3.3.2. Model Fitting
Although there is bi-exponential behavior in the T
data, it creates an additional difficulty in the analysis since it is not possible to resolve the spectral components at lower magnetic fields. Therefore, the relaxometry experimental data were fitted to a single exponential. Furthermore, using different values for the population simply results in a vertical shift of the NMRD profiles, which does not affect any of the conclusions that will follow.
Figure 4 displays the NMRD and model fitting curves for the non-paramagnetic systems, and
Table 4 shows the corresponding parameters. The parameters’ uncertainties were estimated by testing the sensitivity of the fit to each parameter individually. The model fitting to the data was performed using the open-access online platform
fitteia.org, which applies the nonlinear least squares minimization method with a global minimum target [
24].
Regarding rotational diffusion, decreases with increasing DMSO concentration. Looking at the samples having 50% DMSO, it is possible to observe a smaller for the protonated systems as a result of the direct observation of faster moving the DMSO protons. Using the software Avogadro, it was possible to determine the relative position of the intramolecular atoms. This information was the used to calculate the average and consequently the value of . For the 50%DMSO-h6 case, the results from a weighted average of the [Aliquat][Cl] and DMSO values.
For the translational diffusion contribution, the
H spin density and the diffusion coefficient were fixed to the values presented in
Table 1 and
Table 3, respectively. The mean square jump distance was set to the distance
extracted from the X-ray profiles, but allowed to vary within its uncertainty. The distance between spins,
d, has larger values for the 0.1%, 10% DMSO concentrations when compared to the 50% and 99% cases. This differentiates between the diffusion of the [Aliquat]
ion as a whole and the diffusion of the individual aliphatic chains. These results are consistent with the fact that large enough concentrations of DMSO liberate individual chain movements.
Regarding the OPF mechanism, the parameters obtained reveal that increasing DMSO concentrations gradually destroy the locally ordered domains, as seen by the increase in . However, the system with 1% DMSO does not follow this trend, and seems to be one for which this mechanism is most effective. In fact, this result is corroborated both by X-ray results and diffusometry, according to which this DMSO concentration induces larger and smaller diffusion coefficients.
As the OPF and the translational diffusional mechanisms are most effective within the same frequency range, it is possible to have either one as the most important for relaxation. However, by applying the methods and models described above to [Aliquat][Cl] NMRD profiles at different temperatures [
8], and taking into account the fact that
should increase with temperature, it is possible to conclude that the OPF mechanism must dominate over self-diffusion (see
Appendix A.1). These fits can be accessed in the
Supplementary Materials—Figure S1. It is also important to note that, by using viscometry data reported by Litaiem and Dhahbi [
25], it was possible to confirm that the activation energy of the OPF prefactor is related with the viscosity activation energy (
Figure S2) in a manner consistent with the fact that
(see
Appendix A.2).
Cross relaxation was only observed for the systems with 0.1% and 10% DMSO. A single contribution was simultaneously fitted to these three systems given the fact that the cross relaxation peak was mostly masked by other mechanisms and that it seems reasonable to assume that the CR contribution should not vary significantly for these three systems.
Figure 5 displays the NMRD and model fitting curves for the magnetic systems, and
Table 5 shows the corresponding parameters. The “Bulk” curve is the sum of non-paramagnetic mechanisms, which were obtained from the corresponding IL samples and fixed for the NMRD of the MIL samples. The electronic spin
S was fixed to 2.5 throughout the analysis, corresponding to five unpaired electrons, and the solvent mass and density were fixed to the values presented in
Table 1. The factor
q was fixed to 1 because each [FeCl
]
ion is attached to a single [Aliquat]
ion, and the factor
F was fixed to 0.167, based on the assumption that only the nine hydrogen spins closest to the [Aliquat]
nitrogen participate in the IS mechanism.
It was considered that diffusion of the anion occurs by discrete events with an average time between them equal to
. That is, the anion diffuses together with the cation for a time
, then it breaks out from the electrostatic interaction and diffuses with constant
until it settles on another cation’s vicinity. This diffusion by random jumps has an average jump distance relation that is similar to the Torrey model, i.e.,:
where
is a constant that depends on the dimensionality of the diffusion. It was reasoned that the average distance travelled would be the average distance between two contiguous polar regions of the [Aliquat]
ion, which is roughly equal to two times the distance between [Aliquat]
aliphatic chains,
. Since the polar regions of the [Aliquat]
ions define a layer or region with a small thickness, the diffusivity was considered to be two-dimensional, and therefore
. Then, the relationship used for all systems was:
The value of is directly related to the chemical environment surrounding the [FeCl] ion. It was reasoned that it should be the same for every sample, but the fact that it was impossible to fit the system with 99%DMSO-h6 with the same value indicates that the [FeCl] ions in that system are in a fundamentally different environment. This suggests that, in the 99% system, we’re actually seeing the DMSO protons relaxation due to the PM mechanism in which the [FeCl] are surrounded by a larger number of DMSO molecules in its solvation region.
Since we are also able to observe DMSO relaxation in the 50% DMSO-
h6 system, a sum of two different OS contributions was fitted, one with the 50% DMSO-
d6 parameters, to obtain the OS relaxation of the [Aliquat]
protons, and another with the 99% DMSO-
h6 parameters to observe the OS relaxation of the DMSO protons. These two curves were connected by the factor
G, which is a measure of the contribution of the hydrogen spins relaxing due to the solvated [FeCl
]
ions, by Equation (
18):
in which
and
are the OS contributions for 99% DMSO-
h6 and 50% DMSO-
d6 and also
. As the IS mechanism is related to the [FeCl
]
ions in the vicinity of the [Aliquat]
nitrogen, it was considered that the IS contribution in 50% DMSO-
h6 should have the same parameters as in 50% DMSO-
d6. This contribution was multiplied by a factor
, which is the fraction of
H spins in the system that belongs to [Aliquat]
ions.
decreases with increasing DMSO concentration, which shows that the presence of larger concentrations of DMSO results in a larger number of collisions per unit time of the [FeCl] ion with its environment. Again, the 1% DMSO system is the exception, further elucidating the role of small quantities of DMSO in stabilizing the resulting structure.
does not vary uniformly, decreasing slightly up to 10% DMSO concentrations and increasing for larger concentrations. Its increase for 99% DMSO is consistent with the assumption that the [FeCl
]
ions are solvated, since DMSO diffuses faster than the [Aliquat]
ion. The OS distance,
, was determined to be one fourth of
(shown in
Table 2), while allowed to vary within its error. This roughly corresponds to the distance between the polar region and the center of mass of each cation, and the distance decreases with increasing DMSO concentration.
decreases with increasing DMSO concentration, which implies that the presence of more DMSO liberates the rotation of the complex. Within the anion diffusion approximation, the exchange time was estimated to be in the nanosecond range. The IS distance, , seems to be constant for different DMSO concentrations, apart from the 0% case, suggesting that the presence of DMSO, even in small quantities, restricts the relative position of the [FeCl] ion with respect to the cation.