13 C NMR relaxation and reorientation dynamics in imidazolium-based ionic liquids: revising interpretation

The temperature dependencies of 13 C NMR relaxation rates in [bmim]PF 6 ionic liquid have been measured and the characteristic times ( t c ) for the cation reorientation have been recalculated. We found the origin of the incorrect t c temperature dependencies that were earlier reported for ring carbons in a number of imidazolium-based ILs. After a correction of the approach 13 C T 1 , the relaxation data allowed us to obtain the characteristic times for an orientation mobility of each carbon, and a complicated experiment, such as NOE, was not required. Thus the applicability of 13 C NMR relaxation rate measurements to the calculation of the characteristic times for reorientation of all the carbons of the [bmim] + cation was confirmed and our findings have shown that a 13 C NMR relaxation technique allowed its application to ionic liquids to be equally successful as for other liquid systems.


Introduction
Ionic liquids, as a new class of liquid systems, have been investigated actively during the last few decades. NMR relaxation is one of the well known techniques for studying the rotational reorientation of ions and molecules in various liquid systems, including imidazolium-based ionic liquids, see e.g. ref. [1][2][3][4][5][6][7][8][9][10][11][12][13][14] and references therein. In particular, 13 C NMR relaxation has been successfully used for the investigation of the reorientation mobility in a large number of liquid systems. However, its applicability to ionic liquids was recently called into question. In a set of papers W. R. Carper with co-workers [8][9][10][11][12][13] reported on the non-monotonic temperature dependence of the characteristic time (henceforth referred to as correlation time, t c ) for imidazolium-ring carbon reorientation, see e.g. Fig. 2 in ref. 10. The authors attributed the results of the presence of a large contribution of the chemical shift anisotropy (CSA) to ring carbon relaxation, and suggested using Nuclear Overhauser Effect (NOE) experiments in order to correct the relaxation data. However, direct measurements of spin-lattice relaxation time (T 1 ) at two magnetic fields proved undoubtedly that the CSA contribution is negligible for all imidazolium carbons including the ring carbons. 14 As a result, this led to a situation where this well-known and successfully used approach cannot be applied to ionic liquids due to unknown reasons. The aim of our work, therefore, is to analyze the problem and verify the difficulties (if any) in the applicability of NMR T 1 measurements for the investigation of molecular mobility in ionic liquids. With regard to maintaining the relevance and importance of the problem to date, this is evidenced by the following facts. (1) In a recent review 15 in 2011, the conclusion of the studies reported in ref. [8][9][10][11][12][13] was repeated without any comments and without any mention of the results of the work. 14 (2) The authors of ref. 4 (and maybe of other papers) did not use the measured 13 C relaxation data for [bmim] + cations in order to calculate quantitatively t c magnitudes.

Results and discussion
As a first step we have repeated the measurements of 13 C T 1 in the same ionic liquid as in ref. 8-10 namely, [bmim]PF 6 . This appeared to be necessary, as we could not find in ref. 8-10 the primary information, i.e. tables of relaxation times at various temperatures. A chemical structure of the [bmim] + cation is shown in Scheme 1 together with an attribution of the lines.
All measurements were carried out using a NMR AVANCE-400 spectrometer at resonance frequencies ( f 0 ) of 100 MHz for 13 C nuclei. The obtained dependencies shown in Fig. 1   Next, we calculated t c values to verify the ref. 8-10 results by following the same procedure as in the studies reported in ref. 8-13, i.e. by using the well-known BPP (Bloembergen-Purcell-Pound) theory. The theory was first suggested for proton relaxation by Bloembergen-Purcell-Pound. 16 Later it was applied to other nuclei including 13 C nuclei. 17 Namely, a protonated carbon 13 C relaxation rate may be expressed as: 17,18 Here o H and o C are cyclic resonant frequencies (2pf 0 ) of 1 H and 13 C, respectively, and A 0 is a constant, which does not depend on temperature and frequency. For the carbon nucleus this constant is given by the expression: were h is the reduced Plank constant (h/2p), r CH is the length of the C-H chemical bond, g H and g C are the magnetogyric ratios of 1 H and 13 C nuclei; n = 1, 2, and 3 for CH-, CH 2 -and CH 3 -groups, respectively; S 2 r 1 is a constant, the so called order-parameter, see for more details ref. 19. It is useful to note that eqn (1) reveals a BPP-type curve with a maximum only at and this can be verified by simple analytical or digital calculations, see details in the Appendix A. The knowledge of a precise position of the maximum is very important as at this point it is possible to determine, independently, a value of t c using the abscissa magnitude and the ratio in eqn (3) as well as the A 0 value using the ordinate magnitude, see details in Appendix A.
After which, the t c magnitude can be calculated at each temperature using eqn (1) and the obtained value for A 0 . Following this procedure we initially obtained non-monotonic t c dependencies similar to the authors of ref. [8][9][10]. However, a deeper analysis allowed discovery of an error in the calculations. Eqn (1) has two real roots at each temperature and both roots are positive. Calculation details are shown in Appendix B and the temperature dependencies of the roots, i.e. of the calculated t c values, are shown in Fig. 2. The root #1 (t c ) shows an increase in the molecular mobility during an increase in temperature, i.e. it corresponds to conventional models of molecular mobility. The second root, in contrast, shows an increase of the molecular mobility during a decrease in temperature. This contradicts any existing theory, and therefore, this root does not have any physical meaning. It is worth emphasizing that such a situation is not only a feature of ionic liquids as similar solutions of the eqn (1) with two roots will be the result for any fluid system where   10. Alternatively, if one uses the root #1 in the whole temperature range, then the calculations lead to a conventional temperature dependence of t c for all spectral lines, hence, it is not necessary to involve additional experimental procedures, such as NOE, in order to transform 1/T 1 experimental data into t c magnitudes.
Following this approach, we calculated t c temperature dependencies for all spectral lines (functional groups) of the cation with the exception of 4 0 -carbon (CH 3 -group of the butyl chain) as no clear position for a maximum was detected for that line in the temperature range used. It is worth emphasizing once again that no ''set in advance'' temperature dependence of t c was used in the calculation, therefore the received curves represent ''true'' t c values. The dependencies (Fig. 3) did not show any obvious signs of nonlinearity in the rather wide temperature range, therefore we found the use of a simple Arrhenius function possible: for an approximation of obtained data, see Table 1. The approximation was made on the 1000/T scale between 2.80 (357 K) and 3.40 (294 K). In the area of lower temperatures a strong deviation from linearity was observed for 1 0 -and 2 0 -carbons (a-and b-CH 2 -groups of the butyl chain), but this deviation simply reflects a deviation of the experimental 1/T 1 dependencies from the BPP-type, see Fig. 1b as well as Fig. 2a in ref. 8. To be precise, the t c magnitudes calculated in this area are incorrect, and these parts of the curves require a separate description. As regards the deviations of 1/T 1 dependencies themselves, they may be a result of an appearance of an additional low-temperature maximum of 1/T 1 due to an increase in the contribution of the internal rotation inside the cation butyl chain.
The results of Table 1 show that all the main trends observed in ref. [8][9][10][11][12][13] remained. Unfortunately, a more detailed numerical comparison of the parameters of Table 1 with the data of ref. 8-10 was not possible due to two reasons. First, the sets of t c magnitudes in our studies and in ref. 8-10 were calculated using different procedures. Namely, in our work we calculated t c directly from the relaxation data while in ref. 8-10 a complicated procedure was used based on correcting the erroneous t c values calculated from 13 C 1/T 1 by using additional NOE data. We are not certain that the results of this correction are actually equal to our t c set. Second, the authors of ref. 8-10 used a complicated model for data fitting. Their model includes two correlation times, which describe the overall cation reorientation and internal rotations as well as non-Arrhenius temperature   dependence of the overall correlation time. Although valid in essence, such a model overestimates the possibilities of the available set of experimental data for such a detailed description of the mobility of the cation. No particular signs of non-Arrhenius dependence were observed in the temperature range used (see above). In summary, we found the origin of incorrect t c values which were calculated earlier (see ref. [8][9][10] from the temperature dependencies of 13 C NMR relaxation rates of the ring carbons in a number of imidazolium-based ionic liquids. We demonstrated that the calculation using the right procedure leads to conventional temperature dependencies of t c for all carbons of the cation. This means that no additional experiments such as NOE are required to transform the experimental relaxation rates into the correct t c values. Hence, our findings have shown that a 13 C NMR relaxation technique can be applied to ionic liquids equally successfully as for other liquid systems.

A. The confirmation of eqn (3)
In this section, we describe the method of calculation of the temperature dependence of correlation time, t c , from 1/T 1C experimental data. For the methods used, two factors are necessary: (i) the temperature dependence of 1/T 1C has to achieve a maximum, (ii) the dependence has to be described by one correlation time approximation, i.e. by eqn (1). In other words, we calculated t c based on eqn (1) using eqn (3).
First, we demonstrate the correctness of eqn (3). A similar relationship is well known for a 1 H relaxation rate in the form of (t c o H ) E 0.616. However, as we could not find the correct relation for carbon relaxation in the literature, we therefore demonstrate below how the conditions of the function in eqn (1) (1) Eqn (3) allows the constant A 0 to be calculated by using experimental data from 1/T 1C at its maximum point. Substituting the values of x = 0.791 at the point of maximum to eqn (A.1) the constant of A 0 is calculated by the following equation As is evidenced from eqn (A.1) at G(x root ) = 0. As an illustration, the G(x) functions for several experimental points of the peak of carbon 2 are shown in Fig. 5. The values of x root at G(x) = 0 are the roots of eqn (B.2). Each function G(x) = 0 has two different roots with the exception of G(x) for 1/T 1C at the maximum. In this exception, two roots degenerate into one. We solved eqn (B.2) by numerical Fig. 4 The dependence of y(x) function and its components. The position of maximum of y(x) is equal to 0.791. methods and found both t c by eqn (B.3). The results of the calculation for the peak of carbon 2 are shown in Fig. 2. However, only one root is physically justified. For a discussion regarding the choosing of the root see the section: Results and discussion.