request and within the article and its supplementary material.