## 3 Results and Discussion
### Characterization by \({}^{1}\)H TD-NMR
#### 3.2.2 \({}^{1}\)H Transverse Relaxation by CPMG Experiments  
In crosslinked rubbers, the \({}^{1}\)H transverse magnetization decay measured by Hahn echo or CPMG experiments is mainly determined by the dephasing process due to the average residual dipolar interaction of the polymer network chains undergoing non-isotropic fluctuations between constraints; minor contributions arise from faster isotropic motions of dangling chains and possible sol components. While the latter contributions are typically described by exponential functions, different analytical functions have been proposed to describe the contributions from inter-crosslinks chains. In particular, theoretical models that combine the Bloembergen-Purcell-Pound (BBP) theory for fast motions and the Andersen-Weiss formula to take into account the residual \"solid-like\" behavior have been devised [24; 26; 50; 51]. A Gaussian decay of transverse magnetization has been predicted in the short time limit, with a characteristic time constant inversely proportional to the residual dipolar interaction, \(D_{\text{res}}\). However, these functions contain a large number of intercorrelated parameters, which prevent \(D_{\text{res}}\) values from being reliably determined [53].  
In this work, we used a phenomenological approach to the analysis of the CPMG decay curves, which were reproduced using a linear combination of a Weibullian function and an exponential function (Equation (3)) [59; 60; 61], characterized by the decay constants \(T_{\text{2,weib}}\) and \(T_{\text{2,exp}}\), respectively. The Weibullian function represents the contribution of the network chains fluctuating between constraints responsible for the short time decay of transverse magnetization, while the long time decay, ascribable to dangling chains and sol components, is described by the exponential function. Figure 5a shows, as an example, the fitting of the CPMG decay curve of IR_S1_140, which allows a satisfactory reproduction of the experimental data. The best-fitting parameters obtained for all investigated samples are reported in Tables S1 and S2. In all cases, the Weibullian component accounts for the majority of protons in the sample (81-88%) and is characterized by \(T_{\text{2,weib}}\) values of \(\sim\)1 ms. The remaining fraction of protons gives the exponential decay with longer \(T_{\text{2,exp}}\) values of \(\sim\)3 ms.  
As shown in Figure 5b, good linear relationships are found between the relaxation rate \(R_{\text{2,weib}}=1/T_{\text{2,weib}}\) and \(\
u_{\text{c,sw}}\) for all the samples. In particular, \(R_{\text{2,weib}}\) linearly increases by increasing \(\
u_{\text{c,sw}}\), in agreement with the corresponding increase in \(D_{\text{res}}\) (Figure 4b). Similar slopes of about 90 s\({}^{-1}\)/(10\({}^{-5}\) mol/g) with intercepts of about 530 s\({}^{-1}\) are obtained for the IR and NR series.  
Figure 5: **(a)** Fitting of the \({}^{1}\)H CPMG curve of IR_S1_140. Red dots represent the experimental data, while solid lines denote the analytical fitting function (red) and the single contributions of the Weibullian (cyano) and exponential (green) functions; in the inset, the same data are reported, with the solid line representing the linear fit to the first datapoints (up to 1 ms) of the CPMG curve. **(b)**\({}^{1}\)H \(R_{\text{2,weib}}\) obtained from the fitting of the CPMG curves and **(c)** CPMG initial slope vs. \(\
u_{\text{c,sw}}\). Solid lines represent the linear fits to the data.  
Since we observed a good linear trend for the magnetization decay in the 0-1 ms time interval (see the inset of Figure 5a), the slope in this range (CPMG initial slope) was tested as a model-free parameter to be correlated with the crosslink density (see Section 2.5). As for \(R_{2,\text{weib}}\), good linear correlations with \(\
u_{\text{c,sw}}\) were found for the CPMG initial slope (Figure 5c), with slopes of \(-4.3\) and \(-4.9\) s\({}^{-1}\)/(10\({}^{-5}\) mol/g) and intercepts of \(-40\) s\({}^{-1}\) for IR and NR series, respectively.