## 1 Introduction  
The vulcanization of polydiene elastomers by heating with sulfur, accelerators, and other additives is the most widely employed process in the rubber industry to obtain crosslinked materials with thermal, chemical, and mechanical properties suitable for many applications [1; 2; 3; 4], with the production of pneumatic tires being the most famous example. In vulcanization, sulfur reacts with elastomer chains forming crosslinks of different lengths (mainly polysulfide bonds, but also mono- and disulfide bonds), pendant chains, and cyclic sulfides. Usually, at the early stage of vulcanization, long polysulfide links are formed; afterwards, desulfuration and thermal decomposition may occur, which give shorter sulfidic linkages, change the crosslinking degree, and cause main chain modifications. Activators and accelerators are added in order to decrease the vulcanization time,to facilitate the rubber processing, and to improve the vulcanized product properties. In particular, accelerators are able to break the sulfur chains, so that a smaller amount of sulfur is needed for the crosslinking and shorter sulfur fragments are formed in the process, resulting in shorter sulfide bridges in the rubber [5; 6]. Accelerated sulfur vulcanizations are classified as conventional (CV, A/S = 0.1 \(\div\) 0.7), semi-efficient (SEV, A/S = 0.7 \(\div\) 2.5), and efficient (EV, A/S = 2.5 \(\div\) 12), depending on the accelerator/sulfur (A/S) ratio, with the amount of both components being given in phr (parts per hundred rubber) [2]. Generally, by increasing the A/S ratio, the efficiency of the vulcanization reaction is improved and a higher number of sulfur bridges of a shorter length are produced [1; 4; 5; 7; 8]. Moreover, pendant groups containing the accelerator may form, which constitute elastically ineffective modifications of the elastomer chains; these groups can restrict the mobility of the chain segments and stiffen the chains [4; 9; 10]. Another fundamental parameter affecting the crosslinking process during vulcanization is temperature, which influences both the vulcanization kinetics and the possible degradation reactions [11; 12; 13; 14]. In particular, it has been found that by increasing the temperature in the accelerated sulfur vulcanization of natural rubber (NR) the crosslink density decreases, decreasing the polysulfide crosslinks and redistributing the di- and monosulfide linkages [15; 16]. The macroscopic properties of rubbers required for industrial applications are strongly related to the complex structure of the three-dimensional network of crosslinks formed during vulcanization [17; 18; 19; 20; 21; 22]. This structure is defined by the number and spatial distribution of crosslinks and by their functionality and chemical nature, as well as by the presence of defects (pendant chains, loops, and chain modifications) and entanglements. These properties are in turn related to the vulcanization conditions, although in a quite complex manner [7; 8; 9; 10; 13; 14; 15; 16]. Consequently, the investigation of microscopic and macroscopic properties giving access to information on the network structure in relation to vulcanization conditions is extremely important for the optimization of processing and performance of rubbers. Information on the average crosslink density (or on the average mass between consecutive crosslinks, \(M_{\text{c}}\)) of the network formed by the vulcanization of elastomers can be obtained via different experimental approaches; the most commonly used are equilibrium swelling experiments, mechanical measurements, and proton time-domain NMR (TD-NMR) spectroscopy [1; 7; 9; 10; 13; 17; 18; 21; 23; 24; 25; 26; 27; 28; 29; 30; 31; 32; 33; 34; 35; 36; 37]. In several studies, correlations have been drawn between crosslink density measurements obtained by different methods, and origins for discrepancies have been discussed in relation to experimental uncertainties and models adopted in data analysis in the different cases [25; 26; 28; 29; 32; 37].