Heterogeneous Polymer Dynamics Explored Using Static 1H NMR Spectra

NMR spectroscopy continues to provide important molecular level details of dynamics in different polymer materials, ranging from rubbers to highly crosslinked composites. It has been argued that thermoset polymers containing dynamic and chemical heterogeneities can be fully cured at temperatures well below the final glass transition temperature (Tg). In this paper, we described the use of static solid-state 1H NMR spectroscopy to measure the activation of different chain dynamics as a function of temperature. Near Tg, increasing polymer segmental chain fluctuations lead to dynamic averaging of the local homonuclear proton-proton (1H-1H) dipolar couplings, as reflected in the reduction of the NMR line shape second moment (M2) when motions are faster than the magnitude of the dipolar coupling. In general, for polymer systems, distributions in the dynamic correlation times are commonly expected. To help identify the limitations and pitfalls of M2 analyses, the impact of activation energy or, equivalently, correlation time distributions, on the analysis of 1H NMR M2 temperature variations is explored. It is shown by using normalized reference curves that the distributions in dynamic activation energies can be measured from the M2 temperature behavior. An example of the M2 analysis for a series of thermosetting polymers with systematically varied dynamic heterogeneity is presented and discussed.


Introduction
Solid-state NMR spectroscopy remains an important tool for the characterization of the structure and dynamics in a wide range of materials [1][2][3][4][5]. Improvements in magic angle spinning (MAS) spinning speeds; advances in heteronuclear, multiple dimensional, and multiple quantum NMR pulse sequences; plus the sensitivity gains realized from dynamic nuclear polarization (DNP) methods continue to impact the use of NMR to probe a variety of different material phenomena. While many of these advances help resolve additional molecular level details, there is a corresponding increase in the complexity of implementation. One of the oldest and, perhaps, simplest methods for the analysis of NMR spectra involves measuring the second moment (M 2 ) of the line shapes as a function of the temperature or composition [6,7] and has proven to be indispensable in characterizing dynamics and local structures in amorphous and disordered materials. Static (non-spinning) NMR line shapes are broadened by homonuclear and heteronuclear dipolar couplings between different nuclei, quadrupolar couplings (for spin I > 1 2 ), chemical shielding anisotropy, or chemical shift dispersions, thus providing a handle to probe local structures and dynamics. For example, 7 Li, 23 Na, 31 P, and 133 Cs NMR M 2 results have been reported for glasses in analyses of structures and ion dynamics [8][9][10][11][12], as well as 1 H NMR M 2 studies of dynamics, miscibility, and chain orientations in polymers [13][14][15][16][17][18][19][20]. While static NMR is hampered by low spectral resolution, it does benefit from being sensitive to local motions on the order of the line width and can be employed over very wide temperatures ranges.

Results and Discussion
Static solid-state 1 H NMR spectra for the UV-cured BTT-TCDDA networks as a function of temperature and R are shown in Figure 1. At 233 K (−40 • C), a single unresolved broad resonance having a full width at a half-maximum line width of~50 kHz is observed for all R ratios. This broad resonance originates from the strong homonuclear 1 H-1 H dipolar couplings present in these rigid, glassy polymers. At these low temperatures, the local polymer chain dynamics are slow compared to the timescale of the dipolar interaction (i.e., 1/τ c << 2π(M 2 ) 1/2 , where τ c is the correlation time of the chain fluctuations and (M 2 ) 1/2 is related to the spectral line width). Note, for the solid-state 1 H NMR spectra at low temperatures, there are no narrow spectral components overlapping the broad resonances. This lack of a narrow resonance clearly shows that highly mobile polymer fractions (1/τ c >> 2π(M 2 ) 1/2 ) from comparatively low crosslink density regions are not present. Another way to describe this is that, if there are low crosslink density regions, they must have similar dynamic responses to the rest of the network when well below the T g . With increasing temperatures, additional polymer dynamics are activated, leading to the motional averaging of the dipolar coupling (1/τ c~2 π(M 2 ) 1/2 ) and gradual narrowing of the NMR line resonance. This motional averaging dramatically increases near the T g , where the rate and magnitude of the polymer chain fluctuations (α-relaxation) become large enough to ultimately produce the fully dynamically averaged narrow line shapes observed at high temperatures ( Figure 1). With the decreasing R, the temperature at which the α-relaxation becomes activated increases and mirrors the change observed in the dynamic mechanical analysis (DMA) (Supplementary Materials Figure S1). For the BTT-TCDDA networks discussed here, the dynamic heterogeneities (i.e., a distinct mixture of narrow and broad resonances) observed in the 1 H NMR line shape as a function of R near the T g are not as pronounced as those previously noted in related networks [20] and most likely reflect the differences in the UV light source intensity employed and the actual sample temperature during the cure process. the timescale of the dipolar interaction (i.e., 1/τc << 2π(M2) 1/2 , where τc is the correlation time of the chain fluctuations and (M2) 1/2 is related to the spectral line width). Note, for the solid-state 1 H NMR spectra at low temperatures, there are no narrow spectral components overlapping the broad resonances. This lack of a narrow resonance clearly shows that highly mobile polymer fractions (1/τc >> 2π(M2) 1/2 ) from comparatively low crosslink density regions are not present. Another way to describe this is that, if there are low crosslink density regions, they must have similar dynamic responses to the rest of the network when well below the Tg. With increasing temperatures, additional polymer dynamics are activated, leading to the motional averaging of the dipolar coupling (1/τc~2π(M 2 ) 1/2 ) and gradual narrowing of the NMR line resonance. This motional averaging dramatically increases near the Tg, where the rate and magnitude of the polymer chain fluctuations (α-relaxation) become large enough to ultimately produce the fully dynamically averaged narrow line shapes observed at high temperatures ( Figure 1). With the decreasing R, the temperature at which the α-relaxation becomes activated increases and mirrors the change observed in the dynamic mechanical analysis (DMA) (Supplementary Materials Figure S1). For the BTT-TCDDA networks discussed here, the dynamic heterogeneities (i.e., a distinct mixture of narrow and broad resonances) observed in the 1 H NMR line shape as a function of R near the Tg are not as pronounced as those previously noted in related networks [20] and most likely reflect the differences in the UV light source intensity employed and the actual sample temperature during the cure process. While it is common during the analysis of static 1 H NMR line shapes to simply deconvolute the resonance into mobile and immobile fractions, polymer chain dynamics realistically involve a distribution of motional rates, leading to a complex superposition of different motionally averaged line shapes. Here, we describe an analysis of M2 calculated using the spectral intensity at frequency ω over the entire line shape for each NMR spectrum using (see Appendix A for details): Figure 1. Static solid-state 1 H NMR spectra as a function of select sample temperature for the ultraviolet (UV)-cured 1,3,5-benzenetrithiol-tricyclodecane dimethanol diacrylate (BTT-TCDDA) polymer networks of varying thiol-acrylate stoichiometry (R = (SH) 0 /(C=C) 0 ). The reduced temperatures (T-T g )/T g are shown in square brackets, with the transition temperature (T g ) occurring at (T-T g )/T g = 0. Narrowing of the NMR line width is observed for temperatures above the polymer glass transition temperature T g . The middle three rows are at equivalent reduced temperatures, while the upper and lower rows show the lowest and highest temperatures probed for that stoichiometry, respectively.
While it is common during the analysis of static 1 H NMR line shapes to simply deconvolute the resonance into mobile and immobile fractions, polymer chain dynamics realistically involve a distribution of motional rates, leading to a complex superposition of different motionally averaged line shapes. Here, we describe an analysis of M 2 calculated using the spectral intensity at frequency ω over the entire line shape for each NMR spectrum using (see Appendix A for details): The temperature variation of the 1 H NMR M2 for BTT-TCDDA networks as a function of R is shown in Figure 2a. In the low temperature regime M2 for R = 0.00 and 0.09, the networks are similar but are significantly larger than the M2 for R = 0.47 and 0.32, while R = 0.19 is intermediate between these extremes. The larger M2 at a low temperature (increased magnitude of the 1 H-1 H dipolar coupling) may result from multiple factors. The first possibility is that the TCDDA component has an increased intra-or intermolecular dipolar coupling (higher intra-or intermolecular proton density) than the BTT and that higher TCDDA concentrations (decreasing R) will therefore increase the M2. The proton density for the pure monomers is estimated to be 0.087 moles H/cm 3 and 0.047 moles H/cm 3 for BTT and TCDDA, respectively, supporting the argument that proton density increases with decreasing R. While the M2 has not been theoretically calculated for the cured TCDDA-BTT polymers, as the network structures are not known, we argue that the 100-kHz 2 differences in the M2 at 233 K cannot be fully explained by the compositional change. The most reasonable explanation is that, at this temperature, the TCCDA rich compositions have dynamics that are more representative of the rigid glassy state, since, at 233 K, the temperature is further from their respective Tg values. To support this, the M2 is plotted with respect to the scaled temperature (T-Tg)/Tg, as shown in Figure 2b, which reveals that all compositions trend to the same M2 limit. Due to probe hardware limitations, we were not able to investigate temperatures below 233K, preventing us from reaching the rigid lattice limit for all compositions. Between 233 K and 400 K, there is a gradual decrease in the M2 with the increasing temperature (Figure 2a), which is attributed to the β-relaxation process involving local, noncooperative dynamics in the BTT-TCDDA network. The M2 temperature variation for this β-relaxation process is similar for all compositions. As the temperature is further increased through a given Tg, the respective M2 rapidly decreases due to dynamic averaging as the polymer chain motions become faster and more dominant (α-relaxation), until finally reaching the rubbery plateau regions for T > Tg. The expansion of the M2 temperature variation for the BTT-TCDDA networks at reduced temperatures near (T-Tg)/Tg~0 (Figure 2c) reveals distinctly different dynamics, which we will attribute to differences in the activation energy (and corresponding correlation times) for the polymer chain motions in subsequent The temperature variation of the 1 H NMR M 2 for BTT-TCDDA networks as a function of R is shown in Figure 2a. In the low temperature regime M 2 for R = 0.00 and 0.09, the networks are similar but are significantly larger than the M 2 for R = 0.47 and 0.32, while R = 0.19 is intermediate between these extremes. The larger M 2 at a low temperature (increased magnitude of the 1 H-1 H dipolar coupling) may result from multiple factors. The first possibility is that the TCDDA component has an increased intra-or intermolecular dipolar coupling (higher intra-or intermolecular proton density) than the BTT and that higher TCDDA concentrations (decreasing R) will therefore increase the M 2 . The proton density for the pure monomers is estimated to be 0.087 moles H/cm 3 and 0.047 moles H/cm 3 for BTT and TCDDA, respectively, supporting the argument that proton density increases with decreasing R. While the M 2 has not been theoretically calculated for the cured TCDDA-BTT polymers, as the network structures are not known, we argue that the 100-kHz 2 differences in the M 2 at 233 K cannot be fully explained by the compositional change. The most reasonable explanation is that, at this temperature, the TCCDA rich compositions have dynamics that are more representative of the rigid glassy state, since, at 233 K, the temperature is further from their respective T g values. To support this, the M 2 is plotted with respect to the scaled temperature (T-T g )/T g , as shown in Figure 2b, which reveals that all compositions trend to the same M 2 limit. Due to probe hardware limitations, we were not able to investigate temperatures below 233K, preventing us from reaching the rigid lattice limit for all compositions.
The temperature variation of the 1 H NMR M2 for BTT-TCDDA networks as a function of R is shown in Figure 2a. In the low temperature regime M2 for R = 0.00 and 0.09, the networks are similar but are significantly larger than the M2 for R = 0.47 and 0.32, while R = 0.19 is intermediate between these extremes. The larger M2 at a low temperature (increased magnitude of the 1 H-1 H dipolar coupling) may result from multiple factors. The first possibility is that the TCDDA component has an increased intra-or intermolecular dipolar coupling (higher intra-or intermolecular proton density) than the BTT and that higher TCDDA concentrations (decreasing R) will therefore increase the M2. The proton density for the pure monomers is estimated to be 0.087 moles H/cm 3 and 0.047 moles H/cm 3 for BTT and TCDDA, respectively, supporting the argument that proton density increases with decreasing R. While the M2 has not been theoretically calculated for the cured TCDDA-BTT polymers, as the network structures are not known, we argue that the 100-kHz 2 differences in the M2 at 233 K cannot be fully explained by the compositional change. The most reasonable explanation is that, at this temperature, the TCCDA rich compositions have dynamics that are more representative of the rigid glassy state, since, at 233 K, the temperature is further from their respective Tg values. To support this, the M2 is plotted with respect to the scaled temperature (T-Tg)/Tg, as shown in Figure 2b, which reveals that all compositions trend to the same M2 limit. Due to probe hardware limitations, we were not able to investigate temperatures below 233K, preventing us from reaching the rigid lattice limit for all compositions. Between 233 K and 400 K, there is a gradual decrease in the M2 with the increasing temperature (Figure 2a), which is attributed to the β-relaxation process involving local, noncooperative dynamics in the BTT-TCDDA network. The M2 temperature variation for this β-relaxation process is similar for all compositions. As the temperature is further increased through a given Tg, the respective M2 rapidly decreases due to dynamic averaging as the polymer chain motions become faster and more dominant (α-relaxation), until finally reaching the rubbery plateau regions for T > Tg. The expansion of the M2 temperature variation for the BTT-TCDDA networks at reduced temperatures near (T-Tg)/Tg~0 (Figure 2c) reveals distinctly different dynamics, which we will attribute to differences in the activation energy (and corresponding correlation times) for the polymer chain motions in subsequent sections. Additional NMR relaxation experiments, including spin lattice relaxation (T1) and spin-spin Between 233 K and 400 K, there is a gradual decrease in the M 2 with the increasing temperature (Figure 2a), which is attributed to the β-relaxation process involving local, noncooperative dynamics in the BTT-TCDDA network. The M 2 temperature variation for this β-relaxation process is similar for all compositions. As the temperature is further increased through a given T g , the respective M 2 rapidly decreases due to dynamic averaging as the polymer chain motions become faster and more dominant (α-relaxation), until finally reaching the rubbery plateau regions for T > T g . The expansion of the M 2 temperature variation for the BTT-TCDDA networks at reduced temperatures near (T-T g )/T g~0 (Figure 2c) reveals distinctly different dynamics, which we will attribute to differences in the activation energy (and corresponding correlation times) for the polymer chain motions in subsequent sections. Additional NMR relaxation experiments, including spin lattice relaxation (T 1 ) and spin-spin relaxation (T 2 ), were not pursued for the current study due to the desire to match the NMR heating/cooling rates to the heating/cooling rates of the DMA analysis as closely as possible.

Impact of Activation Energy on M 2
To extract additional information regarding polymer chain dynamics from the 1 H NMR spectra, it is instructive to model the effect of different variables on the M 2 behavior. Figure 3 shows simulations of the M 2 temperature variation for varying activation energies (E a ) of the dynamic process leading to the motional averaging. The temperature behavior of the M 2 is defined using Equation (A5) (additional discussion provided in the Appendix A), where we have assumed an Arrhenius behavior for the correlation times τ c using: where τ 0 is the pre-exponential correlation time, E a the activation energy, and R gas is the gas constant. The pre-exponential term was fixed at τ 0 = 0.5905 ns for all simulations, unless otherwise noted. This τ 0 value is in the middle of the range experimentally determined (see Table 1 and later discussion). An example of the simulated M 2 variations for changing τ 0 is shown in Figure S2. A more complicated non-Arrhenius temperature behavior of τ c , such as the Vogel-Fulcher-Tammann (VFT) relationship [30][31][32], could also be considered. As expected, the transition from the rigid lattice M 2 limit to the fully motionally averaged M 2 occurs at higher temperatures with increasing E a . The M 2 temperature variations for this range of E a overlap when scaled to the reduced temperature (T-T g )/T g , as shown in Figure 3b, with this scaling behavior being well-known in the analysis of polymer relaxation processes [33][34][35].
Int. J. Mol. Sci. 2020, 21, x FOR PEER REVIEW 5 of 21 relaxation (T2), were not pursued for the current study due to the desire to match the NMR heating/cooling rates to the heating/cooling rates of the DMA analysis as closely as possible.

Impact of Activation Energy on M2
To extract additional information regarding polymer chain dynamics from the 1 H NMR spectra, it is instructive to model the effect of different variables on the M2 behavior. Figure 3 shows simulations of the M2 temperature variation for varying activation energies (Ea) of the dynamic process leading to the motional averaging. The temperature behavior of the M2 is defined using Equation A5 (additional discussion provided in the Appendix), where we have assumed an Arrhenius behavior for the correlation times τc using: where τ0 is the pre-exponential correlation time, Ea the activation energy, and Rgas is the gas constant. The pre-exponential term was fixed at τ0 = 0.5905 ns for all simulations, unless otherwise noted. This τ0 value is in the middle of the range experimentally determined (see Table 1 and later discussion). An example of the simulated M2 variations for changing τ0 is shown in Figure S2. A more complicated non-Arrhenius temperature behavior of τc, such as the Vogel-Fulcher-Tammann (VFT) relationship [30][31][32], could also be considered. As expected, the transition from the rigid lattice M2 limit to the fully motionally averaged M2 occurs at higher temperatures with increasing Ea. The M2 temperature variations for this range of Ea overlap when scaled to the reduced temperature (T-Tg)/Tg, as shown in Figure 3b, with this scaling behavior being well-known in the analysis of polymer relaxation processes [33][34][35].   Table 1. Activation energies, pre-exponential constant, and distribution parameters for the glass transition in the 1,3,5-benzenetrithiol-tricyclodecane dimethanol diacrylate (BTT-TCCDA) networks obtained from the static 1 H NMR second moment analysis.

Distributions in Dynamic Rates
In amorphous polymers, it is expected that the dynamics include a distribution of chain relaxation correlation times, as previously included in NMR line shapes and relaxation time analyses [36][37][38]. Since a variation in E a moves the temperature at which the T g transition is observed (Figure 3), the presence of a distribution of activation energies will therefore impact the M 2 temperature variation. As an example, Figure 4a shows the simulated M 2 variation with the introduction of a Gaussian distribution of the log mean correlation time τ * or, equivalently, a Gaussian distribution of E a (see Equation (A8)) with different standard deviations σ. Examples of these Gaussian distributions are shown in Figure S3. The results are labeled as a function of σ(E a ) for direct comparison to Figure 3. With the increasing σ, the rate of change of the M 2 with the temperature is reduced, with the width of the T g transition broadened, and is reminiscent of the DMA results shown in Figure S1. This broadening behavior results from the increasing fraction of polymer chains having either slower (higher E a ) or faster (lower E a ) relaxation rates. For a Gaussian distribution, the M 2 intensities at different σ are equivalent at the same T g temperature (here, 442 K), because the mean E a is independent of σ for a symmetric distribution.

Distributions in Dynamic Rates
In amorphous polymers, it is expected that the dynamics include a distribution of chain relaxation correlation times, as previously included in NMR line shapes and relaxation time analyses [36][37][38]. Since a variation in Ea moves the temperature at which the Tg transition is observed ( Figure  3), the presence of a distribution of activation energies will therefore impact the M2 temperature variation. As an example, Figure 4a shows the simulated M2 variation with the introduction of a Gaussian distribution of the log mean correlation time τ * or, equivalently, a Gaussian distribution of Ea (see Equation A8) with different standard deviations σ. Examples of these Gaussian distributions are shown in Figure S3. The results are labeled as a function of σ(Ea) for direct comparison to Figure  3. With the increasing σ, the rate of change of the M2 with the temperature is reduced, with the width of the Tg transition broadened, and is reminiscent of the DMA results shown in Figure S1. This broadening behavior results from the increasing fraction of polymer chains having either slower (higher Ea) or faster (lower Ea) relaxation rates. For a Gaussian distribution, the M2 intensities at different σ are equivalent at the same Tg temperature (here, 442 K), because the mean Ea is independent of σ for a symmetric distribution.  Another distribution commonly employed to describe polymer relaxation is the Davidson-Cole (DC) relationship [39] defined by Equation (A9), with an example of the distribution shown in Figure  S3. The temperature variation of the M 2 as a function of the increasing distribution breadth (decreasing ε) is shown in Figure 4b. The behavior is distinct from that observed for a Gaussian distribution. In the DC distribution model, the M 2 transition is broadened with a decreasing ε for temperatures below the original T g (transition temperature). This broadening results from a larger fraction of polymer chain motions being activated at lower temperatures (1/τ c~2 π(M 2 ) 1/2 ) due to the increased concentration of regions (e.g., defects, reduced crosslink density, etc.) having reduced E a values. For temperatures above the original T g transition, there is limited impact on the M 2 behavior, because the DC distribution (Equation (A9)) is described by the maximum correlation time τ 0 c above which the probability of the slower dynamic processes vanishes. The asymmetry of the DC probability shifts the observed T g (defined here as the midpoint in the M 2 transition) to lower temperatures. The variations in the M 2 as a function of the reduced temperatures for systems with either a Gaussian or a DC distribution are shown in Figure 5a,b, respectively. For small distribution widths, the M 2 temperature behavior is very similar and only begins to deviate for large distributions. S3. The temperature variation of the M2 as a function of the increasing distribution breadth (decreasing ε) is shown in Figure 4b. The behavior is distinct from that observed for a Gaussian distribution. In the DC distribution model, the M2 transition is broadened with a decreasing ε for temperatures below the original Tg (transition temperature). This broadening results from a larger fraction of polymer chain motions being activated at lower temperatures (1/τc~2π(M 2 ) 1/2 ) due to the increased concentration of regions (e.g., defects, reduced crosslink density, etc.) having reduced Ea values. For temperatures above the original Tg transition, there is limited impact on the M2 behavior,  Figure 5a,b, respectively. For small distribution widths, the M2 temperature behavior is very similar and only begins to deviate for large distributions. The comparison of the M2 temperature variation between different network compositions may be complicated by differences in both the Ea (different Tg values) and distribution widths. This issue is explored in Figure 6 for the Gaussian and DC distributions. The M2 temperature behavior ( Figure  6a,b) does not readily allow differences in the distribution to be realized, but, by simply mapping the M2 behavior to a reduced temperature scale, the presence of the distributions is clear, regardless of the different Ea (Figure 6c,d). The M2 behavior is related to the relative magnitude of σ with respect to the Ea, but for similar ranges of the Ea, the M2 curves overlap well for a given value of σ. Therefore, by plotting the M2 with respect to reduced temperatures, it should be possible to experimentally measure distributions of polymer chain dynamics near Tg from static solid-state 1 H NMR experiments. The comparison of the M 2 temperature variation between different network compositions may be complicated by differences in both the E a (different T g values) and distribution widths. This issue is explored in Figure 6 for the Gaussian and DC distributions. The M 2 temperature behavior (Figure 6a,b) does not readily allow differences in the distribution to be realized, but, by simply mapping the M 2 behavior to a reduced temperature scale, the presence of the distributions is clear, regardless of the different E a (Figure 6c,d). The M 2 behavior is related to the relative magnitude of σ with respect to the E a , but for similar ranges of the E a , the M 2 curves overlap well for a given value of σ. Therefore, by plotting the M 2 with respect to reduced temperatures, it should be possible to experimentally measure distributions of polymer chain dynamics near T g from static solid-state 1 H NMR experiments.

Multiple Dynamic Processes
The impact of multiple dynamic processes on the M2 temperature behavior has previously been discussed by several authors [40][41][42][43][44][45][46][47][48]. For two noncorrelated dynamic processes, the M2 behavior is defined by Equation A10. An example is shown in Figure 7 involving a slow dynamic process at lower temperatures (β-relaxation), defined by the activation energy Ea(1), and a second dynamic process (α-relaxation, glass transition) at higher temperatures defined by the activation energy Ea(2). In these simulations, it is very easy to distinguish the two separate dynamic averaging transitions (Figure 7a) by the two-step features that are predicted, consistent with the work of Bilski and coworkers [41]. The addition of Ea(1) distributions to the β-relaxation transition (Figure 7b) begins to obscure the reduction step in the M2 resulting from this dynamic process. This M2 behavior is reminiscent of the M2 variation seen before the main Tg transition for the BTT-TCDDA networks (Figure 2). We were unable to reach the low temperature rigid lattice M2 plateau prior to β-relaxation due to the experimental limitations. The distinct M2 step has only rarely been observed for thermosets (see Figure 4 in Reference [19]) consistent with the presence of significant distributions for βrelaxation dynamics. These limitations result in the poorly constrained fitting of the initial βrelaxation process, such that we will concentrate only on the larger α-relaxation (glass transition) region at higher temperatures.

Multiple Dynamic Processes
The impact of multiple dynamic processes on the M 2 temperature behavior has previously been discussed by several authors [40][41][42][43][44][45][46][47][48]. For two noncorrelated dynamic processes, the M 2 behavior is defined by Equation (A10). An example is shown in Figure 7 involving a slow dynamic process at lower temperatures (β-relaxation), defined by the activation energy E a (1), and a second dynamic process (α-relaxation, glass transition) at higher temperatures defined by the activation energy E a (2). In these simulations, it is very easy to distinguish the two separate dynamic averaging transitions (Figure 7a) by the two-step features that are predicted, consistent with the work of Bilski and coworkers [41]. The addition of E a (1) distributions to the β-relaxation transition (Figure 7b) begins to obscure the reduction step in the M 2 resulting from this dynamic process. This M 2 behavior is reminiscent of the M 2 variation seen before the main T g transition for the BTT-TCDDA networks (Figure 2). We were unable to reach the low temperature rigid lattice M 2 plateau prior to β-relaxation due to the experimental limitations. The distinct M 2 step has only rarely been observed for thermosets (see Figure 4 in Reference [19]) consistent with the presence of significant distributions for β-relaxation dynamics. These limitations result in the poorly constrained fitting of the initial β-relaxation process, such that we will concentrate only on the larger α-relaxation (glass transition) region at higher temperatures.

Distributions in BTT-TCDDA Networks
By considering the M2 variation near Tg and utilizing the reduced temperature, reference curves for the M2 variation with either a Gaussian or a DC distribution were developed, as shown in Figure  8a,b, respectively. With increasing distribution widths (increasing σ or decreasing ε), the M2 variation across the glass transition region is broadened. For the Gaussian distribution, only σ changes on the order of ±1 are distinguishable, while, for the DC distributions, ε variations on the order of ±0.05 can be distinguished, except between ε = 0.5 and 1.0, where the impact on the M2 variation is minimal. The overall M2 behavior in the reference curves is similar for the Gaussian and DC distribution models, but they become slightly more distinct for very large distribution widths. For the current study, the differences in the distribution widths as a function of the composition R are not large enough for the M2 analysis to reliably distinguish between the Gaussian and DC distribution models.

Distributions in BTT-TCDDA Networks
By considering the M 2 variation near T g and utilizing the reduced temperature, reference curves for the M 2 variation with either a Gaussian or a DC distribution were developed, as shown in Figure 8a,b, respectively. With increasing distribution widths (increasing σ or decreasing ε), the M 2 variation across the glass transition region is broadened. For the Gaussian distribution, only σ changes on the order of ±1 are distinguishable, while, for the DC distributions, ε variations on the order of ±0.05 can be distinguished, except between ε = 0.5 and 1.0, where the impact on the M 2 variation is minimal. The overall M 2 behavior in the reference curves is similar for the Gaussian and DC distribution models, but they become slightly more distinct for very large distribution widths. For the current study, the differences in the distribution widths as a function of the composition R are not large enough for the M 2 analysis to reliably distinguish between the Gaussian and DC distribution models.
The M 2 temperature variation for the BTT-TCDDA networks are plotted in Figure 8c,d on the Gaussian and DC distribution reference curves. There is clearly an increase in the distribution with the decreasing R, and assuming a Gaussian distribution, we can assign σ accordingly. Starting from σ < 1 for R = 0.47 and then increasing to σ = 2 for R = 0.19, σ = 3 to 4 for R = 0.09 and σ = 5 for R = 0.0. The M 2 response for the R = 0.47 and R = 0.32 curves were not distinguishable and suggest that the distribution change between these two R compositions is within one sigma. Similarly, assuming a DC distribution, ε > 0.4 for the R = 0.47 and R = 0.32 compositions (recall that, between ε = 0.5 and 1.0, the reference curves are indistinguishable), then the distribution width increases with ε = 0.35 for R = 0.19, ε = 0.3 for R = 0.09, and to ε = 0.25 for the R = 0.0 network. It is also important to note that the M 2 behavior for the R = 0.0 BTT-TCDDA network is not symmetric around T g , which we have attributed to the degradation of this material at the very high temperatures (> 600 K or > 325 • C) achieved in the variable temperature experiments for this composition, and produced irreversible changes in the M 2 behaviors. These results are summarized in Table 1 and reveal that, in the BTT-TCDDA networks, the distribution in E a (and, correspondingly, τ c ) increases with the decreasing R. This is consistent with the picture of increased heterogeneous dynamics due to the chain-growth mechanism being dominant at high acrylate concentrations. The M2 temperature variation for the BTT-TCDDA networks are plotted in Figure 8c,d on the Gaussian and DC distribution reference curves. There is clearly an increase in the distribution with the decreasing R, and assuming a Gaussian distribution, we can assign σ accordingly. Starting from σ < 1 for R = 0.47 and then increasing to σ = 2 for R = 0.19, σ = 3 to 4 for R = 0.09 and σ = 5 for R = 0.0. The M2 response for the R = 0.47 and R = 0.32 curves were not distinguishable and suggest that the distribution change between these two R compositions is within one sigma. Similarly, assuming a DC distribution, ε > 0.4 for the R = 0.47 and R = 0.32 compositions (recall that, between ε = 0.5 and 1.0, the reference curves are indistinguishable), then the distribution width increases with ε = 0.35 for R = 0.19, ε = 0.3 for R = 0.09, and to ε = 0.25 for the R = 0.0 network. It is also important to note that the M2 behavior for the R = 0.0 BTT-TCDDA network is not symmetric around Tg, which we have attributed to the degradation of this material at the very high temperatures (> 600 K or > 325 °C) achieved in the variable temperature experiments for this composition, and produced irreversible changes in the M2 behaviors. These results are summarized in Table 1 and reveal that, in the BTT-TCDDA networks, the distribution in Ea (and, correspondingly, τc) increases with the decreasing R. This is consistent with the picture of increased heterogeneous dynamics due to the chain-growth mechanism being dominant at high acrylate concentrations.

Dynamic Correlation Times
While the reference curves presented above allowed the distribution in activation energies or correlation times to be assessed, we also wanted to explore the impact distributions have on the . For the experimental data, the M 2 was normalized by M 2 = 540 kHz 2 to reflect the partial averaging due to the β-relaxation dynamics present prior to the T g .

Dynamic Correlation Times
While the reference curves presented above allowed the distribution in activation energies or correlation times to be assessed, we also wanted to explore the impact distributions have on the Arrhenius analysis that had assumed a single correlation time when analyzing the M 2 temperature behavior. An "effective" correlation time, τ eff , to describe the polymer chain fluctuations near the T g was determined using Equation (A6) (additional discussion is provided in the Appendix A) [6,7,19,20,49]. Activation energies were then estimated assuming the Arrhenius behavior of τ eff in the high temperature limit. The low temperature plateau shows an invariant τ eff and is an artifact of the dynamics not being fast enough at these low temperatures to average the dipolar coupling (i.e., very slow motions are not probed by the 1 H M 2 ). For a Gaussian distribution of activation energies with small σ (Figure 9a), the linear Arrhenius behavior through the glass transition is easily fit, giving well-defined activation energies (± 0.5 kJ/mol). With the increasing σ, defining the linear region for analysis becomes more difficult. For example, assuming a Gaussian distribution with increasing σ (Figure 9b), the variation of τ eff across the T g shows more curvature. This is intuitively consistent, because there are both dynamic processes that can average the dipolar coupling active at lower temperatures (lower E a ), as well as dynamics requiring higher temperatures (higher E a ) to become active. The curvature in the τ eff high-temperature behavior increases the error in measuring the E a and can become as large as ± 5 kJ/mol (for the variation in distributions shown in Figure 9b), depending on the temperature range for which the linear regression is defined. The average E a for a Gaussian distribution can be calculated from the temperature of the midpoint in the M 2 transition regardless of σ (Figure 3a), if the pre-exponential τ 0 factor is known and is invariant (see Section 2.6), but the M 2 line shape analysis does not allow these two parameters to be distinguished [38].
well-defined activation energies (± 0.5 kJ/mol). With the increasing σ, defining the linear region for analysis becomes more difficult. For example, assuming a Gaussian distribution with increasing σ (Figure 9b), the variation of τeff across the Tg shows more curvature. This is intuitively consistent, because there are both dynamic processes that can average the dipolar coupling active at lower temperatures (lower Ea), as well as dynamics requiring higher temperatures (higher Ea) to become active. The curvature in the τeff high-temperature behavior increases the error in measuring the Ea and can become as large as ± 5 kJ/mol (for the variation in distributions shown in Figure 9b), depending on the temperature range for which the linear regression is defined. The average Ea for a Gaussian distribution can be calculated from the temperature of the midpoint in the M2 transition regardless of σ (Figure 3a), if the pre-exponential τ0 factor is known and is invariant (see Section 2.6), but the M2 line shape analysis does not allow these two parameters to be distinguished [38]. Similarly, increasing the DC distribution also enhances the curvature of the τeff temperature behavior (Figure 9c), but is slightly less pronounced compared to the larger Gaussian distributions and can produce an uncertainty of ± 2 kJ/mol for the range of ε evaluated. Since the DC distribution is nonsymmetric, the average τc varies with the increasing ε and is responsible for the shifting of the midpoint in the M2 transition (Figure 4b). An inspection of Figure 9b,c shows that the difference in the τeff behavior between the Gaussian or DC distribution is subtle and would be difficult to distinguish experimentally.
In the presence of multiple dynamic processes, the temperature behavior of the τeff increases in complexity, as explored in Figure 10. Recall that the τeff estimated using Equation A6 assumes a single dynamic process. Inversion of the M2 variation in the presence of multiple dynamic processes (Equation A10) to define a single effective τeff is not possible, such that Equation A6 is strictly valid only for a single motion. If the M2 transitions are well-separated (the dynamic motions have very different Ea) and one considers each M2 transition separately, then it is possible to obtain more accurate Ea estimates. By estimating a single τeff for systems containing multiple dynamic processes, an incorrect estimate for the Ea describing the initial transition (β-relaxation) is observed for both the Similarly, increasing the DC distribution also enhances the curvature of the τ eff temperature behavior (Figure 9c), but is slightly less pronounced compared to the larger Gaussian distributions and can produce an uncertainty of ± 2 kJ/mol for the range of ε evaluated. Since the DC distribution is nonsymmetric, the average τ c varies with the increasing ε and is responsible for the shifting of the midpoint in the M 2 transition (Figure 4b). An inspection of Figure 9b,c shows that the difference in the τ eff behavior between the Gaussian or DC distribution is subtle and would be difficult to distinguish experimentally.
In the presence of multiple dynamic processes, the temperature behavior of the τ eff increases in complexity, as explored in Figure 10. Recall that the τ eff estimated using Equation (A6) assumes a single dynamic process. Inversion of the M 2 variation in the presence of multiple dynamic processes (Equation (A10)) to define a single effective τ eff is not possible, such that Equation (A6) is strictly valid only for a single motion. If the M 2 transitions are well-separated (the dynamic motions have very different E a ) and one considers each M 2 transition separately, then it is possible to obtain more accurate E a estimates. By estimating a single τ eff for systems containing multiple dynamic processes, an incorrect estimate for the E a describing the initial transition (β-relaxation) is observed for both the Gaussian and DC distributions results (Figure 9b,c) and should be avoided. If the entire β transition in the M 2 could be experimentally measured, then it would be possible to correctly determine the E a (1). As noted previously, experimentally, our NMR probe limits us from reaching very low temperatures and prevents a complete analysis of the β transition. Figure 10a,c do reveal that using a τ eff for the α-relaxation process allows the correct measurement of the E a (2) if the first dynamic process (i.e., β-relaxation) is well-separated (Ea > 40% different). Therefore, the activation energy corresponding to the glass transition temperature can be measured and will be the focus of the subsequent analysis for the BTT-TCDDA networks.
Ea(1). As noted previously, experimentally, our NMR probe limits us from reaching very low temperatures and prevents a complete analysis of the β transition. Figure 10a,c do reveal that using a τeff for the α-relaxation process allows the correct measurement of the Ea(2) if the first dynamic process (i.e., β-relaxation) is well-separated (Ea > 40% different). Therefore, the activation energy corresponding to the glass transition temperature can be measured and will be the focus of the subsequent analysis for the BTT-TCDDA networks.

Arrhenius Behavior for TCDDA-BTT Networks
Using the insight from the discussion above, the Arrhenius behavior of τeff for the BTT-TCDDA networks for different compositions is shown in Figure 11 and summarized in Table 1. Only the hightemperature glass transition (α-relaxation) was evaluated, with the Ea for the R = 0.0 network not reported because of an insufficient number of high-temperature data points. The Ea values are ~2 times smaller than those we reported for epoxy thermosets [19] and reflect the lower Tg values in these thiol-acrylate networks. When decreasing R, the activation energy decreases from 29.2 to 16.7 kJ (even though the Tg is increasing), while the pre-exponential term (τ0) increases by a factor of 10 over the same R range (Table 1). Both Ea and τ0 reveal a linear dependence in the composition ratio (R) ( Figure S4). The activation energy (Ea) increases for networks with larger R, while the entropy of activation (ln(τ0)) decreases for larger R. This behavior may be related to differences in the free

Arrhenius Behavior for TCDDA-BTT Networks
Using the insight from the discussion above, the Arrhenius behavior of τ eff for the BTT-TCDDA networks for different compositions is shown in Figure 11 and summarized in Table 1. Only the high-temperature glass transition (α-relaxation) was evaluated, with the E a for the R = 0.0 network not reported because of an insufficient number of high-temperature data points. The E a values arẽ 2 times smaller than those we reported for epoxy thermosets [19] and reflect the lower T g values in these thiol-acrylate networks. When decreasing R, the activation energy decreases from 29.2 to 16.7 kJ (even though the T g is increasing), while the pre-exponential term (τ 0 ) increases by a factor of 10 over the same R range (Table 1). Both E a and τ 0 reveal a linear dependence in the composition ratio (R) ( Figure S4). The activation energy (E a ) increases for networks with larger R, while the entropy of activation (ln(τ 0 )) decreases for larger R. This behavior may be related to differences in the free volume fraction of the BTT and TCDDA moieties and the relative contributions of each to the network free volume with changing R compositions. Similar trends in E a and τ 0 have been observed for the polymer relaxation in nano-patterned polymer films [50] and polymer thin films [51]. From the Arrhenius behavior of τ eff , a compensation or isokinetic temperature (T comp ) was determined (Figure 11b) to be~333 K (60 • C), which is below the T g for all R compositions but falls near the cure temperature (25 • C). The compensation temperature corresponds to that temperature at which the chain dynamics are equivalent for all R networks. free volume with changing R compositions. Similar trends in Ea and τ0 have been observed for the polymer relaxation in nano-patterned polymer films [50] and polymer thin films [51]. From the Arrhenius behavior of τeff, a compensation or isokinetic temperature (Tcomp) was determined ( Figure  11b) to be ~333 K (60 °C), which is below the Tg for all R compositions but falls near the cure temperature (25 °C). The compensation temperature corresponds to that temperature at which the chain dynamics are equivalent for all R networks. A linear correlation between the pre-exponential factor and the activation energy is observed for these BTT-TCDDA networks ( Figure 12a) and is an example of the compensation effect or enthalpy entropy compensation (EEC) [52]. The EEC phenomena is commonly reported in glass-forming polymer liquids [53][54][55]. The physical significance of compensation in polymers is still under discussion but has been related to the presence of cooperative dynamics during the glass transition [56,57], along with local structural heterogeneity and distributions of corresponding dynamic activation energies [58]. A complete analysis of the ECC effect is beyond the scope of this manuscript, but it is suggested that local dynamic heterogeneity plays a role in the ECC behavior in BTT-TCDDA networks. A linear correlation between the pre-exponential factor and the activation energy is observed for these BTT-TCDDA networks ( Figure 12a) and is an example of the compensation effect or enthalpy entropy compensation (EEC) [52]. The EEC phenomena is commonly reported in glass-forming polymer liquids [53][54][55]. The physical significance of compensation in polymers is still under discussion but has been related to the presence of cooperative dynamics during the glass transition [56,57], along with local structural heterogeneity and distributions of corresponding dynamic activation energies [58]. A complete analysis of the ECC effect is beyond the scope of this manuscript, but it is suggested that local dynamic heterogeneity plays a role in the ECC behavior in BTT-TCDDA networks.  Table 1 for the different BTT-TCDDA networks. For the R = 0.0 composition, Ea and τ0 were estimated from the linear dependence of these parameters with R, as shown in Figure  S4.
Using Table 1, the distributions in the correlation times (τc) for the different network compositions are shown in Figure 12b. These results support the notion that, during the cure of a homogeneous network, such as R = 0.47, the chain dynamics are very uniform, as the polymer chains are trapped in the glassy phase during sample vitrification. In contrast, during the cure of heterogeneous networks, such as R = 0.0, there is a distribution of chain dynamics where a population  (2)) and the activation energy (E a ) and (b) the τ c distributions from the experimental results in Table 1 for the different BTT-TCDDA networks. For the R = 0.0 composition, E a and τ 0 were estimated from the linear dependence of these parameters with R, as shown in Figure S4.
Using Table 1, the distributions in the correlation times (τ c ) for the different network compositions are shown in Figure 12b. These results support the notion that, during the cure of a homogeneous network, such as R = 0.47, the chain dynamics are very uniform, as the polymer chains are trapped in the glassy phase during sample vitrification. In contrast, during the cure of heterogeneous networks, such as R = 0.0, there is a distribution of chain dynamics where a population of mobile chains is retained as the material enters the glassy phase. This mobile fraction allows for residual functional groups to become spatially near each other due to chain diffusion, leading to additional crosslinking at a given cure temperature and, subsequently, higher T g values. Continued crosslinking (and increasing T g ) will shift the distribution of polymer chain dynamics to slower τ c values, until there are no longer any mobile chain fragments available that would permit further crosslinking. The remaining R compositions are intermediate between these limiting networks.

NMR Spectroscopy
Wide-line solid-state 1 H NMR spectra were acquired on a Bruker Avance III instrument operating at an observe frequency of 400.14 MHz using a 7-mm high-temperature DOTY MAS probe (DOTY Scientific Inc, Columbia, SC, USA) under static (non-spinning) conditions. All 1D static 1 H NMR spectra used a Hahn echo (HE) pulse sequence with an inter-pulse delay of 10 µs, 16 scan averages, and a 5-s recycle delay. Variable temperature experiments were conducted between 233 K (−40 • C) and 673 K (+400 • C) using a 5-min temperature equilibration prior to acquisition. The second moment (M 2 ) of the 1 H NMR spectra was evaluated using Equation (1) using MATLAB (MathWorks, Inc., Natick, MA, USA). The NMR observed glass transition temperature T g (NMR) was obtained by fitting the M 2 temperature variation using: where M 2 is the motionally averaged moment at temperatures just prior to the T g transition, M 2 is the residual second moment above the T g transition, and 1/b is the rate of the temperature-induced change occurring at T g . The limits were chosen near T g to separate the transition from the other motional (i.e., β-relaxation) M 2 variations. The distribution analysis used reduced temperatures ((T-T g )/T g ) incorporating T g (NMR) [19]. In general, T g (NMR) was found to be higher than the T g determined from DMA analysis and reflects the NMR time probed through the averaging of the dipolar coupling.

Materials
The preparation of these polymer materials has recently been detailed [20]. Briefly, tricyclodecane dimethanol diacrylate (TCDDA) and the aromatic comonomer 1,3,5-benzenetrithiol (BTT) were mixed with different ratios of the reactive functional groups defined by the ratio R = (SH) 0 /(C=C) 0 , where (SH) 0 and (C=C) 0 were the initial concentrations of the thiol and acrylate functional groups, respectively. The photo initiator p-xylylene bis(N,N-diethyldithiocarbamate) (XDT) was added to the mixture at a 1 wt% concentration. The samples were photocured at room temperature (RT) using a Henkel (Düsseldorf, Germany) Loctite Zeta 7400 UV lamp. The light intensity was 100 mW/cm 2 at 365 nm (300 mW/cm 2 at 254 nm) measured using a Thorlabs (Newton, NJ, USA) PM100D power meter with a S120VC photodiode sensor. The sample temperature was observed to increase to~80 • C during the UV cure. After photocuring, the samples were further thermally post-cured (in the absence of UV light) by heating to 250 • C at 3 • C/min.

Conclusions
The impact of distributions in activation energies and correlation times describing dynamic motions on the temperature behavior of the NMR spectral second moment, M 2 , were evaluated and allowed methods to be developed for the experimental measurement of distributions and activation energies. It was demonstrated that, with the use of a T g -reduced temperature scale, it is possible to directly compare the M 2 behavior during the glass transition of polymers over a wide range of conditions. A set of reference curves were developed to address the impact of distribution widths, assuming a Gaussian distribution of activation energies or a Davidson-Cole distribution of dynamic correlation times. These reference curves were used to estimate changes in the distributions from the 1 H NMR M 2 temperature variations for a series of BTT-TCDDA networks. These NMR M 2 analyses demonstrated that there is an increase in the distribution of dynamic relaxation rates for the polymer chain motion near T g for networks dominated by chain-growth polymerization chemistry and that this leads to increasing T g values at a given cure temperature.  Figure S1: DMA analysis from the BTT-TCDDA networks. Figure S2: Simulated M 2 behavior with variation in pre-exponential correlation time. Figure S3. Simulated probability distributions. Figure S4. Correlations between R and E a or τ 0 .  Acknowledgments: Sandia National Laboratories is a multi-mission laboratory managed and operated by National Technology and Engineering Solutions of Sandia, LLC., a wholly owned subsidiary of Honeywell International, Inc., for the U.S. Department of Energy's National Nuclear Security Administration under contract DE-NA0003525. This paper describes objective technical results and analyses. Any subjective views or opinions that might be expressed in the paper do not necessarily represent the views of the U.S. Department of Energy or the United States Government.

Conflicts of Interest:
The authors declare no conflicts of interest.  For static NMR spectra whose line width is dominated by dipolar couplings, M 2 was first derived by Van Vleck [59], with the impact of molecular reorientation further described by Powles and Gutowsky [49,60]:

NMR
Nuclear Magnetic Resonance TCDDA Tricyclodecane dimethonal diacrylate Tg Glass transition temperature Tg(NMR) Glass transition temperature obtained from NMR M2 analysis UV Ultraviolet

A.1. NMR Second Moment
The second moment ( ) 2 M of the NMR spectral line shape ( ) f ω as a function of frequency ω, around a maximum at frequency 0 ω , is defined by [6,7] ( ) ( ) ( ) For static NMR spectra whose line width is dominated by dipolar couplings, For a single dynamic process, the second moment is a function of the motional correlation time and can be defined by [49,60,61] (