Relative Cation-Anion Diffusion in Alkyltriethylammonium-Based Ionic Liquids

19F Nuclear Magnetic Resonance spin-lattice relaxation experiments have been performed for a series of ionic liquids including the same anion, bis(trifluoromethanesulfonyl)imide, and cations with alkyl chains of different lengths: triethylhexylammonium, triethyloctylammonium, decyltriethylammonium, dodecyltriethylammonium, decyltriethylammonium, and hexadecyltriethylammonium. The experiments have been carried out in a frequency range of 10 kHz to 10 MHz versus temperature. A thorough analysis of the relaxation data has led to the determination of the cation–anion as a relative translation diffusion coefficient. The diffusion coefficients have been compared with the corresponding cation–cation and anion–anion diffusion coefficients, revealing a correlation in the relative translation movement of the anion and the triethylhexylammonium, triethyloctylammonium, decyltriethylammonium, and dodecyltriethylammonium cations, whereas the relative translation diffusion between the anion and the cations with the longer alkyl chains, decyltriethylammonium and hexadecyltriethylammonium, remains rather uncorrelated (correlated to a much lesser extent).


Introduction
Mechanisms concerning the translation diffusion of ionic liquids raise a great deal of interest in the field of fundamental science, as well as in terms of its use in applications. In both cases, one of the most important questions concerns the correlation effects in the ionic movement. One wonders to what extent the translation displacement of one ion (cation or anion) affects the movement of other ions (of the same and of the opposite sign) as a result of their mutual interactions.
It is very difficult to address this subject experimentally. Nuclear Magnetic Resonance (NMR) methods are able to give insight into ionic motion. NMR diffusometry probes the translational movement of individual ions in a magnetic field gradient [1,2]. For ionic liquids composed of 1 H containing cations and 19 F containing anions, one can obtain translation diffusion coefficients of the cation and the anion, respectively. The diffusion coefficients describe self-diffusion of the ions. This information is highly valuable, but it does not give insight into motional correlation. NMR relaxometry probes relative translational dynamics [3][4][5][6][7][8][9][10][11][12][13]. "Classical" NMR experiments are carried out in a single, high magnetic field. The Fast Field Cycling technology [14,15] applied in NMR relaxometry enables NMR relaxation experiments to be performed in a very broad range of magnetic fields (and, hence, resonance frequencies are proportional to the magnetic field)-in the present work they are performed in a range of 10 kHz to 10 MHz (referring to 1 H resonance frequency). The principle of the relaxation process is as follows. 1 H ( 19 F) nuclei can occupy two energy levels in an external magnetic field, which are associated with parallel and anti-parallel orientations of the magnetic moments of the nuclei with respect to the direction of the magnetic field. The molecular or ionic system gains magnetization as a result of the difference in the populations of the two energy levels, as according to the Boltzmann distribution, the population of the lower energy level is higher. When the magnetic field is altered, the populations of the two energy levels change, thus reaching a new equilibrium (according to the Boltzmann distribution). This process is referred to as spin-lattice relaxation. The repopulation of the energy levels is observed as being an evolution of magnetization over time. In most cases, there is a single-exponential evolution, and the characteristic time constant of the magnetization dependence on time is referred to as the spin-lattice relaxation time (and its reciprocal value is called the spin-lattice relaxation rate). The repopulation of the energy levels (i.e., the relaxation process) requires exchanging energy with surrounding molecules or ions, which is referred to as a lattice. The exchange occurs via magnetic dipole-dipole interactions with neighbouring NMR active nuclei. The interactions fluctuate in time as a result of the molecular or ionic motion. The strength of the interactions, as well as the timescale of the motion and its mechanism, determine the relaxation rates.
According to spin relaxation theory [16][17][18], at a given resonance frequency, the dominating contribution to the spin-lattice relaxation is associated with a dynamic process occurring on a timescale matching the reciprocal frequency. This statement is somewhat oversimplified as it does not account for the corresponding interaction strengths; however it captures the essential feature: with increasing frequencies, one obtains progressively faster dynamics. Consequently, NMR relaxometry gives access to molecular motion on the timescale from ms to ns, whereas "classical" NMR relaxation experiments give access to a much faster motion.
Dipole-dipole interactions leading to the relaxation process can be of inter-molecular (inter-ionic) or intra-molecular (intra-ionic) origin. The inter-molecular (inter-ionic) interactions fluctuate in time due to the relative translation diffusion of the interacting species, whereas the intra-molecular (intra-ionic) ones are modulated by rotational dynamics. Consequently, NMR relaxometry is a very valuable source of information in terms of understanding dynamic processes in condensed matter. This method has extensively been exploited for investigating the translational and rotational dynamics of molecular liquids; significantly fewer studies have been performed for ionic liquids. The applications of NMR relaxometry for ionic liquids in bulk [4,[6][7][8]10] and confinement [8,12,13] have given a very valuable insight into the dynamics of cations and anions, especially for liquids composed of 1 H containing cations and 19 F containing anions.
In this work, we exploit NMR relaxometry to enquire into the relative cation-anion dynamics in a series of ionic liquids composed of the same ( 19  Moreover, one can expect (and this has proven to be true) that by increasing the length of the alkyl chain (and, hence, increasing the ratio between the number of 1 H and 19 F nuclei), the 19 F-19 F relaxation contribution will become negligible. This creates an extraordinary opportunity to enquire into the relative cation-anion translation diffusion. One should stress at this stage, that with separate NMR diffusometry probes, the cation and the anion self-diffuse, but this does not give any insight into the relative cation-anion motion. To our knowledge, this is the first example of a direct, experimental determination of the relative cation-anion translation diffusion coefficient for ionic liquids. Moreover, the 19 F spin-lattice relaxation experiments for the series of ionic liquids, complemented by the results for [TEA-C4] [TFSI] [4], give insight into how the structure of the cation affects the relative cation-anion movement.
In cases of isotropic (three-dimensional) translation diffusion (characteristic of liquids in bulk), at low frequencies, one observes a linear dependence of the spin-lattice relaxation rate at the squared root of the resonance frequency [19][20][21][22][23]. In cases of inter-molecular (inter-ionic) interactions, the relaxation rate is dominated by a relaxation contribution originating from dipole-dipole interactions between one kind of NMR active nuclei ( 1 H-1 H or 19 F-19 F). Moreover, the translation diffusion coefficient can be determined from the low frequency slope of the relaxation rate versus the squared root of the frequency [7,23]. In this work, a corresponding expression has been described and applied for the case of 1 H-19 F dipole-dipole interactions, enabling a straightforward determination of the cation-anion relative translation diffusion coefficient.  Table 1. The data have been analysed in terms of Equation (1) of Section 4. The obtained cation-anion relative translation diffusion coefficients, , and the cation-anion distance of closest approach, , are listed in Table 2. The obtained cation-anion relative translation diffusion coefficients, D CA trans , and the cation-anion distance of closest approach, d CA , are listed in Table 2. In addition, the relative translation diffusion coefficients have been obtained from the low frequency slope of the relaxation rates plotted, versus the squared root of the resonance frequency, in terms of Equation (3). Figure 2a, (Figures A1-A5). The cation-anion translation diffusion coefficients obtained from the low frequency slopes are included in Table 2. They are in very good agreement with those obtained from the fits of the relaxation data in terms of Equation (3).

308
5.14 × 10 −12 8.8 6.05 × 10 −11 5.28 × 10 −12 313 6.77 × 10 −12 9.7 7.57 × 10 −11 6.78 × 10 −12 318 8.26 × 10 −12 11.0 9.24 × 10 −11 8.27 × 10 −12 In addition, the relative translation diffusion coefficients have been obtained from the low frequency slope of the relaxation rates plotted, versus the squared root of the resonance frequency, in terms of Equation (3). Figure 2a, plotted, versus the squared root of the resonance frequency, accompanied by the corresponding low frequency fits, are shown in Appendix B (Figures A1-A5). The cation-anion translation diffusion coefficients obtained from the low frequency slopes are included in Table 2. They are in very good agreement with those obtained from the fits of the relaxation data in terms of Equation (3). The translation diffusion coefficients, obtained from the full fits, are plotted in Figure  3 versus the reciprocal temperature. The data are complemented by the relative cationanion translation diffusion coefficients for [TEA-C4] [TFSI], which are taken from [4]. The translation diffusion coefficients, obtained from the full fits, are plotted in Figure 3 versus the reciprocal temperature. The data are complemented by the relative cation-anion translation diffusion coefficients for [TEA-C4] [TFSI], which are taken from [4]. 308 5.14 × 10 −12 8.8 6.05 × 10 −11 5.28 × 10 −12 313 6.77 × 10 −12 9.7 7.57 × 10 −11 6.78 × 10 −12 318 8.26 × 10 −12 11.0 9.24 × 10 −11 8.27 × 10 −12 In addition, the relative translation diffusion coefficients have been obtained from the low frequency slope of the relaxation rates plotted, versus the squared root of the resonance frequency, in terms of Equation (3). Figure 2a, plotted, versus the squared root of the resonance frequency, accompanied by the corresponding low frequency fits, are shown in Appendix B (Figures A1-A5). The cation-anion translation diffusion coefficients obtained from the low frequency slopes are included in Table 2. They are in very good agreement with those obtained from the fits of the relaxation data in terms of Equation (3). The translation diffusion coefficients, obtained from the full fits, are plotted in Figure  3 versus the reciprocal temperature. The data are complemented by the relative cationanion translation diffusion coefficients for [TEA-C4] [TFSI], which are taken from [4].   [6]. The comparison is shown in Figure 4.  which were obtained in [6]. The comparison is shown in Figure 4.
In [6] the relative cation-cation, anion-anion, and cation-anion translational diffusion coefficients, [TEA-C14] [TFSI] was determined. It has been shown that the cation-cation diffusion is considerably slower than the anion-anion relative translation movement. This was expected, taking into account the size of the cation and anion. It is, however, worth to noting that at lower temperatures, the diffusion coefficients tend to converge. As NMR relaxometry probes the relative translation diffusion coefficients, in [6], the cation-cation diffusion coefficients have been compared with the cation self-diffusion coefficients, showing that the relative diffusion coefficients are somewhat lower than twice the self-diffusion ones. This has been attributed to the correlation effects in the cation-cation relative movement. It might be worth saying at this stage, that for uncorrelated dynamics, the relative diffusion coefficient is given as a sum of self-diffusion coefficients of the interacting species (so twice the self-diffusion coefficient in case they are identical). The same observation has  Table 1 includes the list of the liquids with their molecular mass and density. Moreover, the numbers of 1 H and 19 F nuclei per unit volume have been provided. The number of 1 H nuclei (hydrogen atoms) per unit volume has been obtained from the relationship: where n H denotes the number of hydrogen atoms per cation, N A is the Avogadro number, denotes density of the ionic liquid, and M is its molecular mass. The N F number is given as: N F = n F n H N H , where n F denotes the number of fluorine atoms per anion.
The synthesis procedure of the series of ionic liquids has been described in detail in [24]. The applied theoretical model is presented below. 19 F relaxation processes in ionic liquids composed of 1 H containing cations and 19 F containing anions stem from three relaxation channels: 1 H-19 F magnetic dipole-dipole interactions modulated by relative cation-anion translation diffusion, 19 F-19 F dipole-dipole interactions between anions (inter-anionic), modulated by a relative anion-anion translation motion, and 19 F-19 F dipole-dipole interactions within the anion (intra-anionic, provided the anion includes more than one 19 F nuclei), modulated by the rotational dynamics of the anion. For the investigated series of ionic liquids, the 19 F spin-lattice relaxation rate, R 1F (ω F ) (ω F denotes 19 F resonance frequency in angular frequency units), is dominated by the relaxation contribution originating from the 1 H-19 F cation-anion dipole-dipole interactions. Consequently, the relaxation rate, R 1F (ω F ), is given as [4,7]:  19 F nuclei per unit volume, N F . This explains why this relaxation contribution is much lower than that originating from 1 H-19 F interactions-the N H number is much larger than N F (Table 1). It has been shown already, that for [TEA-C4] [TFSI], the anion-anion relaxation contribution to 19 F spin-lattice relaxation is much lower than the cation-anion term [4]. By increasing the chain length, and given the N H number compared with N F , the 19 F-19 F relaxation contribution becomes negligible. As far as the intra-anionic relaxation contribution is concerned, the relaxation rates are low because of the fast rotational dynamics of the TFSI cations, which has already been shown in [4]. This creates a unique opportunity to solely probe the relative cation-anion translation movement.
The translational spectral density: (here, we used the symbols τ trans , d , and N, which demonstrate the general meaning of this formulae) can be expanded into the Taylor series in the range of ωτ < 1. The expansion takes the form [20,21,23]: where D trans denotes a relative translation diffusion coefficient; a is a frequency independent term. This implies that for isotropic translational diffusion, one observes a linear dependence of the spin-lattice relaxation rates on the square root of the resonance frequency. Moreover, from the low frequency slope, one can straightforwardly determine the translation diffusion coefficient [7,23]. An expression derived on this basis has been used for molecular and ionic liquids when the dominating relaxation contribution originates from 1 H-1 H dipole-dipole interactions. Combining Equation (3) with Equation (1), one can determine a relationship between the low frequency slope, B, and the relative cation-anion translation diffusion coefficient for the present case: Taking into account that ω H ∼ = ω F , one can simplify the relationship to form: A detailed derivation of the expression can be found in Appendix A.

Conclusions
It has turned out that the 19  can be fully attributed, in the frequency range from 10kHz to 10MHz, to the cation-anion 1 H-19 F dipole-dipole interactions. Consequently, this paved the way for the determination of the relative cation-anion translation diffusion coefficients. In addition to the analysis of the relaxation data in terms of a full fit, a formula allowing the straightforward determination of the relative translation diffusion coefficient, from the slow, low frequency of the relaxation rates plotted, versus the squared root of the resonance frequency, has been ascertained. The values of the diffusion coefficients, obtained in both ways, are in good agreement with one another. The cation-anion diffusion coefficients have been compared with the corresponding values of the cation-cation diffusion coefficients taken from [9]. With the knowledge that the anion diffusion is not slower than the translation motion of the   This leads to Equation (4), which can be simplified to Equation (5)     This leads to Equation (4), which can be simplified to Equation (5)