## 3 Results and Discussion
### Protonic Conductivity  
The protonic conductivity of the nanocomposite membranes was assessed in the in-plane configuration. Figure 7 shows the impedance spectra of the two membranes collected at 80 \({}^{\circ}\)C and two different RH (20% and 98%) and normalized to the maximum in Z\({}^{\prime}\) to facilitate comparison. The spectra are in all cases dominated by one single semicircle at high frequency, with an amplitude corresponding to the ohmic resistance (\(R\)) of the membrane, which was used to calculate the conductivity. At high humidity, the membrane resistance decreases substantially, giving rise to the onset of a second contribution at low frequency, which is ascribed to the electrode impedance (that varies under variable test signal amplitude [20]). The capacitance associated to the high frequency relaxation is mostly the stray capacitance (\(C_{\text{stray}}\), ca. 20-30 pF) due to the platinum wires of the sample holder (see Reference [20], including electronic supplementary data, and Reference [23] for further details on the analysis of impedance data). In fact, the capacitance of the sample in the in-plane direction is much lower due to the small cross-sectional area determined essentially by the thickness of the membrane (\(C_{\text{in-plane}}=\epsilon_{0}\epsilon_{\tau}w\delta{L_{0}}^{-1}\), where \(\epsilon_{0}\) and \(\epsilon_{\tau}\) are the vacuum and relative permittivities, respectively, of the membrane material, and \(w\), \(\delta\) and \(L_{0}\) are the width, thickness and length of the membrane). The shift of the relaxation (peak) frequency (given by \(\
u_{0}=2\pi(RC_{\text{stray}})^{-1}\), assuming a parallel \(RC\) equivalent electric circuit) towards higher values, which is apparent in the spectrum of P(_bis_MEP)/BC_2 obtained at 98% RH, is due to the decrease in the membrane resistance, since \(C_{\text{stray}}\) remains unaltered by the external conditions. Therefore, for high humidity, the \(R\) corresponds to the Z\({}^{\prime}\) value at the minimum of \(Z^{\prime\prime}\) observed on the transition between the high frequency semicircle and the low frequency electrode plate (Figure 7). Figure 8 shows the Arrhenius plots of the in-plane protonic conductivity of both nanocomposite membranes (P(_bis_MEP)/BC_1 and P(_bis_MEP)/BC_2) measured under variable temperature (30 \({}^{\circ}\)C, 40 \({}^{\circ}\)C, 60 \({}^{\circ}\)C, 80 \({}^{\circ}\)C and 94 \({}^{\circ}\)C) and RH (20%, 40%, 60%, 80% and 98%). The data clearly demonstrate the dependence of the protonic conductivity on temperature, RH and polyelectrolyte content, with RH as the parameter responsible for the most significant differences. In fact, the protonic conductivity of membrane P(_bis_MEP)/BC_1 increased 4 orders of magnitude from 35.8 \(\times\) 10\({}^{-4}\) mS cm\({}^{-1}\) to 22.4 mS cm \begin{table}
\begin{tabular}{c c c} \hline \hline
**Sample \({}^{*}\)** & _WUI_\%** & **IEC/mmol g\({}^{-1}\)** \\ \hline BC & 121 \(\pm\) 11 & – \\ P(_bis_MEP) & – & 3.5 \(\pm\) 0.02 \\ P(_bis_MEP)/BC\_1 & 79 \(\pm\) 6 & 1.1 \(\pm\) 0.12 \\ P(_bis_MEP)/BC\_2 & 155 \(\pm\) 8 & 3.0 \(\pm\) 0.05 \\ \hline \hline \end{tabular} \({}^{*}\) See Table 1 for sample identification.  
\end{table} 
Table 4: Water-uptake (_WUI_) and ion exchange capacity (IEC) of the P(_bis_MEP)/BC nanocomposite membranes and the corresponding individual components. after raising the RH from 20% to almost saturated conditions (98%) at a temperature of 94 \({}^{\circ}\)C (Figure 8a). The impact of temperature is less pronounced with the protonic conductivity of membrane P(_bis_MEP)/BC_2 increasing less than one order of magnitude from 6.21 mS cm\({}^{-1}\) to 27.2 mS cm\({}^{-1}\) with the temperature augment from 30 \({}^{\circ}\)C to 80 \({}^{\circ}\)C at 98% RH (Figure 8b).