Elucidating the ¹H NMR relaxation mechanism in polydisperse polymers and bitumen using measurements, MD simulations, and models

The polymers used in this study are listed in Table 1. The average molecular weight $M_{w}$ and poly-dispersivity index $M_{w}/M_{n}$ were measured using gel permeation chromatography (GPC) using an Agilent Technologies 1200 module. The data in Table 1 indicate that the polymers are highly dispersive, up to $M_{w}/M_{n}\simeq 3.11$ in the case of B360000 poly(isobutene). The large polydispersivity of the polymers make them ideal for comparing with crude-oils, which are also highly dispersed as evidenced by their wide $T_{2}$ distributions freedman:spe2001 . In the case of the three poly(isobutene) polymers in Table 1, the viscosity fit well to the functional form $\eta\simeq A\,M_{w}^{\alpha}$ holden:japs1965 , with $\alpha\simeq 2.4$ and $A\simeq 1.07\times 10^{-4}$ in units of (cP) and (g/mol) at ambient singer:EF2018 . An illustration of a section of poly(isobutene) is shown in Fig. 2, which was used for molecular dynamics (MD) simulations.

The viscosity measurements were made using a Brookfield AMETEK viscometer. The viscosities did not depend on shear-rate (within experimental uncertainties), thereby making them suitable viscosity “standards” for comparing with NMR measurements which are measured at zero shear-rate. The viscosity measurements were made at both ambient temperature ($\simeq$ 25 ^∘C) and at equilibrated temperatures of 40 ^∘C using a circulation heat-bath. The viscosity data at 40 ^∘C is used as a proxy for the NMR data at 38.4 ^∘C.

A 1 GHz electron paramagnetic resonance (EPR) apparatus was used to measure the concentration of paramagnetic ions plus the (weight equivalent) concentration of free radicals (i.e. unpaired valence electrons), which both contribute to NMR paramagnetic relaxation. The EPR data on the Brookfield viscosity standards indicated $<100$ ppm paramagnetic impurities (i.e. the signal was below the detection limit of the apparatus). The paramagnetic concentration in the polymers is at least an order of magnitude less than the estimated $\simeq$ 1,000 ppm for Athabasca bitumen zhao:ef2007 ; singer:EF2018 .

¹H NMR $T_{1}$ and $T_{2}$ measurements at a resonance frequency of $\omega_{0}/2\pi=f_{0}$ = 2.3 MHz were made with a GeoSpec2 from Oxford Instruments, with a 29 mm diameter probe. The samples were measured at ambient conditions ($\simeq$ 25^∘C) and after temperature equilibration to $\simeq$ 30^∘C. Additional measurements at $\simeq$ 35^∘C were made by turning off the chiller and equilibrating to the magnet temperature. $T_{1}$ and $T_{2}$ measurements at $f_{0}$ = 22 MHz and $\simeq$ 30^∘C were made with a special spectrometer from MR Cores at Core Laboratories, with a 30 mm diameter probe. $T_{1}$ and $T_{2}$ measurements at $f_{0}$ = 400 MHz at $\simeq$ 25^∘C were made with a Bruker Avance spectrometer, in a 5 mm diameter probe.

¹H NMR $T_{1}$, $T_{1\rho}$, and $T_{2}$ measurements at $f_{0}$ = 21 MHz and 38.4^∘C were made at IFP-EN on a Maran Ultra, with a 18 mm diameter probe. The $T_{1\rho}$ measurements in the rotating frame kimmich:book ; steiner:cpl2010 were made with a spin-locking frequency $\omega_{1}/2\pi=f_{1}=$ 1.7 kHz $\leftrightarrow$ 42 kHz. Field cycling measurements were made on a Stelar fast field-cycling (FC) relaxometer from $f_{0}=0.01\leftrightarrow$ 30 MHz at 38.4^∘C, with a 10 mm diameter probe. All of the above $T_{2}$ measurements were measured using a CPMG sequence with echo spacings of $T_{E}=0.1$ ms or less, except at $f_{0}$ = 400 MHz where $T_{2}$ was estimated from $T_{2}\simeq 1/\pi\Delta\!f$, where $\Delta\!f$ is the width of the NMR spectrum. The hydrogen index ($HI\simeq$ 1.17) of the polymers is discussed in singer:EF2018 .

The $T_{1}$, $T_{1\rho}$, and $T_{2}$ distributions of the pure polymers were determined using inverse Laplace transforms venkataramanan:ieee2002 ; song:jmr2002 . The FC $T_{1}$ distributions shown in Figs. 3 and 4 tend to narrow with increasing frequency due to larger (absolute) longitudinal cross-relaxation kalk:jmr1976 ; kowalewski:book (a.k.a. spin-diffusion). Figs. 3 and 4 also show the finite ramp-time of 3 ms required to ramp the field up and down. While in theory this does not effect the $T_{1}$ acquisition kimmich:pnmrs2004 , it has been noted that it does effect broad $T_{1}$ distributions with fast relaxing components roos:jbio2015 ; ward:jpc2018 . Our relaxation model indicates that this is likely the case for the 102,000 cP polymer, where the log-mean $T_{1LM}$ is most likely overestimated by a factor $\simeq 2^{1/2}$ at low frequencies.

The $T_{1\rho}$ distributions at $f_{1}$ = 1.7 kHz and $f_{1}$ = 42 kHz are shown in Fig. 5, alongside the $T_{2}$ distribution. $T_{1\rho}$ and $T_{2}$ distributions tend to narrow with increasing viscosity, which is opposite to the trend in polydispersivity index $M_{w}/M_{n}$ in Table 1. The narrowing may therefore be a result of larger transverse cross-relaxation (in the low frequency limit) with increasing viscosity (i.e. increasing correlation time) kowalewski:book . In the case of the most viscous polymer, Fig. 5 shows that $T_{1\rho}$ increases when going from $f_{1}$ = 1.7 kHz to 42 kHz, indicating the presence of molecular correlation times shorter than $\tau\lesssim(2\omega_{1})^{-1}\simeq$ 1 $\mu$s.

The log-mean values $T_{1LM}$, $T_{1\rho LM}$ and $T_{2LM}$ of the distributions are used for data analysis and fitting to the model, where for example $T_{1LM}=\exp\!\left<{\rm ln}\,T_{1}\right>$, which is justified from the constituent viscosity model freedman:spe2001 . As shown in the Supporting Information, the effects of dissolved oxygen on $T_{1}$ as a function of frequency were measured for $n$-heptane, and the concentration of dissolved oxygen $C_{\rm O_{2}}$ in the polymer-heptane mix was predicted by MD simulations. The results indicate that the effects of dissolved oxygen on $T_{1}$ (and $T_{2}$) are negligible for all the polymers at all frequencies.

The underlying expressions for $T_{1}$, $T_{2}$ and $T_{1\rho}$ in an isotropic system are given by mcconnell:book ; cowan:book :

$J(\omega_{0})$ is the spectral density at the resonance frequency $\omega_{0}=2\pi f_{0}$. The expression for $T_{1}$ in the rotating frame, $T_{1\rho}$, is similar to $T_{2}$ except that the zero-frequency term $J(0)$ is replaced by $J(2\omega_{1})$ where $\omega_{1}=2\pi f_{1}$ is the spin-locking frequency kimmich:book . Note that Eq. 1 does not assume a model for the spectral density $J(\omega)$.

Molecular dynamics (MD) simulations of monodisperse polymers were conducted. Four different chain lengths of poly(isobutene) (see Fig. 2) were simulated at 25^∘C: a 16-mer, 8-mer, 4-mer and 2-mer. The viscosity of the 16-mer and 8-mer were estimated using the relation $\eta=A\,M_{w}^{\alpha}$ holden:japs1965 , with $\alpha\simeq 2.4$ and $A\simeq 1.07\times 10^{-4}$ in units of (cP) and (g/mol) at ambient singer:EF2018 . For example, in the case of the 16-mer where $M_{w}=912$ g/mol, this predicts a viscosity of $\eta\simeq$ 1,000 cP. The viscosities of the 4-mer and 2-mer were predicted using the empirical relation $T_{2LM}\simeq 9.56\left(\eta/T\right)^{-1}$ lo:SPE2002 in units of (ms) and (cP/K).

MD simulations of the intra-molecular ($T_{1R,2R}$) bloembergen:pr1948 and inter-molecular ($T_{1T,2T}$) torrey:pr1953 ; hwang:JCP1975 ¹H-¹H dipole-dipole relaxation were then computed for the polymers, from which the total relaxation times ($T_{1,2}$) are calculated:

The procedure for the MD simulations are the same as reported elsewhere singer:jmr2017 ; asthagiri:seg2018 ; singer:jcp2018 , and details are given in the Supporting Information.

Fig. 6(a) shows the ratio of inter-molecular to intra-molecular relaxation times $T_{1T}/T_{1R}$ as a function of frequency for the four polymers. A value larger than unity $T_{1T}/T_{1R}>1$ indicates that intra-molecular relaxation dominates, while $T_{1T}/T_{1R}<1$ indicates that inter-molecular relaxation dominates. We find that $T_{1T}/T_{1R}>1$ increases with increasing viscosity, implying that intra-molecular relaxation dominates (by at least an order of magnitude) at high viscosities ($\eta\gtrsim$ 1,000 cP). $T_{2T}/T_{2R}$ (not shown) show similar results. These findings justify the assumption in our model (Eq. 8) that intra-molecular relaxation dominates over inter-molecular relaxation.

Fig 6(b) shows the total relaxation $T_{1}$ (Eq. 12) as a function of frequency for the four polymers. The lowest viscosity (1 cP) polymer shows high $T_{1}$ values and minimal dispersion (i.e. minimal frequency dependence). On the other hand, the highest viscosity polymers show significant dispersion. The 189 cP and 1,000 cP polymers merge above $f_{0}>$ 100 MHz into a linear relation $T_{1}\propto f_{0}$. This behavior is exactly predicted by the model (Eq. 10), namely that $T_{1}\propto f_{0}$ is independent of viscosity. The $f_{0}>$ 100 MHz region for the 1,000 cP polymer is compared with measurements and the model below.

We note that a similar relation was previously reported from MD simulations of heptane confined in a polymer matrix, where the surface relaxation $T_{1S}$ of heptane followed $T_{1S}\propto f_{0}$ under confinement parambathu:arxiv2020 . This implies a connection between the molecular dynamics of high-viscosity fluids and low-viscosity fluids under confinement.

The results for $T_{1LM}$ for crude oils (including bitumen) and polydisperse polymers are shown in Fig. 7(a). The crude-oil data are taken from various sources listed in the legend. The most recent addition is the bitumen data at $f_{0}=$ 400 MHz by Kausik et al. kausik:petro2019 , measured over a range of temperatures (30^∘C $\leftrightarrow$ 90^∘C).

Also shown in Fig. 7(a) is the BPP prediction bloembergen:pr1948 at $f_{0}$ = 2.3 MHz and 400 MHz from Eq. 2. The crude oils roughly follow the BPP prediction $T_{1LM}\propto(\eta/T)^{-1}$ at low viscosities, however $T_{1LM}$ clearly plateaus at high viscosity. Also shown are the MD simulations at 400 MHz for the polymers, which are consistent with the polydisperse polymer measurements.

Fig. 7(b) shows the same data as Fig. 7(a) but on a frequency normalized scale. More specifically, the $x$-axis ($\eta/T$) is multiplied by $f_{0}/2.3$ with $f_{0}$ in units of MHz, while the $y$-axis ($T_{1LM}$) is divided by $f_{0}/2.3$. Frequency normalizing has the effect of collapsing the frequency dependence of the BPP model onto one universal curve zhang:spwla2002 . It also has the effect of collapsing the bitumen and polymer data in the slow-motion regime onto one plateau value given by $T_{1LM}\times 2.3/f_{0}\simeq$ 3 ms, i.e. $T_{1LM}\propto f_{0}$. The more recent bitumen data at 400 MHz shows a slight departure from the low frequency data, namely the plateau value is lower than at lower frequencies. As shown below, the model takes this departure into account with the $\omega_{0}\tau_{L}$ term in Eq. 10.

Fig. 8 shows $T_{1LM}$ data for the polydisperse polymers and the bitumen in the slow-motion regime (i.e. high-viscosity) regime, which corresponds to data within the horizontal lines (the log-average) in Fig. 7(a). The best fit to the new model using Eq. 9 and Eqs. 1 are shown for both polydisperse polymers and bitumen, and the best fit parameters are shown in Table 2. The second moment is fixed to $\Delta\omega_{R}/2\pi$ = 20.0 kHz, which is the value for $n$-heptane singer:EF2018 . A Stokes radius of $R=$ 1.85 Å is used for the polymers, which is the value needed to match the correlation $T_{1LM,2LM}=9.56\,(\eta/T)^{-1}$ in the low-viscosity regime lo:SPE2002 . A slightly larger Stokes radius of $R=$ 2.47 Å is used, which is the value needed to match the correlation $T_{1LM,2LM}=4.0\,(\eta/T)^{-1}$ in the low-viscosity regime freedman:spe2001 .

We note that the fact that $T_{1LM,2LM}$ are consistent with a constant Stokes radius $R$ in the low-viscosity regime, i.e. $R$ is independent of molecular size (and therefore viscosity), clearly shows that $T_{1,2}$ are probes of the local molecular dynamics. This is in stark contrast to the radius of gyration $R_{g}$ which depends on the molecular size singer:jmr2017 .

The first free-parameter in the model is the order parameter $S^{2}$, which characterizes the rigidity of the molecule. $S^{2}\simeq 0.147$ is found for the polydisperse polymers, which is consistent with previously reported data for monodisperse polymers at high molecular-weights $M_{w}>$ 4,000 g/mol graf:prl1998 ; kariyo:prl2006 ; kariyo:macro2008b . The fit to bitumen indicates a lower $S^{2}\simeq 0.085$, implying a less-rigid molecule (i.e. more isotropic internal-motions). The second free-parameter is the local correlation-time $\tau_{L}$, which characterizes the fast $\tau_{L}$ ($\simeq$ 10’s ps) motions of the molecular branches. The fit indicates a local correlation time of $\tau_{L}\simeq 20$ ns for the polydisperse polymers, and a longer $\tau_{L}\simeq 53$ ns for bitumen. We note that unlike bitumen, the fit for the polydisperse polymers is not very sensitive to $\tau_{L}$, therefore an upper bound $\tau_{L}\lesssim 20$ ns may be more appropriate for the polydisperse polymers.

Also shown in Fig. 8 are the MD simulations of the 1,000 cP polymer above $f_{0}>$ 100 MHz, corresponding viscosity independent region (see Fig. 6) where $T_{1}\propto f_{0}$. The agreement between simulation and data/model in Fig. 8 is remarkable given that only a monodisperse model of poly(isobutene) is used in the simulations. This suggests that the $T_{1}\propto f_{0}$ behavior is generic at high-viscosities, and that ¹H-¹H dipole-dipole relaxation dominates over paramagnetism at high-viscosities.

The results for $T_{2LM}$ for the crude oils are shown in Fig. 9(a), taken from various sources listed in the legend, with the most recent addition is the bitumen data at $f_{0}=$ 400 MHz by Kausik et al. kausik:petro2019 . $T_{2LM}$ for the polydisperse polymers are shown in Fig. 7(b), along with de-oxygenated $n$-alkane data shikhov:amr2016 , and the MD simulation results for the polymers at 400 MHz.

Solid lines are fits using the model in Eq. 11 with fitting parameters shown in Table 2, restricted to the high-viscosity region $\eta/T>0.3$ cP/K, or $\eta>100$ cP at ambient equivalently. Both the polydisperse polymers and bitumen data are consistent with $T_{2LM}\propto\left(\eta/T\right)^{-1/2}$ for viscosities higher than $\eta/T>0.3$ cP/K. The model indicates that $\tau_{0}$ is a factor $\simeq$ 6 larger for bitumen than for the polydisperse polymers. As discussed in Section II.2.2, there are two explanations for $\tau_{R}=(\tau_{\eta}\tau_{0})^{1/2}\propto(\eta/T)^{1/2}$ and the interpretation of constant $\tau_{0}$ korb:jpcc2015 ; kausik:petro2019 , and more investigations are required to narrow down the theory.

The BPP model predicts a “kink” in $T_{2}$ during the transition from the low- to high-viscosity regimes, where $T_{2}$ is shifted up by a factor 10/3 with increasing viscosity. The kink is supposed to occur at a viscosity corresponding to $\omega_{0}\tau_{\eta}=0.615$, which as shown in Fig. 9 predicts am intermittent spread between low frequency (2.3 MHz) and high frequency (400 MHz) data. However, no spread between the 2.3 MHz and 400 MHz $T_{2LM}$ data is apparent (within uncertainties), for both polydisperse polymers and bitumen. In other words, there is no apparent frequency dependence in $T_{2LM}$ (within uncertainties) during the low- to high-viscosity regime, at least up to 400 MHz. We also note that such a kink in $T_{2LM}$ has never been reported before for polydisperse fluids with a broad $T_{2}$ distribution.

We also note that the MD simulations of the polymers agree well with the measurements, which again suggests that ¹H-¹H dipole-dipole relaxation dominates over paramagnetism at high-viscosities.

The full frequency dependence in $T_{1LM}$ for the polydisperse polymer are presented in Fig. 10. Also shown are the fits from the full expression of the model in Eq. 8 using fixed values of $S^{2}$ and $\tau_{L}$ listed in Table 2, and optimized values of $\tau_{0}$ listed in Table 3. The optimized values of $\tau_{0}$ in Table 3 tend to decrease with increasing viscosity, however the average agrees with the value in Table 2 determined from the entire set of $T_{2LM}$ data. The factor $\simeq$ 2 smaller $\tau_{0}$ (and factor $\simeq\,2^{1/2}$ smaller $\tau_{R}$) for 102,000 cP is likely due to FC ramp-time effects which overestimate $T_{1LM}$ by a factor $\simeq\,2^{1/2}$.

The full expression of the model shows good agreement with the data over the entire frequency range $f_{0}=0.01\leftrightarrow 400$ MHz, including the coalescence at high frequencies corresponding to the slow-motion regime $\omega_{0}\tau_{R}\gg 1$. Likewise, the MD simulation of the 1,000 cP polymer above $f_{0}>$ 100 MHz shows $T_{1}\propto f_{0}$ behavior, consistent with the data and the model.

Fig. 11 shows the same $T_{1LM}$ data as Fig. 10, along with $T_{2LM}$ and $T_{1\rho LM}$ (at $f_{1}$ = 1.7 kHz) which show similar values to $T_{1LM}$ at $f_{0}$ = 0.01 MHz. This implies that $T_{2LM}$ and $T_{1\rho LM}$ ($f_{1}$ = 1.7 kHz) have no dependence on $f_{0}$ (within uncertainties), and remain in the fast-motion regime even when $\omega_{0}\tau_{R}\gg 1$. We speculate this is due to the broad distribution in correlation times, which for $T_{1}$ is more efficiently narrowed by longitudinal cross-relaxation (a.k.a. spin-diffusion) than for $T_{2}$ by transverse cross-relaxation. This stems from the fact that transverse cross-relaxation for $T_{2}$ is only effective in the “non-secular” limit, i.e. the low frequency limit kowalewski:book . As such, we speculate that the transition from the fast- to slow-motion regime may occur over a much broader frequency range for $T_{2}$ than for $T_{1}$. This is supported by the lack of frequency dependence (i.e. the lack of a kink) in $T_{2LM}$ shown in Fig. 9(b). As such, we retain the factor 10/3 in Eq. 11 even in the slow-motion regime, at least up to 400 MHz.

The measured $T_{1,2}$ of a bulk fluid at atmospheric conditions are given by the following:

$T_{1B,2B}$ are the bulk contributions from ¹H-¹H dipole-dipole relaxation. For de-oxygenated $n$-alkanes at ambient it was previously shown that $T_{1B,2B}=9.56\,(\eta/T)^{-1}$ lo:SPE2002 , which for $n$-heptane is $T_{1B,2B}$ = 7,320 ms at 25 ^∘C (where $\eta$ = 0.39 cP).

Given the measured $T_{1,2}$ and the known $T_{1B,2B}$ for $n$-heptane at ambient, Eq. 13 yields the paramagnetic contribution $T_{1{\rm O_{2}}}$ from dissolved oxygen. As shown in Fig. 12(a), $T_{1{\rm O_{2}}}$ increases from $T_{1{\rm O_{2}}}=$ 2,500 ms at $f_{0}=2.3$ MHz to $T_{1{\rm O_{2}}}=$ 5,600 ms at $f_{0}=400$ MHz. $T_{1{\rm O_{2}}}$ depends on the electron correlation time $\tau_{e}$ and the concentration $C_{\rm O_{2}}$ of dissolved oxygen through the following relations mcconnell:book :

where the BPP spectral density is given by:

$\tau_{e}$ is the electron correlation time of the paramagnetic O₂ molecule, and the second moment (i.e. strength) is given by $\Delta\omega^{2}_{\rm O_{2}}\propto C_{\rm O_{2}}$. Note that the Larmor frequency of the electron $\omega_{e}$ is larger than ¹H by a factor $\omega_{e}\simeq 659\,\omega_{0}$. The BPP model for the spectral density $J_{{\rm O_{2}}}(\omega)$ was previously shown to work well for a variety of solvents of various molecular weights teng:jmr2001 . Fig. 12(a) shows that the fit to $T_{1{\rm O_{2}}}$ using Eqs. 14 and 16 work well, with best fit parameters $\tau_{e}=1.15$ ps (which is consistent with teng:jmr2001 ) and $\Delta\omega_{\rm O_{2}}/2\pi$ = 62.8 kHz.

Given that the concentration of dissolved oxygen in $n$-heptane is $C_{\rm O_{2}}=$ 132 mg/L (or 4.12 mM equivalently) at ambient ($p_{\rm O_{2}}$ = 0.21 atm) implies that the relaxivity (defined in MRI terminology lauffer:cr1987 ) of O₂ is $r_{1}=0.10$ mM^-1s^-1 at low frequencies $f_{0}\lesssim$ 100 MHz.

As shown in teng:jmr2001 , $\tau_{e}$ is roughly constant for solvents with the molecular weight of heptane or higher, and also appears roughly constant at lower molecular weights ward:jpc2018 . Furthermore as shown in Fig. 12(b), MD simulations indicate that $C_{\rm O_{2}}$ is $\sim$30 % less for the 1,000 cP polymer than for heptane parambathu:arxiv2020 , implying that $T_{1{\rm O_{2}}}$ is $\sim$30 % larger for the polymer than for heptane. This predicts that $T_{1{\rm O_{2}}}\simeq$ 8,000 ms for the polymer at $f_{0}=400$ MHz, which is much larger than the measured $T_{1}\simeq$ 300 ms. In other words, the effects of oxygen on $T_{1,2}$ are negligible for the polymer at 400 MHz. Below 400 MHz, the effects of oxygen on the polymer are even less since the measured $T_{1,2}\ll$ 300 ms.

The only cases in this report where the effects of dissolved oxygen are apparent are for the low-viscosity crude-oils at $f_{0}\lesssim$ 100 MHz where $T_{1LM,2LM}\gtrsim$ 1,000 ms. In such cases $T_{1LM,2LM}$ deviate slightly from the BPP correlation line $T_{1LM,2LM}=4.0\left(\eta/T\right)^{-1}$ in the low-viscosity regime.

The autocorrelation function $G(t)$ for fluctuating magnetic ¹H-¹H dipole-dipole interactions is central to the development of the NMR relaxation theory in liquids bloembergen:pr1948 ; torrey:pr1953 ; abragam:book ; hwang:JCP1975 ; mcconnell:book ; cowan:book ; kimmich:book . From $G(t)$, one can determine the spectral density function $J(\omega)$ by Fourier transform as such:

for $G(t)$ in units of s^-2 mcconnell:book . As shown in Fig. 13(a), $G(t)$ for the $n$-alkanes has a multi-exponential (i.e. “stretched”) decay rather than the single exponential decay predicted by BPP. Furthermore, the degree of stretching increases with increasing carbon number (i.e. increasingly non-spherical geometry). MD simulations indeed confirm that $n$-pentane and $n$-hexane have a more stretched intra-molecular $G_{R}(t)$ autocorrelation function (i.e. a broader distribution of correlation times) than the symmetric neo-pentane and benzene molecules, respectively singer:jcp2018 . The degree of stretching can be further analyzed using inverse Laplace transforms (see also singer:jcp2018b and supporting information in singer:jcp2018 ).

By comparison, Fig 13(b) shows the effect of decreasing the exponent $\beta$ in the spectral density function $J^{\left(\beta\right)}\!(\omega)$ as such:

where the real part of the inverse Fourier transform $G^{\left(\beta\right)}\!(t)$ is also defined. An analytical expression for $G^{(\beta)}(t)$ exists for the two extreme cases:

The case of $\beta=1$, $G^{\left(1\right)}\!(t)$, is the BPP model with a single-exponential decay. The case of $\beta=1/2$, $G^{\left(1/2\right)}\!(t)$ abramowitz:1964 , is the stretched case used in the phenomenological model to account for the independence of $T_{1LM}$ on $\tau$, i.e. the independence on $\eta/T$. As demonstrated in Fig 13, increasing the carbon number increases the degree of stretching in Fig 13(a), which corresponds to decreasing $\beta$ in Fig 13(b).

It is interesting to note that the long time behavior of Eq. 21 is $G^{\left(1/2\right)}\!(t)\propto t^{-2}$ for $t/\tau\gg 10$. This is analogous to the Mittag-Leffler function, which starts off as a stretched exponential at short times, and turns into a power-law decay at long times magin:jmr2011 .

We note that $G^{\left(1/2\right)}\!(t)$ starts to diverge below $t/\tau=0.14$, therefore the model is valid for times $t/\tau\gtrsim 0.14$. Correspondingly, this implies that the model for $J^{\left(1/2\right)}\!(\omega)$ is valid for frequencies $2\omega\tau\lesssim 7.1$ (where the factor 2 reflects the $J(2\omega_{0})$ terms in Eq. 1). Given the local correlation-time $\tau_{L}\simeq 50$ ps, this corresponds to validity below frequencies $f_{0}\lesssim$ 10 GHz.

The Cole-Davidson function is widely used to describe the dielectric response davidson:jcp1951 ; lindesy:jcp1980 ; beckmann:prep1988 and its departure from the traditional Debye model. The function is defined as such:

which reduces to the following expression:

The Cole-Davidson exponent is bound by $0<\beta_{CD}\leq 1$, where $\beta_{CD}=1$ is the Debye model. As $\beta_{CD}$ is decreased, the underlying distribution $P_{CD}(\tau)$ in molecular correlation times $\tau$ grows wider.

The Cole-Davidson function has also been successfully used to describe NMR relaxation of glycerol and monodispersed polymers through the relation $\chi_{DD}^{\prime\prime}(\omega)\propto\omega/T_{1}$ kruk:pnmrs2012 ; flamig:jpcb2020 . The NMR Cole-Davidson spectral density $J_{CD}(\omega)$ is given by the following:

which reduces to the following expression:

For the purposes of comparison with our model, we have added a factor $\frac{1}{3}\Delta\omega_{R}^{2}/\beta_{CD}\pi/2$ to the definition of $J_{CD}(\omega)$. The factor $\frac{1}{3}\Delta\omega_{R}^{2}$ is related to the units convention of the spectral density cowan:book . The factor $1/\beta_{CD}\pi/2$ is introduced to highlight the similarity with our model. $J_{CD}(\omega)$ as defined in Eq. 25 becomes independent of $\beta_{CD}$ in the limit $\beta_{CD}\lesssim 10^{-3}$, and tends towards: