## 2 Materials and Methods
### Analysis of the \({}^{1}h\) FC NMRD Curves  
\({}^{1}\)H NMR dispersion (NMRD) curves were obtained by plotting \(R_{1}\) as a function of the Larmor frequency. \({}^{1}\)H NMRD curves were converted to a (non-normalized) susceptibility representation \(\chi^{\prime\prime}(\omega)=\omega R_{1}(\omega)\)[62]. The \(\chi^{\prime\prime}(\omega)\) curves present a maximum, usually observed at high frequencies and low temperatures in the case of elastomers, when the condition \(\omega\tau_{\text{s}}\cong 1\) is fulfilled, where \(\tau_{\text{s}}\) is the characteristic time of the segmental motions within the Kuhn segments related to glassy dynamics. Under the assumption that the FTS principle holds true [63; 64], master curves of \(\chi^{\prime\prime}(\omega\tau_{\text{s}})\) can be obtained by shifting curves recorded at different temperatures along their frequency axis until they overlap. The values of \(\tau_{\text{s}}\) can be determined directly from the condition \(\omega\tau_{\text{s}}\cong 1\) for the temperatures at which the \(\chi^{\prime\prime}(\omega)\) maximum is visible and from the shift factors for the other temperatures.  
Using a similar procedure, in this work the \(\chi^{\prime\prime}(\omega)\) curves of the vulcanized samples were shifted along the frequency axis until they overlapped the \(\chi^{\prime\prime}(\omega)\) curve of the corresponding unvulcanized sample, taken as a reference, to obtain \(\chi^{\prime\prime}(\omega k)\) master curves, where \(k\) is the shift factor (\(k=1\) for the reference sample). Although this procedure does not allow a direct determination of \(\tau_{\text{s}}\) values, it provides values of \(k\), which are proportional to \(\tau_{\text{s}}\) through the unknown proportionality constant \(\alpha\), which is equal for all samples based on the same elastomer. As shown in [42], at a given temperature crosslinking has the effectof slowing down glassy dynamics, so that \(\tau_{\rm s}\) progressively becomes longer by increasing the crosslink density; the same trend is expected for the \(k\) factor.  
Based on a reworked version of the Vogel\(-\)Fulcher\(-\)Tammann equation reported by [PERSON] et al. [65], polymers with the same pre-exponential factor, \(\tau_{0}\), and fragility index, \(m\), display the same behavior if Log\(\tau_{\rm s}\) is plotted as a function of (\(T/T_{\rm g}-1\)) [42]:  
\[\text{Log}\bigg{(}\frac{\tau_{\rm s}(T)}{\tau_{0}}\bigg{)}\ =\ \frac{\text{Log}^{2} \bigg{(}\frac{\tau_{\rm s}\left(T_{\rm g}\right)}{\tau_{0}}\bigg{)}}{m\Big{(} \frac{T}{T_{\rm g}}-1\Big{)}+\text{Log}\bigg{(}\frac{\tau_{\rm s}\left(T_{\rm g }\right)}{\tau_{0}}\bigg{)}} \tag{4}\]  
Equation (4) can also be written as a function of \(k\) and \(\alpha\):  
\[\text{Log}\bigg{(}\frac{k(T)}{\alpha\tau_{0}}\bigg{)}\ =\ \frac{\text{Log}^{2} \bigg{(}\frac{\tau_{\rm s}\left(T_{\rm g}\right)}{\tau_{0}}\bigg{)}}{m\Big{(} \frac{T}{T_{\rm g}}-1\Big{)}+\text{Log}\bigg{(}\frac{\tau_{\rm s}\left(T_{\rm g }\right)}{\tau_{0}}\bigg{)}} \tag{5}\]  
Therefore, if we assume that \(\tau_{0}\) and \(m\) do not change, all samples obtained using the same polymer matrix should display the same trend also if Log\(k\) is plotted as a function of (\(T/T_{\rm g}-1\)).