## 3 Results and Discussion
### Broadband Dielectric Spectroscopy
[36] presents a study of the same ionic liquid, [C\({}_{3}\)C\({}_{1}\)Pyrr][TFSI], with LiTFSI mixed with different concentrations of other organic solvents (acetonitrile, \(\gamma\)-butylrolactone, and dimethyl sulfoxide), obtaining ionic conductivities up to 3.1 S m\({}^{-1}\) for mixtures with acetonitrile. The ionic conductivity was also fitted against the molar fraction with the Casteel-Amis equation (Equation (1)) and represented in Figure 5a:  
\[\sigma=\sigma_{max}\Big{(}\frac{x}{x_{max}}\Big{)}^{a}exp\Big{(}b(x-x_{max})^{ 2}-\frac{a}{x_{max}}(x-x_{max})\Big{)} \tag{1}\]  
where \(\sigma_{\max}\) is the maximum ionic conductivity, x\({}_{\max}\) is the [C\({}_{3}\)C\({}_{1}\)Pyrr][TFSI] molar fraction at which the maximum conductivity is reached, and \(a\) and \(b\) are empirical parameters [39,40]. The fitting parameters are summarised in Table 5.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Sample** & **273 K** & **283 K** & **293 K** & **298 K** & **313 K** & **323 K** \\ \hline
**x** & & & Mixtures (1 \(-\) x) EC + x [C\({}_{3}\)C\({}_{1}\)Pyrr][TFSI] & & & \\
1.0 & 0.1138(37) & 0.200(11) & 0.308(21) & 0.397(38) & 0.645(59) & 0.77(11) \\
0.85 & 0.1878(24) & 0.3010(52) & 0.4542(99) & 0.538(12) & 0.837(32) & 1.023(43) \\
0.7 & 0.2253(31) & 0.3688(70) & 0.553(13) & 0.645(16) & 1.007(43) & 1.242(61) \\
0.5 & 0.3783(86) & 0.573(24) & 0.806(48) & 0.960(29) & 1.36(18) & 1.681(86) \\
0.4 & 0.467(11) & 0.689(22) & 0.927(35) & 1.084(40) & 1.531(75) & 1.809(98) \\
0.3 & 0.588(15) & 0.832(28) & 1.092(44) & 1.261(63) & 1.704(99) & 2.03(13) \\
0.25 & 0.618(13) & 0.875(21) & 1.153(34) & 1.306(40) & 1.783(69) & 2.026(83) \\
0.2 & 0.677(28) & 0.921(42) & 1.180(66) & 1.351(83) & 1.76(15) & 2.02(18) \\
0.15 & 0.713(15) & 0.994(29) & 1.291(43) & 1.461(50) & 1.903(87) & 2.21(10) \\
0.1 & 0.1160(21) & 0.924(76) &  
As shown, the Casteel-Amis equation provides a good fit for all temperature data series, except for the lowest IL molar fraction (x\({}_{\text{IL}}\) = 0.10) at 273 K, whose ionic conductivity is significantly lower than the expected value. After performing the measurement, the sample was found to be completely solid, which explains the observed low ionic conductivity. When comparing different temperature data sets, it is found that increasing the temperature shifts the maximum ionic conductivity to higher IL molar fractions (Table 5). This shift is attributed to the reduction in viscosity and ion aggregation, which enhances the mobility of charge carriers and causes a shift in the maximum ionic conductivity [41]. To further study the conduction mechanism and the ionic conductivity dependence with IL concentration, the 298 K data series was fitted with the pseudolattice theory [42], the resulting fit is presented in Figure 5b. This theory gives a universal curve without fitting parameters, predicting the normalised electrical conductivity of IL with polar solvents as shown in Equation (2) [43]:  
\[\frac{\sigma}{\sigma_{max}}=\xi_{IL}(2-\xi_{IL}) \tag{2}\]  
where, similarly with the previous equation, \(\sigma_{\text{max}}\) is the maximum of the ionic conductivity and \(\xi_{\text{IL}}\) is the reduced concentration, defined by Equation (3) as follows:  
\[\xi_{IL}=\frac{c}{c_{max}} \tag{3}\]  
Being c the ionic liquid concentration, in moles per litre, and \(c_{\text{max}}\) the concentration at which the maximum conductivity takes place. Even though the data follow quite well the universal curve, the fitting fails to determine the position and the maximum in ionic conductivity due to a lack of a parameter describing the high effects of high ion concentrations. To compute the effect of concentrated mixtures, an additional term can be added to the universal curve as follows:  
\[\frac{\sigma}{\sigma_{max}}=\xi_{IL}(2-\xi_{IL})+D\xi_{IL}(2-\xi_{IL})^{2} \tag{4}\]  
where \(D\) is a fitting parameter that considers the effect of high ion concentrations.