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The ion dynamics in the polymer matrix is examined by evaluating the ionic conductivity and is expressed by relation; (n_i is number of free charge carriers, z_i, is ion charge, and & μ_i is ion mobility). Ionic conductivity is linked with number of free charge carriers available in the polymer matrix and ion mobility. The ionic conductivity is examined via complex impedance spectroscopy (CIS) technique by applying ac signal (10-100 mV) across the cell assembly SS||PE||SS (SS refers to stainless steel electrode). From the obtained Nyquist plot (Z″ vs. Z′), bulk resistance (R_b) is extracted from the intercept on the real axis and ionic conductivity is obtained through this equation; ; where ‘t’ is the thickness of the polymer electrolyte (PE) film, A is the area of the SS electrodes and R_b is the bulk resistance.

The ionic conductivity also varies with temperature. The increase of temperature thermally activates the charge carriers and lowers the activation energy or potential barrier required for ion migration. The variation of ionic conductivity with temperature follows three behaviors depending upon temperature range: (i) Arrhenius behavior, (ii) Vogel-Tamman-Fulcher (VTF) behavior, and (iii) Williams-Landel-Ferry (WLF) behavior [17–22].

Arrhenius behavior

The increase of temperature in the polymer matrix thermally activates the charge carriers and increase in flexibility leads to fast ion migration via coordinating sites. This collectively favors the ion dynamics and Arrhenius’s behavior suggests the ion transport occurs via hopping mechanism. This behavior dominates when the temperature is lower than the glass transition temperature (T_g) [18]. To explore it further, activation energy is evaluated and the lower value of the activation energy is favorable for fast ion dynamics and hence promotes higher ionic conductivity. The activation energy (Ea) is slope of linear-least square fitting of the log σvs. 1/T plot by Arrhenius equation and is expressed as; [Here, σo is pre-exponential factor, k is Boltzmann constant].

Vogel-Tamman-Fulcher (VTF) behavior

The VTF σ vs. 1/T plot is the non-linear plot and ion transport occurs via the segmental motion of the polymer chain coupled with hopping. The ion diffusion within the polymer matrix occurs via the availability of free volume that is delivered by the polymer chains. The thermally activated charge carriers cross the potential barrier and contribute to conduction [19, 20]. The VTF equation is; . Here, σ is the ionic conductivity, A is the pre-exponential factor, B is a constant, and T_o is the temperature close to the T_g of material (where entropy is zero).