## 1 Introduction
Network formation by vulcanization also affects the dynamics of elastomer chains, by imposing constraints to segmental and chain motions and causing strong connectivity between chains. In particular, crosslinking has been found to influence the segmental dynamics connected to the rubber-glass transition (\(\alpha\)-relaxation), as shown by dynamic mechanical analysis, dielectric relaxation, and \({}^{1}\)H field cycling NMR (FC NMR) relaxometry [33; 38; 39; 40; 41; 42; 43]. In the present work, rubbers vulcanized under different conditions were investigated by exploiting both methods that are routinely employed in industrial analyses (i.e., equilibrium swelling, moving die rheometry, MDR, and differential scanning calorimetry, DSC) and TD-NMR experiments, which respectively give access to macroscopic and microscopic properties correlated with the crosslink density. The final aim of the work was to find correlations between NMR observables and the crosslink density or macroscopic properties of the network that depend on this quantity. Three different \({}^{1}\)H TD-NMR methods were employed: analysis of transverse relaxation, multiple-quantum (MQ) NMR, and analysis of longitudinal relaxation times measured by FC NMR relaxometry. All NMR observables depend on the modulation of \({}^{1}\)H-\({}^{1}\)H dipolar couplings by segmental motions. In elastomers, segmental motions are quite fast above \(T_{\text{g}}\), but due to topological constraints imposed by entanglements and crosslinks, they are anisotropic. Therefore, dipolar interactions cannot be fully averaged by segmental dynamics, and weak dipolar couplings remain, which can be eventually averaged over longer temporal and spatial scales by slower polymer motions. On the other hand, isotropic motions can be undergone by terminal dangling chains and small molecules present in the rubbers. In anisotropic systems, \({}^{1}\)H transverse magnetization decay is mainly induced by residual \({}^{1}\)H-\({}^{1}\)H dipolar interactions, with effective \(T_{2}\) time constants that monotonically increase with the degree of mobility. The analysis of transverse magnetization decays acquired using different methods has been extensively used to investigate the phase and dynamic properties of polymeric materials for different applications [44; 45; 46; 47; 48]. In the case of rubbers, the presence of solid-like and liquid-like components results in a typical non-exponential decay of the transverse magnetization, which depends on network characteristics [24; 26; 49; 50; 51; 52], such as the crosslink density and fractions of network chains and dangling chain ends, although the extraction of reliable information on these parameters is not straightforward and strongly depends on both the experimental conditions and model used to fit the decay [53]. Double-quantum (DQ) and more generally MQ NMR techniques allow average residual dipolar couplings (\(D_{\rm res}\)) and their distributions, due to network inhomogeneities, to be quantitatively measured, independently of the time scale of segmental motions. The obtained residual couplings are directly proportional to the crosslink density, although a highly model-dependent procedure is required to extract \(M_{\rm c}\) from \(D_{\rm res}\) values [25; 27; 28; 29; 37]. FC NMR relaxometry has been seldom employed to measure \({}^{1}\)H longitudinal relaxation times, \(T_{1}\) (or rates \(R_{1}=1/T_{1}\)), over a broad range of Larmor frequencies (\(\
u=0.01\)-\(40\) MHz) on vulcanized elastomers and to extract information on segmental (also referred to as \"glassy\") and polymer dynamics [39; 40; 41; 42]. The relaxation rate was found to increase with the crosslink density, as also revealed by measurements at a single frequency [28].