Glassy and Polymer Dynamics of Elastomers by 1H Field-Cycling NMR Relaxometry: Effects of Cross-Linking

1H spin lattice relaxation rate (R1) dispersions were acquired by field-cycling (FC) NMR relaxometry between 0.01 and 35 MHz over a wide temperature range on polyisoprene (IR), polybutadiene (BR), and poly(styrene-co-butadiene) (SBR) rubbers, obtained by vulcanization under different conditions, and on the corresponding uncured elastomers. By exploiting the frequency–temperature superposition principle, χ″(ωτs) master curves were constructed by shifting the total FC NMR susceptibility, χ″(ω) = ωR1(ω), curves along the frequency axis by the correlation times for glassy dynamics, τs. Longer τs values and, correspondingly, higher glass transition temperatures were determined for the sulfur-cured elastomers with respect to the uncured ones, which increased by increasing the cross-link density, whereas no significant changes were found for fragility. The contribution of polymer dynamics, χpol″(ω), to χ″(ω) was singled out by subtracting the contribution of glassy dynamics, χglass″(ω), well represented using a Cole–Davidson spectral density. For all elastomers, χpol″(ω) was found to represent a small fraction, on the order of 0.05–0.14, of the total χ″(ω), which did not show a significant dependence on cross-link density. In the investigated temperature and frequency ranges, polymer dynamics was found to encompass regimes I (Rouse dynamics) and II (constrained Rouse dynamics) of the tube reptation model for the uncured elastomers and only regime I for the vulcanized ones. This is clear evidence that chemical cross-links impose constraints on chain dynamics on a larger space and time scale than free Rouse modes.


■ INTRODUCTION
Linear polymer melts show dynamics over a wide frequency scale, from the fast scale of local segmental dynamics and internal motions to the very slow scale of diffusive motions of the center of mass of the polymer chains. Segmental motions within the so-called Kuhn segment with characteristic time τ s are responsible for "glassy dynamics" connected with the structural relaxation or α-process (τ s = τ α ). If mobile side groups exist in the polymer, the main-chain local segmental motions may be supplemented and superimposed to reorientations of such groups. The connection of segments into chains results in slower polymer-specific collective dynamics, indicated in the following as "polymer dynamics". For polymers with molar mass M below the critical value M e (average molar mass between entanglements), Rouse theory 1 well describes polymer dynamics. Longer chains (M > M e ) form physical entanglements, and their dynamics has been described by the tube reptation (TR) model developed by De Gennes 2 and Doi and Edwards, 3 by the n-renormalized Rouse model, 4−7 and by the mode−mode coupling (MC) model. 8,9 In the most accepted TR model, a polymer chain feels the topological constraints exerted by neighboring chains, which are represented by a fictitious curved tube. All the models predict characteristic power law dependences of the segmental mean-squared displacement (MSD, ⟨r 2 (t)⟩) on time. In the TR model, four power law regions (I−IV) for the MSD are predicted in succession from high to low frequencies: regimes I (Rouse dynamics), II (constrained Rouse dynamics or local reptation), and III (reptation) reflect subdiffusive motions characterized by ⟨r 2 (t)⟩ ∝ t α with α ≤ 0.5, whereas in the fourth regime (regime IV) molecules undergo Fickian diffusion (α = 1). 10 Sulfur curing (vulcanization) is employed in tire industry to obtain rubbers with desirable mechanical properties. The formation of sulfur bridges results in permanent cross-linking of polymer chains, which imposes geometric constraints on chain dynamics, in addition to transient physical entanglements. For entangled polymer networks, Lang and Sommer 11,12 proposed a model in which a phantom network with cross-links attached to an elastic background is considered and segmental fluctuations of chain strands between cross-links are confined within a tube determined by entanglements with neighboring chains. The chain segments slide along the tube, similarly to the case of polymer melts, but reach a constant finite value of the MSD in the long time limit. Moreover, a different distribution of fluctuation amplitudes for segments in different positions of the chain was found by computer simulations for polymer networks and polymer melts. Therefore, different mode distributions could be expected for melts and networks in the Rouse regime and entangled dynamics could be hampered in networks, also depending on cross-link density. Upon vulcanization, glassy dynamics slows down because of an increase in the molecular friction coefficient. All this considered, cross-linking strongly affects polymer elasticity and other important properties in polymer technology, such as thermal properties and the glass transition behavior. 13−20 The investigation of the effects of cross-linking on elastomer chain dynamics is therefore of interest from a fundamental point of view and for establishing relationships with macroscopic properties that are important in order to design new materials for specific applications.
Nuclear magnetic resonance (NMR) offers many versatile techniques for the characterization of dynamics in polymers. Among them, 1 H field-cycling (FC) NMR relaxometry allows a broad range of frequencies to be investigated. 21−23 It thereby represents an ideal tool for the characterization of polymer dynamics, which takes place over a huge range of characteristic times. 24 In fact, this technique measures the dependence on Larmor frequency (ν or ω = 2πν) of the proton longitudinal relaxation rate, R 1 = 1/T 1 (T 1 is the longitudinal relaxation time), also referred to as nuclear magnetic relaxation dispersion (NMRD). To a fair approximation, NMRD curves reflect the spectrum of reorientational and translational dynamics of polymer segments (see the Theoretical Background) and may be analyzed to get information on both glassy and polymer dynamics. 1 H FC NMR relaxometry has been indeed extensively applied to study the dynamics of linear polymer melts, both below and above the entanglement limit, especially focusing on testing models on ideal polymers, that is, simple unbranched polymers in the molten state at favorable experimental temperatures. 24−36 In particular, 1 H FC NMR relaxometry measurements on several linear polymers have been reviewed by Kimmich and Fatkullin,24 which, at ωτ s ≪ 1, appear to reveal an universal dispersion behavior of R 1 with characteristic power law regimes R 1 (ω) ∝ ω −γ . These dispersions have been interpreted within Rouse and renormalized Rouse theories for nonentangled and entangled polymers, respectively. The 1 H R 1 dispersion behavior has been more recently reinvestigated by Rossler and co-authors. 25,[27][28][29][30][31][32]35 By exploiting the frequency− temperature superposition (FTS) principle, which applies for polymers at T ≫ T g (where T g is the glass transition temperature), 37,38 R 1 data have been transformed to the susceptibility representation χ″(ω) = ωR 1 (ω) and master curves of χ″(ωτ s ) have been built (see the Theoretical Background) and interpreted in a new fashion in order to extract contributions from glassy and polymer dynamics, to investigate the crossover from simple liquid to Rouse and to entanglement dynamics as a function of M, to confirm the existence of an universal polymer dynamics, and to disentangle contributions to relaxation arising at different frequencies from reorientational and translational motions.
Notwithstanding the importance of elastomers in materials science and industrial applications and the great utility of 1 H FC NMR relaxometry in studying dynamics, a small number of papers have been reported in the literature concerning the investigation of chemically cross-linked polymer networks using this technique. 10,39−43 In particular, in the papers by Kariyo and Stapf, cross-linked elastomers were considered as polymer melts to which sparse fixed geometrical constraints are added, and the observed differences with respect to un-crosslinked elastomers were ascribed to a slowdown of both glassy and polymer dynamics, the motional spectrum remaining virtually unchanged.
In the present work, NMRD curves were acquired over a wide range of frequencies (0.01−35 MHz) and temperatures on three elastomers of high technical importance in the tire industry, that is, isoprene rubber (IR), butadiene rubber (BR), and styrene-co-butadiene rubber (SBR) both uncured and vulcanized under different conditions. By passing to the susceptibility representation, master curves were built, which allowed dynamics to be investigated over a broad frequency range. The τ s correlation times, characterizing glassy dynamics, were determined for all samples at different temperatures and used to investigate the effects of the cross-linking density and cross-link type on the glass transition behavior. The contributions of polymer and glassy dynamics to the total motional spectrum were extracted and restrictions to polymer dynamics from chemical cross-links were highlighted for the different rubbers.

■ THEORETICAL BACKGROUND
For protons, longitudinal relaxation is governed by the modulation of the magnetic dipole−dipole interaction of 1 H− 1 H spin pairs by molecular motions. The interaction Hamiltonian results from the sum of all pairs of spins i,j in the sample, separated by vectors r ij . For each pair, the interaction term depends on the orientation of r ij with respect to the magnetic field and on the distance r ij . In the case of polymers, contributions to R 1 arise from both intrachain (or intrasegment) dipolar couplings, modulated by reorientations (R 1,intra ), and interchain (or intersegment) couplings, modulated by both reorientational and translational dynamics (R 1,inter ). Although the intrachain contribution dominates at high frequency, the interchain contribution becomes increasingly important and exceeds the intrachain one at low frequencies. 34−36 The R 1 dispersion reflects the spectrum of motions of 1 H− 1 H spin pairs and can be expressed as a linear combination of spectral densities, J(ω). The latter is the Fourier transform where K is a proportionality constant depending on the second moment of the relevant intra-or intersegment dipolar interactions. The spectral density for the reorientations within the Kuhn segment (glassy dynamics) can be phenomenologically described by the Cole−Davidson (CD) function 45 where 0< β CD ≤ 1 and the characteristic time τ CD is related to τ s by the expression τ s = β CD τ CD . According to Kimmich and Fatkullin, 24 glassy dynamics is mostly responsible for the decay of the dipolar autocorrelation function. As a consequence, the temperature at which T 1 reaches the minimum value (or maximum for the corresponding R 1 ), usually observed at high frequency in FC NMR experiments at low temperature, is predominantly determined by this component and indicates the value of τ s via the "minimum condition" ωτ s ≃ 1 (actually ωτ s ranges from ∼0.42 for β CD ⊙ 0 to ∼0.62 for β CD = 1). However, glassy dynamics covers only a restricted solid-angle range of the 1 H− 1 H interdipolar vector, so that the dipolar autocorrelation function does not decay to zero. The residual correlation decays by polymer dynamics.
For polymer dynamics, dipolar translational (C trans (t)) and rotational (C rot (t)) autocorrelation functions can be written for t ≫ τ s as proportional to power laws of ⟨r 2 (t)⟩. Thereby, depending on the chosen model, different power dependences of C trans (t) and C rot (t) on time and, correspondingly, of R 1,intra and R 1,inter on frequency are found in the different regimes. 34,35 In particular, in the TR model, C rot (t) ∝ t −1 , t −1/4 , and t −1/2 in regimes I, II, and III, respectively.
On the basis of the fluctuation−dissipation theorem, 25,28,31 the spectral density, J(ω), of thermal equilibrium fluctuations of a property is related to the linear response of that property to a weak external perturbation, that is, to a susceptibility function. In particular, the imaginary part of the susceptibility is given by χ″(ω) = ωJ(ω). This relationship is of help in the analysis of FC NMR data and in comparisons with other techniques for the investigation of dynamics that give access to susceptibility, such as dielectric spectroscopy and rheology, as pointed out by Rossler and co-authors. 35 On the basis of eq 1, one can write Considering 2ω to be not too different from ω, ωR 1 (ω) ≅ 3Kχ′(ω), where χ′(ω) is the "normalized NMR susceptibility" and 3 is a normalization factor to provide an integral over χ′(ω) of π/2. Moreover, a (non-normalized) total FC NMR susceptibility χ DD ″ (ω) = ωR 1 (ω) has been introduced to analyze FC NMR data.
Assuming the validity of the FTS principle, master curves of χ″(ωτ s ) can be built as a function of frequency reduced by the correlation time for glassy dynamics by shifting χ″(ω) curves collected at different temperatures along the frequency axis until they overlap. Under the assumption that K does not change significantly with temperature, master curves can also be obtained for χ DD ″ (ω).
Considering that polymer and glassy dynamics are statistically independent and well separated in time, the FC NMR susceptibility χ DD ″ (ω), in the following simply indicated as χ″(ω), can be expressed as the sum of glassy and polymer contributions f represents the fractional contribution of polymer dynamics to the total spectrum of motions and is given by , which is a measure of the spatial restrictions imposed on segmental motions by chain connectivity and entanglements. However, being derived from a 1 H− 1 H dipolar autocorrelation function, its value also depends on the angles (θ ij ) formed between the internuclear vectors of 1 H− 1 H spin pairs and the contour of the polymer chain (S = 1/2 (3 cos 2 θ ij − 1) S chain , where S chain is the chain order parameter).
Because the total spectrum of motions is dominated by glassy dynamics, the spectrum of polymer dynamics χ pol ″ (ω) can be correctly interpreted only after subtracting χ glass ″ (ω) from χ″(ω). χ glass ″ (ω) can be determined experimentally as the total χ″(ω) for polymers with very small molar mass, which do not undergo Rouse or reptation polymer dynamics. Furthermore, it can be theoretically expressed as a function of the CD spectral density (eq 2) as χ glass ″ (ω) coincides with χ″(ω) for frequencies larger than ≈1/τ s . Therefore, τ s values can be determined by fitting the high-frequency branch of the χ″(ω) curves, ascribable to the sole glassy dynamics, to eq 6 with J CD (ω) as in eq 2. τ s correlation times exhibit non-Arrhenius temperature dependence, as expected for the structural relaxation. The Vogel−Fulcher−Tammann (VFT) empirical equation 37

DT T T exp
is generally used to describe these data, where τ 0 is a preexponential factor, T 0 is the temperature at which the correlation time diverges to infinity, and D is a fragility factor. The glass transition temperature T g can therefore be determined as the temperature at which τ s is 100 s. 46 A measure of polymer fragility, that is, of the rapidity with which glassy dynamics slows down on cooling the polymer toward the glass transition, can be given by determining the fragility index m defined as On the basis of this equation, polymers with same τ 0 and m show a superposition of Logτ s versus (T/T g − 1) curves.
For the preparation of samples denominated IR_Sphr_150, BR_Sphr_150, and SBR_Sphr_150, IR, BR, and SBR were mixed with 3 phr (parts per hundred rubber) of ZnO, 2 phr of stearic acid, 1, 2, or 3 phr of sulfur, 3 phr of N-tert-butyl-2-benzothiazole sulfenamide (TBBS) as an accelerator, and 2 phr of N-(1,3-dimethylbutyl)-N′phenyl-p-phenylenediamine (6PPD) as an antioxidant and were vulcanized into sheets of 1 mm thickness at T vulc = 150°C. Three vulcanized IR-based samples were also prepared with approximately the same cross-link density but with a different content of mono-(C− S−C), di-(C−S−S−C), and polysulfidic (C−S x −C, x ≥ 3) bridges. To this aim, IR was mixed with 3 phr of ZnO, 2 phr of stearic acid, 2 phr of 6PPD, and appropriate amounts of sulfur and other additives ( Table 1) and vulcanized into sheets of 1 mm thickness at T vulc = 170°C . These samples are indicated as IR_29, IR_60, and IR_80, with the number indicating the percentage of cross-links constituted by polysulfidic bridges, determined using the thiol-amine analysis. 48 Samples vulcanized at the maximum degree of cross-linking were considered for FC NMR measurements.
Cross-link density, defined as 1/M c , where M c is the average molar mass between two adjacent cross-links, was determined from equilibrium swelling data in toluene using the Flory−Rehner equation; 48 1/M c values of all vulcanized elastomers are reported in Table 1.
Glass transition temperatures (T g ) were determined by differential scanning calorimetry (DSC) using a DSC Mettler-Toledo 822e instrument. Thermal cycles between 148 and 323 K were performed for BR-based samples, while for IR-and SBR-based samples, thermal cycles were performed from 183 to 323 K. The cooling/heating rate was 10 K/min. T g was determined as the intersection point of the two tangents to the DSC curve at the endothermic step. For uncured IR, BR, and SBR, T g was 208, 167, and 225 (±1) K, respectively; T g values of vulcanized rubbers are reported in Table 1.    21,22 were used below and above 12 MHz, respectively. The polarizing and detection frequencies were 25.0 and 16.3 MHz, respectively. The switching time was 3 ms and the 90°pulse duration was 9.8 μs. A single scan was acquired. All the other experimental parameters were optimized for each measurement. All the 1 H magnetization curves versus time were monoexponential within the experimental error and the errors on R 1 were always lower than 2%. For measurements, samples were cut into small pieces and introduced in a 10 mm diameter glass tube. The temperature was controlled within ±0.1°C with a Stelar VTC90 variable temperature controller.
Data Analysis. Fittings of the susceptibility curves were performed using the least-squares minimization procedure implemented in the Fitteia environment. 49,50 The spectral decomposition of the susceptibility master curves and the determination of power law exponents were carried out using purposely written programs in the  (Table 1); a selection of NMRD curves is reported in Figure 1, while NMR susceptibility (χ″(ω) = ωR 1 (ω)) curves are shown in Figure  S1.
Uncured IR shows NMRD curves with three different power law dependences of R 1 on Larmor frequency (R 1 (ω)∝ω −γ ), depending on the temperature (Figure 1a). At all the investigated temperatures, a regime with γ ≅ 0.16 is observed. For T = 303 and 313 K, a regime characterized by γ ≅ 0.51 is observed at high frequencies; γ progressively increases by decreasing the temperature, reaching values of 1.1 at the lowest temperatures. For T ≥ 303 K, a small region with γ ≅ 0.37 (determinable only for the highest temperatures) is found at low frequencies. The crossover points between regimes are shifted toward higher frequencies with increasing temperature.
The observed exponents are similar to those determined by other authors for polyisoprene with molar mass above the entanglement limit 10,40,41 and attributed by Kimmich and Fatkullin to regimes I (high mode number limit; intrasegment dipolar interaction; γ = 0.5 ± 0.05), II (low mode number limit; intrasegment dipolar interaction; γ = 0.25 ± 0.05), and III (low mode number limit; intersegment dipolar interaction; γ = 0.44 ± 0.05) of renormalized Rouse theory. 24 The anomalous value of γ found for IR in regime II (0.16) was already pointed out and discussed in the literature by several research groups (vide infra). 29,52 In terms of the TR model, which we will refer to hereafter, the regime with γ ≅ 0.16 is ascribable to the Rouse regime (regime I, γ = 0), while that with γ ≅ 0.37 is ascribable to an incipient constrained Rouse regime (regime II, γ = 0.75), as better discussed below. The increase in the exponent observed on cooling at high frequencies and low temperature results from the overlap of the time scales of reorientations within the Kuhn segments in the glassy dynamics regime (regime 0) and motions of the Kuhn segments themselves in the incipient regime I. In fact, at the lowest temperatures, the high-frequency region is dominated by glassy dynamics; the maximum observed for R 1 on changing the temperature is due to the matching between the Larmor frequency and the frequency of glassy dynamics (i.e., the condition ωτ s ≅ 1 is satisfied).
The vulcanized IR-based samples show NMRD curves analogous to those of uncured IR in regimes I and 0 ( Figure  1a), whereas power law dependences typical of regime II are never detected for them, despite the high temperatures investigated (up to 383 K). The values of γ are equal to 0.17−0.18 in regime I at all the temperatures. Higher R 1 values in all frequency regions at each temperature are observed for cross-linked rubbers with respect to IR. This behavior indicates a progressive slowdown of motions with increasing cross-link density of the rubbers, analogous to that observed for IR on decreasing the temperature. A similar behavior was observed by Stapf and co-workers in natural rubber, an elastomer essentially made of polyisoprene. 40,41 Moreover, for the sulfurcured samples, the temperatures at the R 1 maxima are higher than those of uncured IR, indicating the slowing down of glassy dynamics with increasing cross-link density. The matching between the glassy dynamics time scale and the experimental Larmor frequency (ωτ s ≅ 1) gives rise to a maximum of the χ″(ω) curves for T ≤ 273 K, the maximum shifting to lower frequencies with increasing cross-link density. The power laws found for the R 1 dispersions are reflected in χ″(ω) ∝ ω 1−γ dependences ( Figure S1a).
A selection of NMRD and NMR susceptibility curves of BRbased samples is reported in Figures 1b and S1b, respectively. Between 213 and 233 K, uncured BR shows curves mainly arising from glassy dynamics and polymer dynamics in regime I of the TR model. At the lowest temperatures, polymer dynamics in regime I occurs on a time scale overlapping that of glassy dynamics; the resulting effective γ is about 1.1. In regime I, NMRD curves follow a power law with γ = 0.24−0.25. This regime is found to dominate the NMRD curves between 253 and 303 K. For T ≥ 283 K, a more pronounced dispersion region is observed at low frequencies, with γ ≅ 0.65, ascribable to regime II of TR. By increasing the temperature, the crossover points between regimes are shifted toward higher frequencies. These findings are in accordance with those reported in the literature for high-molar mass polybutadiene melts. 39,53 NMRD and susceptibility curves analogous to those of uncured BR are observed for the sulfur-cured BR samples, with γ values similar to those of regimes I and 0. BR_S1_150, the rubber with the lowest cross-link density, additionally shows a more pronounced dispersion below 30 kHz at T ≥ 363 K. The R 1 values at each investigated temperature increase with increasing cross-link density. A similar behavior was reported by Kariyo and Stapf for rubbers obtained by vulcanization of polybutadiene at 160°C with 1, 2, 3, and 4 phr of sulfur. 42 Selected FC NMR data acquired on uncross-linked and cross-linked SBR are shown in Figures 1c and S1c. At T = 273 K, NMRD and corresponding susceptibility curves essentially due to glassy dynamics are observed for uncured SBR. By increasing the temperature, regimes I and II of the TR model are found at lower frequencies with regime II being observed at very low frequencies only for the highest investigated temperatures (T ≥ 363 K). In the high-frequency regime, R 1 shows a maximum as a function of temperature because of the matching of the condition ωτ s ≅ 1 for the glassy dynamics; correspondingly, maxima are observed in the χ″(ω) curves. The γ values are 0.23−0.26 in regime I and ≅ 0.48 in regime II. The vulcanized SBR-based samples show NMRD and susceptibility curves similar to those of uncured SBR in Macromolecules pubs.acs.org/Macromolecules Article regimes I and 0. The maxima of R 1 as a function of temperature at high frequencies occur at higher temperatures for samples with higher cross-link density. Correspondingly, the maxima of the χ″(ω) curves observed at the lower temperatures are shifted to lower frequencies ( Figure S1c). In order to better discern the effect of sulfur bridges with different lengths on dynamics in IR-based rubbers with similar cross-link densities, 1 H NMRD curves were also acquired in the 263−363 K temperature range on samples IR_29, IR_60, and IR_80, characterized by an increasing content of polysulfidic bridges; representative results are shown in Figure  S2. NMRD and NMR susceptibility curves are similar to those of the other IR-based samples, showing characteristic features of glassy dynamics at high frequencies and lower temperatures and of regime I of TR at low frequencies and higher temperatures. By increasing the content of polysulfidic bridges, a small increase in R 1 is observed at all temperatures in the NMRD curves, while a slight shift to lower frequencies is found for the maxima in the susceptibility curves. These features indicate a slowdown of glassy dynamics in rubbers with a higher content of polysulfidic bridges.
NMR Susceptibility Master Curves. NMR susceptibility, χ″(ω), curves obtained for the different samples at the investigated temperatures (a selection is shown in Figures S1 and S2b) were employed to build χ″(ωτ s ) master curves by shifting along the frequency axis by the correlation times for glassy dynamics, τ s . For the χ″(ω) curves collected at the lowest temperatures, clearly showing the maximum and both low-and high-frequency branches, the high-frequency branch was fitted to eq 6 with the Cole−Davidson spectral density (eq 2). In particular, for samples in the IR series, the highfrequency branch of χ″(ω) curves at 263 and 273 K was fitted  The χ″(ωτ s ) master curves obtained for the investigated samples cover wide ranges of frequency (7 orders of magnitude of ωτ s , almost 6 at ωτ s < 1) and amplitude (about 4 orders of magnitude) and show characteristic shapes, reflecting the hierarchy of motions of polymer chains in the different regimes: at high reduced frequency, glassy dynamics dominates, whereas for ωτ s < 10 −1 , the contribution of polymer dynamics becomes progressively more important.
Almost identical master curves were obtained for all the sulfur-cured IR rubbers, which differ from those of uncured IR only at very low frequency, where regime II of the TR model is observed for the latter sample (Figure 2a). The master curve obtained for uncured IR is quite similar to that reported in the literature for samples of linear high-molar mass polyisoprene. 29,30,54 The χ″(ωτ s ) curve of uncured BR results from the contribution of glassy dynamics and polymer dynamics in regimes I and II, in analogy to that reported in the literature for linear high-molar mass polybutadiene samples. 25,27,29−31 On the other hand, the χ″(ωτ s ) curves of cross-linked BR-based rubbers show the contribution from glassy dynamics and regime I of polymer dynamics. A deviation from the slope of regime I is observed at the lowest ωτ s values for BR_S1_150 (Figure 2b), as already evidenced by the NMRD and susceptibility curves at the highest temperatures. Master curves are very similar for all the vulcanized SBR-based rubbers, which, however, differ from those of uncured SBR at low frequencies, where regime II is found for the latter sample ( Figure 2c). The differences between cross-linked and corresponding uncross-linked elastomers can be better highlighted by dividing χ″(ωτ s ) curves by the reduced frequency ωτ s , as shown in Figure S5; in this way, master curves of R 1 (ωτ s )/τ s are obtained, which can be considered as a sort of spectral density (J(ωτ s ) = χ″(ωτ s )/ωτ s ) master curves.
By applying the derivative method described in refs 32 and 55, the exponents of the power law dependences of χ″(ωτ s ) on the reduced frequency were determined for all samples. Exponent values of 0.85 ± 0.10, 0.78 ± 0.10, and 0.75 ± 0.05 were found in regime I for uncured IR, BR, and SBR, respectively. Considering the rotational dynamics in the framework of TR theory, the exponent in regime I arising from Rouse dynamics should be 1, although values close to ours were found for entangled polymer melts both in experimental studies by FC NMR 28,32 and DQ NMR 55,56 and in computer simulations 57,58 and ascribed to a reduced number of Rouse modes due to chain local stiffness. In regime II, exponent values of ∼0.66, 0.30, and 0.51 were determined for uncross-linked IR, BR, and SBR, respectively, which are higher than those predicted by the TR model for rotations in the constrained Rouse regime (0.25). Although the contribution of interchain dipolar interactions cannot be excluded at low frequencies, this discrepancy can be mainly ascribed to the fact that a restricted frequency range was investigated for regime II because of technical limitations.
Cross-linked elastomers show exponent values in the same ranges of those determined for the corresponding uncured ones in regime I. The lack of a deviation from this power law dependence at low reduced frequencies indicates that geometrical constraints introduced by vulcanization hamper longrange coherent chain motions, the chain dynamics remaining limited to local motions in the free Rouse regime.
Effects of Cross-Linking on Glassy Dynamics. Within each series of elastomers, the cross-link density, as determined from equilibrium swelling experiments, 48 increases linearly with the sulfur amount used in vulcanization ( Figure S6), with intercept values of 2.2 × 10 −5 , 4.8 × 10 −5 , and 1.2 × 10 −5 mol/ g for IR-, BR-, and SBR-based rubbers, respectively; these  (Table 1) with the corresponding M e values determined from the plateau modulus for uncured elastomers with a similar microstructure (3120 g/mol for IR, 59 2350 g/mol for BR, 59 and 3000 g/mol for SBR 37 ) it can be inferred that the investigated rubbers are moderately crosslinked. As already observed from NMRD and NMR susceptibility curves, a slowdown of glassy dynamics occurs for all the investigated elastomers after vulcanization, which increases with the increasing amount of sulfur and cross-link density. Restrictions imposed by the network junctions reduce the configurational freedom of the chains and increase the local friction. This effect is more quantitatively described by the trends of τ s reported in Figure 3. Within each series of elastomers, τ s values progressively increase with increasing cross-link density at the same temperature, in analogy to what observed by Hernańdez et al. for natural rubber using broadband dielectric spectroscopy. 20 Moreover, slightly longer τ s values were found for IR samples showing a very similar cross-link density but a higher fraction of polysulfidic links ( Figure S4a For all samples, τ s shows a non-Arrhenius temperature dependence as expected for glassy dynamics. Unfortunately, the limited range of temperature investigated hampered a robust analysis of τ s in terms of the VFT equation (eq 7) and, thereby, the determination of dynamic T g and the fragility index for the elastomers. By considering the static T g determined by DSC, plots of Logτ s versus (T/T g − 1), reported in Figures 4 and S4b, show a good superposition of curves for uncross-linked and cross-linked elastomers for IRand BR-based samples; the superposition is almost perfect for elastomers of the SBR series. Slight deviations in Logτ s versus (T/T g − 1) curves for vulcanized and uncured elastomers can be accounted for by possible errors in determining T g and τ s and on a different effect of cross-links on static and dynamic T g . All this considered, these results indicate that if τ 0 remains unchanged, vulcanization does not appreciably affect polymer fragility (see eq 9), in agreement with results on other crosslinked polymers with relatively low cross-link densities. 14,15,17 A linear relationship was found between T g determined by DSC and 1/M c values for all sulfur-cured rubbers ( Figure 5) with slopes of (1.1 ± 0.1) × 10 5 , (0.5 ± 0.5) × 10 5 , and (1.9 ± 0.7) × 10 5 K·g/mol and intercepts of 207 ± 1, 166 ± 5, and 226 ± 3 K, quite close to the T g of the corresponding uncured elastomers, for IR-, BR-, and SBR-based rubbers, respectively. The observed linear trends are in agreement with the literature, 60−65 and the slope values are in the order of those reported for other polymers. 61,64,65 An increase in T g was also found for vulcanized IR-based rubbers with a higher percentage of polysulfidic bridges and a very similar cross-link density (Table 1). In fact, not only the cross-link density but also the cross-link type and the presence of sulfur-polymer or curative-polymer adducts can affect the glassy dynamics of vulcanized elastomers. 13,20,61,64,65 In particular, polysulfidic cross-links and cyclic sulfide structures, favored when vulcanization is carried out at higher sulfur/ accelerant ratios, have a more profound effect on T g than monosulfidic cross-links because they restrict the reorientation possibilities of polymer chains. On the other hand, when present in large amounts and endowed with a bulky and inflexible structure, accelerants considerably stiffen polymer chains by hampering their reorientations and/or forming transient cross-links by intermolecular interactions. In our case, because a low-molecular weight accelerant is used (TBBS) and the sulfur/accelerant ratio increases with the percentage of polysulfidic cross-links in our samples, the observed effect is ascribed to polysulfidic cross-links and, possibly, cyclic sulfide structures.
Effects of Cross-Linking on Polymer Dynamics. In order to inspect in more detail the effect of cross-linking on the dynamics of the polymer chains, the spectra due to the sole polymer dynamics, called from now on "polymer spectra", were singled out from the susceptibility master curves shown in Figure 2, following a spectral decomposition procedure similar to that described in the previous literature. 25 As shown in Figure S7 for uncured IR, BR, and SBR, the polymer spectra (χ pol ″ (ωτ s )) were obtained by subtracting from each χ″(ωτ s ) master curve the contribution of glassy dynamics (χ glass ″ (ωτ s )), described using a Cole−Davidson function, under the assumption of statistical independence and time-scale separation between glassy and polymer dynamics, such that eq 4 holds true. Moreover, the contribution of polymer dynamics at high frequencies (ω ≳ 1/τ s ) was assumed to be negligible, so that the high-frequency branch of χ″(ωτ s ) was practically coincident with χ glass ″ (ωτ s ). This procedure allowed the relaxation strength of polymer dynamics, f, to be calculated on the basis of eq 5. Notably, although a similar approach was previously used for disentangling glassy and polymer spectra of polyisoprene, polybutadiene, and polydimethylsiloxane melts, 25,27,29 to the best of our knowledge, it is here applied for the first time to SBR and to cross-linked elastomers.
The polymer spectra, normalized to provide an integral equal to π/2, χp ol ″ (ωτ s ), are shown in Figure 6 for a representative selection of samples. For uncured IR, BR, and SBR (Figure 6a), similar power law dependences of χp ol ″ (ωτ s ) Macromolecules pubs.acs.org/Macromolecules Article on reduced frequency are observed. In particular, regime I (with power law exponents between 0.70 and 0.75) and an incipient regime II (with power law exponents between 0.30 and 0.60) of the TR model can be recognized at intermediate and low reduced frequencies, respectively, while a maximum is observed at high frequency, which signs the cut off of the polymer spectrum. The exponents determined from the polymer spectra are similar but not equal to those found from the total NMRD and susceptibility curves, indicating that the contribution of glassy dynamics to 1 H T 1 relaxation at low frequencies is not completely negligible. As expected, the major deviations are observed for IR-based samples, for which the lowest values of f are found (vide infra). These results are in agreement with those previously obtained for polyisoprene and polybutadiene melts. 29 Sulfur-cured IR-, BR-, and SBR-based elastomers show polymer spectra with a frequency dependence typical of regime I down to the lowest explored frequencies, with power law exponents of 0.75 ± 0.05 for rubbers of the BR and IR series and of 0.70 ± 0.03 for those of the SBR one, similar to those found for the corresponding uncured elastomers (Figure 6b). A slight decrease in the slope at low reduced frequencies is only observed for BR_S1_150, suggesting that longer range motions can occur in rubbers with low cross-link densities when temperature is far above T g .
The values of f obtained for all the samples studied here, reported in Table 2, range between 0.05 and 0.14 (corresponding to values of the dynamic order parameter S between 0.22 and 0.37), confirming that the contribution of polymer dynamics is only a very small fraction of the total spectrum of motions. Although according to Kimmich and Fatkullin f should be 10 −3 to 10 −2 , values similar to ours were also found by Rossler and co-workers for linear polymers with M > M e . 27,29,30 The smaller value of f obtained for IR with respect to BR and SBR elastomers can be explained with the absence in polyisoprene of a spin pair along the double bond, for which the orientation of the internuclear vector, closer to the contour of the chain, results in a higher value of the dynamic order parameter S (see the Theoretical Background). These results agree with those obtained by Herrmann et al. and Kariyo et al. for polyisoprene and polybutadiene polymer melts. 25,27,29 The values of f do not significantly change in passing from the uncured elastomers to the corresponding vulcanized ones, and no clear dependence on cross-link density is found. This can be due to the fact that f, as the corresponding dynamic order parameter S, depends on both free Rouse and entangled dynamics, at variance with the dynamic order parameter S e measured by double quantum (DQ) NMR experiments, 66,67 which contains the sole contribution of entangled dynamics and represents only a small fraction (less than 10%) of S. 27 As a consequence, the increase in S e , usually observed by DQ NMR experiments by increasing the cross-link density, could not be detected by our approach because it is in the order of the experimental error on S. The different sensitivity and precision of the FC NMR and DQ NMR approaches are inherent in the measured observables. Although R 1 arises from the fluctuations of the dipolar interaction and S is derived indirectly as the difference between the whole spectral density and the strongly Figure 6. Normalized polymer spectra χ̃p ol ″ (ωτ s ) of: (a) uncured IR, BR, and SBR; (b) IR_S3_150, BR_S3_150, and SBR_S3_150 rubbers; and (c) BR_S1_150, BR_S2_150, and BR_S3_150 rubbers. Each spectrum was normalized to provide an integral equal to π/2, multiplying χ pol ″ (ωτ s ) by the factor π/(2f I), 27 where I stands for the integrated intensity of χ″(ωτ s ). Dashed lines indicate the power law dependences recognized for regimes I and II with the reported values of the exponent.

■ CONCLUSIONS
Glassy and polymer dynamics were investigated on IR, BR, and SBR, three elastomers of technological interest for the tire industry, both uncured and cross-linked by sulfur curing under different conditions. 1 H FC NMR relaxometry measurements over a wide range of temperatures and frequencies, combined on the basis of the FTS principle, allowed dynamics to be carefully investigated over a quite broad time scale, ranging from local segmental motions within the Kuhn segment to chain dynamics within the constraints imposed by entanglements and sulfidic cross-links. Slowing down of glassy dynamics was found for all sulfur-cured elastomers with respect to the uncured ones because of an increase in the frictional coefficient. Correspondingly, higher glass transition temperatures were estimated for vulcanized elastomers, which linearly increased with the cross-link density. On the other hand, polymer fragility seems to be substantially unaffected by vulcanization. The effect of cross-linking on collective chain dynamics was investigated by separating the contribution of polymer dynamics to 1 H longitudinal relaxation from that of glassy dynamics for uncured and, for the first time in this work, for vulcanized elastomers. In all cases, the fractional contribution of polymer dynamics was found to be very small (0.05−0.14), and no significant dependence on the cross-link density was observed because of an intrinsic insensitivity of the procedure. A larger contribution of polymer dynamics was found for elastomers of the BR and SBR series with respect to those of the IR one, ascribable to the different structures of the monomeric units, which in BR and SBR present a spin pair along the double bonds, resulting in an increased dynamic order parameter. In the investigated frequency range, polymer dynamics in regimes I and II of tube reptation theory was observed for all uncross-linked elastomers, regime II being better observed for BR because of its lower T g . On the contrary, only regime I was detected for most cross-linked elastomers, indicating that permanent cross-links impose constraints on dynamics occurring on a larger scale than free Rouse modes, while they do not significantly affect the spectrum of Rouse dynamics. The deviation from regime I observed at the lowest frequencies and highest temperatures for the BR sample with the lowest cross-link density suggests that for lightly cross-linked rubbers, larger space and time scale dynamics could occur far above T g . In this respect, because FC NMR measurements at higher temperatures are not possible because of sample stability, it could be useful to perform experiments at much lower frequencies, exploiting home-built relaxometers with compensation for earth and stray magnetic fields. 31 ■ ASSOCIATED CONTENT * sı Supporting Information The Supporting Information is available free of charge at https://pubs.acs.org/doi/10.1021/acs.macromol.0c01439. 1