## 3 Results and Discussion
### Characterization by \({}^{1}\)H TD-NMR  
#### 3.2.1 \({}^{1}\)H MQ NMR Experiments  
\({}^{1}\)H MQ NMR represents one of the most quantitative methods used to determine the average values of \(D_{\rm res}\)[29; 54]. In crosslinked rubbers, \(D_{\rm res}\) is the result of the incomplete motional average of the \({}^{1}\)H dipolar interactions due to the presence of chemical crosslinks and physical entanglements, which constrain the motions of polymer segments and induce the persistence of a residual local order even at temperatures far above the glass transition. The values of \(D_{\rm res}\) for all vulcanized IR and NR samples were obtained by analyzing the initial build-up of the normalized DQ signal (\(I_{\rm nDQ}\)) using Equation (2), as described in Section 2.4. The fitting of the \(I_{\rm nDQ}\) build-up curve of IR_140_S1 is shown as an example in Figure 4a. In all cases, a linear dependence of \(D_{\rm res}\) on \(v_{\rm c,sw}\) was found (Figure 4b), with slopes of 19 and 21 Hz/(10\({}^{-5}\) mol/g) and intercepts of 120 and 103 Hz for IR and NR samples, respectively. The linear correlations found for all the crosslinked samples indicate that neither \(D_{\rm res}\) nor \(v_{\rm c,sw}\) are affected by the chemical modifications of the polymer chains, which were found to mainly occur at high vulcanization temperatures. These  
Figure 3: \(T_{\rm g}\) values measured by DSC vs. \(v_{\rm c,sw}\) for **(a)** IR and **(b)** NR samples. Solid, dashed, and dash-dotted lines represent the linear fits to the data for the IR(NR)_140, IR(NR)_150, and IR(NR)_170 series, respectively. results are in line with those reported in the literature for different kinds of crosslinked systems [25; 27; 28; 32; 69; 34]. In our case, similar slopes were obtained for NR and IR samples, as expected, since polyisoprene is the main component of NR. On the contrary, other authors found significantly different linear relationships for series of IR and NR samples sulfur-cured in the same conditions, and attributed this behavior to microstructural differences between the two kinds of polymers, which lead to the different conformational statistics and stiffness levels of the polymer chains [34]. The non-zero intercept is due to the fact that all the elastically active constraints, i.e., both chemical crosslinks and physical entanglements, contribute to \(D_{\text{res}}\), while only chemical crosslinks and trapped entanglements that do not relax upon swelling contribute to \(\upsilon_{\text{c,sw}}\)[25; 27; 32; 34]. A decay of \(D_{\text{res}}\) toward zero has been theoretically predicted in the very high temperature limit and for crosslink densities approaching zero due to large-scale chain motions [70], which are too slow at the experimental temperature to produce observable effects. An NMR crosslink density can be defined as \(\upsilon_{\text{c,NMR}}=1/(2M_{\text{c,NMR}})\), where \(M_{\text{c,NMR}}\) can be calculated from \(D_{\text{res}}\) using Equation (6) [25]:  
\[M_{\text{c,NMR}}\ =\ \frac{2\pi d_{\text{ref}}}{D_{\text{res}}}\left[\frac{kg}{mol}\right] \tag{6}\]  
This equation is derived under the assumption of Gaussian chain statistics for the end-to-end vector between constraints. Here, \(d_{\text{ref}}\) is a parameter that depends on the kind of polymer and on the theoretical model and approximations used to estimate the static-limit reference dipolar coupling (\(D_{\text{stat}}/\text{k}\)) and chain stiffness. For all calculations of \(\upsilon_{\text{c,NMR}}\), \(d_{\text{ref}}=617\) Hz was used, a value estimated from spin dynamics simulations for poly(_cis_-1,4-isoprene) [25]. The linear correlations between \(\upsilon_{\text{c,NMR}}\) and \(\upsilon_{\text{c,sw}}\) (Figure 4c) are characterized by slopes higher than 1 (1.6 for IR and 1.7 for NR), similar to those found by other authors for sulfur-cured IR and NR networks [25; 27; 32; 34]. This result can be explained by considering that values of both NMR and swelling crosslink densities are model-dependent and can be affected by systematic errors.