## 3 Results and Discussion
### Characterization of Dynamics  
The dynamic properties of all samples were investigated by means of FC NMR experiments for the measurement of \({}^{1}\)H spin-lattice relaxation rates (\(R_{1}=1/T_{1}\)) at variable Larmor frequencies, from 0.01 to 35 MHz. In fact, \({}^{1}\)H spin-lattice relaxation is driven by the modulation of the \({}^{1}\)H-\({}^{1}\)H dipolar couplings by molecular motions. The dependence of \({}^{1}\)H \(R_{1}\) on frequency (v or \(\omega=2\pi\
u\)), called nuclear magnetic relaxation dispersion (NMRD), can be described by a linear combination of spectral densities, \(J(\omega)\), which are the Fourier transform of the autocorrelation functions of motion [43]. For polymers far above \(T_{\text{g}}\), \(R_{1}\) dispersions are mainly governed by segmental dynamics, that is local motions within the Kuhn segment, activated at the glass transition. Segmental dynamics, also referred to as \"glassy\" dynamics, is responsible for the so-called \(\alpha\)-relaxation. Additional contributions to \(R_{1}\) dispersions arise from collective motions involving longer and longer chain portions, indicated as polymer dynamics. The contribution to \(R_{1}(\omega)\) arising from segmental dynamics can be well represented using the Cole-Davidson spectral density, while contributions from polymer dynamics result in power law dependences of the type \(R_{1}(\omega)\propto\omega^{-\gamma}\), with different values of the \(\gamma\) exponent depending on the motional regime as defined by the mostly accepted tube-reptation theory [44; 45; 46]. Therefore, FC NMR has proven to be an effective technique to characterize the dynamics of polymers far above  
Figure 2: \({}^{13}\)C DE/MAS spectra of (**a**) vSBR and (**b**) vSBR_k recorded using a recycle delay of 20 s. \({}^{1}\)H-\({}^{13}\)C CP/MAS spectra of (**c**) vSBR, (**d**) vSBR_k and (**e**) Kristalex(tm) 5140 recorded using a contact time of 0.2 ms and a recycle delay of 4 s. Spinning sidebands are marked with an asterisk; impurities are marked with a filled triangle. All spectra were recorded using an MAS frequency of 5 kHz. glass transition over a broad range of characteristic motion times [47, 48]. In particular, it has been successfully applied to obtain information on segmental and polymer dynamics in polymer melts and vulcanized rubbers [49, 50, 51, 52, 53, 54]. For this aim, \(R_{1}(\omega)\) data are usually transformed to the susceptibility representation, \(\chi^{\prime\prime}(\omega)=\omega\)\(R_{1}(\omega)\), and \(\chi^{\prime\prime}(\omega\tau_{\rm s})\) master curves are built from \(\chi^{\prime\prime}(\omega)\) curves acquired at different temperatures by exploiting the frequency-temperature superposition (FTS) principle [55, 56] to determine the values of the correlation time for segmental dynamics, \(\tau_{\rm s}\). At low temperatures, \(\chi^{\prime\prime}(\omega)\) curves show a maximum, and the high-frequency branch of the curves, which mainly arises from the contribution of segmental dynamics (\(\chi^{\prime\prime}\,_{sg}(\omega)\)), can be fitted to Equation (2):  
\[\chi^{\prime\prime}\,_{sg}(\omega)=\omega K_{CD}[J_{CD}(\omega)+4J_{CD}(2 \omega)] \tag{2}\]  
using the Cole-Davidson spectral density function:  
\[J_{CD}(\omega)=\frac{2\sin[\beta_{CD}\text{arctan}(\omega\tau_{CD})]}{\omega [1+\omega\tau_{CD}]^{\beta_{CD}/2}} \tag{3}\]  
with \(0<\beta_{\rm CD}\leq 1\). Thus, \(\tau_{\rm s}\) values can be obtained from \(\tau_{\rm CD}\) using the relation \(\tau_{\rm s}=\beta_{\rm CD}\tau_{\rm CD}\). At higher temperatures, the \(\chi^{\prime\prime}(\omega)\) curves do not show a maximum, and the frequency axis is scaled until they overlap the curves obtained at lower temperatures to build the master curve. In this case, \(\tau_{\rm s}\) can be determined as the frequency scaling factor. Here, \({}^{1}\)H NMRD curves were acquired for all the investigated samples at different temperatures in the 303-393 K range. In Figure 3a, the curves recorded at 303, 333 and 363 K are reported as examples.