# Correlated Dynamics in Ionic Liquids by Means of NMR Relaxometry: Butyltriethylammonium bis(Trifluoromethanesulfonyl)imide as an Example

^{1}

^{2}

^{*}

## Abstract

**:**

^{1}H and

^{19}F spin-lattice relaxation experiments have been performed for butyltriethylammonium bis(trifluoromethanesulfonyl)imide in the temperature range from 258 to 298 K and the frequency range from 10 kHz to 10 MHz. The results have thoroughly been analysed in terms of a relaxation model taking into account relaxation pathways associated with

^{1}H–

^{1}H,

^{19}F–

^{19}F and

^{1}H–

^{19}F dipole–dipole interactions, rendering relative translational diffusion coefficients for the pairs of ions: cation–cation, anion–anion and cation–anion, as well as the rotational correlation time of the cation. The relevance of the

^{1}H–

^{19}F relaxation contribution to the

^{1}H and

^{19}F relaxation has been demonstrated. A comparison of the diffusion coefficients has revealed correlation effects in the relative cation–anion translational movement. It has also turned out that the translational movement of the anions is faster than of cations, especially at high temperatures. Moreover, the relative cation–cation diffusion coefficients have been compared with self-diffusion coefficients obtained by means of NMR (Nuclear Magnetic Resonance) gradient diffusometry. The comparison indicates correlation effects in the relative cation–cation translational dynamics—the effects become more pronounced with decreasing temperature.

## 1. Introduction

^{1}H and

^{19}F typically for ionic liquids) versus time, as a result of changes in the resonance frequency. The most important characteristic of NMR diffusometry is that this method provides values of self-diffusion coefficients [3,4,5], in contrast to NMR relaxometry, exploited in this work, which probes relative translation motion of ions (molecules).

^{1}H resonance frequency) [15,16]. Consequently, one can probe in a single experiment molecular (ionic) motion on the time scale from ms to ns. NMR relaxometry not only gives access to the value of the diffusion coefficients but also allows identification of the mechanism (dimensionality) of the motion [12,13,17,18,19,20,21].

^{1}H (

^{19}F) relaxation rates are given as linear combinations of spectral density functions being Fourier transforms of corresponding correlation functions characterising the dynamical processes that give rise to stochastic fluctuations of magnetic dipole–dipole interactions causing the relaxation processes [22,23,24,25,26,27]. As the mathematical form of the correlation function (and, hence, the spectral density) depends on the mechanism of the motion [13,17,18,28,29,30,31], the relaxation dispersion profiles (spin-lattice relaxation rates versus the resonance frequency) are a direct fingerprint of this mechanism. In this way, one can unambiguously distinguish between translational and rotational dynamics and reveal the isotropy/anisotropy of the motion [32,33].

^{1}H and

^{19}F). To enquire into the dynamical properties of

^{1}H containing cations and

^{19}F containing anions in ionic liquids, one has to properly model

^{1}H and

^{19}F relaxation processes accounting for the role of

^{1}H–

^{19}F (cation–anion) magnetic dipole–dipole interactions.

^{1}H and

^{19}F spin-lattice relaxation data butyltriethylammonium bis(trifluoromethanesulfonyl)imide ([TEA-C4][TFSI]), taking into account all relevant relaxation pathways, especially the role of

^{1}H–

^{19}F (cation–anion) mutual dipole–dipole coupling. In this way, we quantitatively describe the translational and rotational dynamics of the ions and enquire into correlation effects in the translation movement. Consequently, the work has two intertwined goals: to present the methodology that enables probing translation diffusion of ions in ionic liquids by means of NMR relaxometry and to reveal the scenario of the translation movement in [TEA-C4][TFSI]. A deep insight into the dynamical properties of ionic liquids is necessary for revealing factors determining conductivity of liquid electrolytes and, consequently, their tailoring for specific applications.

## 2. Theory

^{1}H and

^{19}F relaxation processes are caused by magnetic dipole–dipole interactions that can be of intra-molecular (intra-ionic) and inter-molecular (inter-ionic) interactions. For ionic liquids composed of

^{1}H containing cations and

^{19}F containing anions, the

^{1}H and

^{19}F spin-lattice relaxation rates, ${R}_{1,H}\left({\omega}_{H}\right)$ and ${R}_{1,F}\left({\omega}_{F}\right)$, respectively, (${\omega}_{H}$ and ${\omega}_{F}$ denote

^{1}H and

^{19}F resonance frequencies, respectively, in angular frequency units), include the following relaxation contributions:

^{1}H–

^{1}H dipole–dipole interactions within the cation (${R}_{1,H}^{intra}\left({\omega}_{H}\right))$ and

^{19}F–

^{19}F dipole–dipole interactions within the anion (${R}_{1,F}^{intra}\left({\omega}_{F}\right)$). These interactions fluctuate in time as a result of rotational dynamics of the ions. Consequently, the relaxation contributions ${R}_{1,H}^{intra}\left({\omega}_{H}\right)$ and ${R}_{1,F}^{intra}\left({\omega}_{F}\right)$ can be expressed as [6,8,9,12,13,21]:

^{1}H–

^{1}H (cation–cation) and

^{19}F–

^{19}F (anion–anion) interactions and can be expressed as [6,9,12,21,31,32]:

^{1}H and

^{19}F nuclei per unit volume, respectively. They can be obtained from the relationship: ${N}_{H}=\frac{{n}_{H}{N}_{A}\varrho}{M}$ and ${N}_{F}=\frac{{n}_{F}{N}_{A}\varrho}{M}$, where ${n}_{H}$ and ${n}_{F}$ denote the number of hydrogen atoms per cation and the number of fluorine atoms per anion, respectively, ${N}_{A}$ is the Avogadro number, $\varrho $ denotes density of the ionic liquid, while $M$ is its molecular mass; ${\gamma}_{H}$ and ${\gamma}_{F}$ are

^{1}H and

^{19}F gyromagnetic factors, other symbols have their obvious meaning. Equations (1) and (2) clearly show that the

^{1}H and

^{19}F relaxation processes are not independent, and in this sense, they are both affected by the

^{1}H–

^{19}F (cation–anion) interactions. The corresponding relaxation contributions can be expressed as [6,12]:

## 3. Results and Analysis

^{1}H and

^{19}F spin-lattice relaxation data for [TEA-C4][TFSI] are shown in Figure 1a,b, respectively. The figures include fits performed in terms of the model outlined in Section 2.

^{1}H spin-lattice relaxation data can be reproduced without the relaxation contribution, ${R}_{1,H}^{inter,HF}\left({\omega}_{H}\right)$, associated with the cation–anion,

^{1}H–

^{19}F, dipole–dipole interactions. Consequently, the fits of the

^{1}H relaxation data include four adjustable parameters: ${D}_{trans}^{C}$, ${d}_{CC}$, ${C}_{DD}^{HH}$ and ${\tau}_{rot}^{C}$. The number of

^{1}H nuclei per unit volume, ${N}_{H}$, has been calculated as described in Section 2; for [TEA-C4][TFSI] (C

_{12}H

_{24}F

_{6}N

_{2}O

_{2}S

_{2}) one gets: $M$ = 438.45 g/mol, $\varrho $ = 1.332 g/mol, ${n}_{H}$ = 24; consequently ${N}_{H}$ = 4.39·10

^{28}/m

^{3}. The parameters are collected in Table 1.

^{28}/m

^{3}(${N}_{F}$/${N}_{H}$ = 1/4), the ${R}_{1,H}^{inter,HF}\left({\omega}_{H}\right)$ relaxation contribution to the ${R}_{1,H}\left({\omega}_{H}\right)$ relaxation rates can indeed be small; however, the contribution also depends on other factors—we shall come back to this subject later. Figure 2 shows the

^{1}H spin-lattice relaxation rates, ${R}_{1,H}\left({\omega}_{H}\right)$, decomposed into the individual relaxation contributions: ${R}_{1,H}^{intra}\left({\omega}_{H}\right)$ and ${R}_{1,H}^{inter,HF}\left({\omega}_{H}\right)$.

^{−9}s (or shorter), the relaxation contribution ${R}_{1,H}^{intra}\left({\omega}_{H}\right)$ becomes frequency independent (as then the condition ${\omega}_{H}{\tau}_{rot}^{C}\ll 1$ is approached). However, as the dipolar relaxation constant ${C}_{DD}^{HH}$ has unambiguously been determined from the analysis of the relaxation data at lower temperatures and kept unchanged with temperature, the values of ${\tau}_{rot}^{C}$ can be determined even when the ${R}_{1,H}^{intra}\left({\omega}_{H}\right)$ is frequency independent. In Table 2, the ratio ${\tau}_{trans}^{C}/{\tau}_{rot}^{C}$ has been calculated. The value monotonically decreases with temperature from 13.3 to 7.0. Following this line, the

^{19}F spin-lattice relaxation data have been analysed in terms of the model presented in Section 2 (Figure 3). In this case, the

^{1}H–

^{19}F relaxation term, ${R}_{1,F}^{inter,FH}\left({\omega}_{F}\right)$, gives a considerable contribution to the overall

^{19}F spin-lattice relaxation rates, ${R}_{1,F}\left({\omega}_{F}\right)$. This is not surprising considering that ${N}_{H}$/${N}_{F}$ = 4 and ${R}_{1,F}^{inter,FH}\left({\omega}_{F}\right)$ is proportional to ${N}_{H}$ (Equation (8)). However, as a result of fast rotational dynamics of TFSI anions and, presumably, a relatively small dipolar relaxation constant (${C}_{DD}^{FF})$, the relaxation contribution ${R}_{1,F}^{intra}\left({\omega}_{F}\right)$ has turned out to be negligible. The obtained parameters are collected in Table 2.

^{1}H–

^{19}F, relaxation contribution, ${R}_{1,F}^{inter,FH}\left({\omega}_{F}\right)$ dominates the

^{19}F spin-lattice relaxation rate, ${R}_{1,F}\left({\omega}_{F}\right)$, and its importance increases with increasing temperature.

^{1}H spin-lattice relaxation data have been fitted again, including the ${R}_{1,F}^{inter,HF}\left({\omega}_{F}\right)$ relaxation contribution with ${D}_{trans}^{CA}$ and ${d}_{CA}$ fixed to the values obtained from the analysis of the

^{19}F spin-lattice relaxation data (Figure 4).

## 4. Discussion

^{1}H and

^{19}F spin-lattice relaxation experiments have been performed in the temperature range from 258 to 298 K. As the melting point of [TEA-C4][TFSI] is 289.1 K, the data have been collected (except at the highest temperature of 298K) in the supercooled state. In the first step, the

^{1}H spin-lattice relaxation data have been analysed considering only

^{1}H–

^{1}H dipole–dipole interactions, i.e., neglecting the cation–anion

^{1}H–

^{19}F dipole–dipole coupling. The reason for neglecting the relaxation contribution associated with the

^{1}H–

^{19}F dipole–dipole interactions is ${N}_{F}$ being four times smaller than ${N}_{H}$. The attempt has turned out to be successful yielding the values of the translation diffusion coefficient for the cation from 1.92·10

^{−13}m

^{2}/s at 258 K to 5.70·10

^{−12}m

^{2}/s at 298 K. Combining these values with the cation–cation distance of the closest approach of 4.30 Å, for the translational correlation, the range from 4.82·10

^{−7}s (258 K) to 1.62·10

^{−8}s (298 K) has been obtained. The analysis has also revealed the rotational correlation time for the cation—it ranges from 3.63·10

^{−8}s at 258 K to 2.31·10

^{−9}s at 298 K. Notably, the ratio ${\tau}_{trans}^{C}/{\tau}_{rot}^{C}$ decreases monotonically from 13.3 at 258 K to 7.0 at 298 K. The Stokes equation predicts for spherical molecules the ratio between the translational and the rotational correlation times equal to 9 [27], while for “real” molecular liquids, values in the range of 20–40 have been obtained [35].

^{19}F relaxation data are concerned, the relaxation contribution associated with the cation–anion

^{1}H–

^{19}F dipole–dipole interactions (mediated by the cation–anion relative translational diffusion) is not only non-negligible, but it dominates the relaxation contribution caused by the anion–anion

^{19}F–

^{19}F interactions, as shown in Figure 3. The diffusion coefficient for the anion ranges from 3.14·10

^{−13}m

^{2}/s at 258 K and 2.51·10

^{−11}m

^{2}/s at 298 K, which means that the anions diffuse faster than the cations. With the anion–anion distance of the closest approach, 2.60 Å, the translational correlation time for the anion ranges from 1.08·10

^{−7}s at 258 K to 1.35·10

^{−9}s at 298 K. At the same time, the relative cation–anion translation diffusion coefficient ranges from 3.05·10

^{−13}m

^{2}/s at 258 K to 7.76·10

^{−12}m

^{2}/s at 298 K, which gives (for the cation–anion distance of the closet approach of 3.88 Å) the range of the corresponding correlation times from 4.94·10

^{−7}to 1.83·10

^{−8}s. Due to fast rotation of the anions, the relaxation contribution associated with the intra-anionic

^{19}F–

^{19}F interactions turned out to be negligible.

^{1}H spin-lattice relaxation data have been analysed again, accounting for the (known)

^{1}H–

^{19}F relaxation contribution. The analysis has led to somewhat larger translation diffusion coefficients of the cation and a considerably larger value of the cation–cation distance of the closest approach, 5.95 Å. Consequently, the translational correlation time has become longer (from 8.19·10

^{−7}s at 258 K to 2.33·10

^{−8}s at 298 K). Taking into account that the rotational correlation time has only slightly been affected by the extended relaxation scenario, the ratio between the correlation times has become larger, yielding 23.0 at 258 K and monotonically decreasing to 10.4 at 298 K. As one can see in Figure 4, the

^{1}H–

^{19}F relaxation contribution to the

^{1}H relaxation has turned out to be of importance.

^{1}H–

^{19}F relaxation contribution is accounted for) yields from 3.30 at 298 K to 1.45 at 258 K—this indicates that the diffusion coefficients tend to converge at low temperatures. In order to compare the relative cation–anion diffusion coefficients with the cation and anion diffusion coefficients, the last two values have been multiplied by two in Figure 5; moreover, a sum of the diffusion coefficients of the action and of the anion has been plotted. One can clearly see from the comparison that the relative cation–anion translation diffusion coefficients are smaller than the sum—the ratio yields from 0.40 at 258 K to 0.56 at 298 K.

^{1}H spin-lattice relaxation data is, in fact, the relative cation–cation translation diffusion coefficient that has been treated as equal to twice the self-diffusion coefficient ${D}_{trans}^{C*}$ (or ${D}_{trans}^{C}$). However, with decreasing temperature, the values of ${D}_{trans}^{C*}$ (being, in fact equal to a half of the relative translation diffusion coefficient) become progressively smaller compared to those obtained by means of NMR diffusometry, suggesting that the relative cation–cation translation movement becomes more correlated. This might suggest that translation movement of one cation triggers (to some extend) a displacement of neighbouring cations in the direction of the first one (consequently, their relative diffusion becomes slower as their distance changes less in time than in the case of uncorrelated dynamics).

## 5. Materials and Methods

^{1}H and

^{19}F spin-lattice relaxation measurements have been performed for [TEA-C4][TFSI] in the frequency range 10 kHz to 10 MHz (referring to the

^{1}H resonance frequency) versus temperature, from 258 to 298 K using an NMR relaxometer, produced by Stelar s.r.l. (Mede (PV), Italy). The temperature was controlled with an accuracy of 0.5 K. The experiments started at a higher temperature, which was progressively decreased. For each resonance frequency, 32 magnetisation values versus time in a logarithmic time scale have been recorded. Below 4 MHz, pre-polarisation at 0.19 T was applied. The switching time of the magnet was set to 3 ms.

^{1}H and

^{19}F) turned out to be single-exponential for all measured temperatures in the whole frequency range, as shown in the Supplementary Materials (Figures S1–S8 for

^{1}H and Figures S9–S16 for

^{19}F).

## 6. Conclusions

^{1}H and

^{19}F spin-lattice relaxation data for [TEA-C4][TFSI] collected at a broad range of resonance frequencies (from about 10 kHz to 10 MHz) and a temperature range from 258 to 298 K has revealed relative (cation-cation, anion–anion and cation–anion) translation diffusion coefficients. A comparison of the cation–cation diffusion coefficients with corresponding values obtained by means of NMR gradient diffusometry indicates correlation effects in the relative cation–cation translation movement that become more pronounced with decreasing temperature (at 263 K, the ratio between the self-diffusion coefficient of the cation obtained by means of NMR diffusometry and a half of the relative cation–cation translation diffusion coefficient reaches about 1.6, while at 298 K, the two quantities are very similar). It has also turned out that the translation motion of the anions is faster than that of the cations (the ratio between the relative anion–anion and cation–cation translation diffusion coefficients ranges from 3.30 at 298 K to 1.45 at 258 K), although with decreasing temperature, the values tend to converge. At the same time, the relative cation–anion translation diffusion coefficients are smaller than the sum of the diffusion coefficients of the cation and of the anion, indicating a correlated cation–anion movement. The analysis has also allowed determining the rotational correlation time of the action. The ratio between the translational and rotational correlation times monotonically decreases with increasing temperature and lies in the range from about 23 to about 10, similarly to molecular liquids.

## Supplementary Materials

^{1}H and

^{19}F magnetization curves (magnetization versus time) for butyltriethylammonium bis(trifluoromethylsulfonyl) imide. Solid lines denote single exponential fits (Figures S1–S16).

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Conflicts of Interest

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**Figure 1.**(

**a**)

^{1}H and (

**b**)

^{19}F spin-lattice relaxation data for [TEA-C4][TFSI]; solid lines—fits in terms of the model of Section 2.

**Figure 2.**

^{1}H spin-lattice relaxation rates, ${R}_{1,H}\left({\omega}_{H}\right)$, for [TEA-C4][TFSI]; solid lines—theoretical fits decomposed into ${R}_{1,H}^{intra}\left({\omega}_{H}\right)$ (dashed-dotted lines) and ${R}_{1,H}^{inter,HH}\left({\omega}_{H}\right)$ (dashed lines) at different temperatures (

**a**–

**h**).

**Figure 3.**

^{19}F spin-lattice relaxation rates, ${R}_{1,F}\left({\omega}_{F}\right)$, for [TEA-C4][TFSI]; solid lines—theoretical fits decomposed into ${R}_{1,F}^{inter,FF}\left({\omega}_{F}\right)$ (dashed lines) and ${R}_{1,F}^{inter,FH}\left({\omega}_{F}\right)$ (dashed-dotted-dotted lines) at different temperatures (

**a**–

**h**).

**Figure 4.**

^{1}H spin-lattice relaxation rates, ${R}_{1,H}\left({\omega}_{H}\right)$, for [TEA-C4][TFSI]; solid lines—theoretical fits decomposed into ${R}_{1,H}^{inter,HH}\left({\omega}_{H}\right)$ (dashed lines), ${R}_{1,H}^{inter,HF}\left({\omega}_{H}\right)$ (dashed-dotted-dotted lines) and ${R}_{1,H}^{intra}\left({\omega}_{H}\right)$ (dashed-dotted lines) at different temperatures (

**a**–

**h**).

**Figure 5.**(

**a**) Translation diffusion coefficients for [TEA-C4] cation and [TFSI] anion, solid lines—fits according to the Arrhenius law; (

**b**) translational and rotational correlation times with a corresponding fit of the Arrhenius law.

**Table 1.**Parameters characterising the translational and rotational dynamics of the TEA-C4 cation in [TEA-C4][TFSI]; ${C}_{DD}^{HH}$ = 1.49·10

^{9}Hz

^{2}, ${d}_{CC}$ = 4.30 Å. The correlation time ${\tau}_{trans}^{C}$ has been calculated from the relationship:${\tau}_{trans}^{C}=\frac{{d}_{CC}^{2}}{2{D}_{trans}^{C}}$.

Temp. (K) | ${\mathit{D}}_{\mathit{t}\mathit{r}\mathit{a}\mathit{n}\mathit{s}}^{\mathit{C}}\text{}({\mathbf{m}}^{2}/\mathbf{s})$ | ${\mathit{\tau}}_{\mathit{r}\mathit{o}\mathit{t}}^{\mathit{C}}\left(\mathbf{s}\right)$ | Rel. Error (%) | ${\mathit{\tau}}_{\mathit{t}\mathit{r}\mathit{a}\mathit{n}\mathit{s}}^{\mathit{C}}\text{}\left(\mathbf{s}\right)$ | ${\mathit{\tau}}_{\mathit{t}\mathit{r}\mathit{a}\mathit{n}\mathit{s}}^{\mathit{C}}/{\mathit{\tau}}_{\mathit{r}\mathit{o}\mathit{t}}^{\mathit{C}}$ |
---|---|---|---|---|---|

258 | 1.92·10^{−13} | 3.63·10^{−8} | 10.7 | 4.82·10^{−7} | 13.3 |

263 | 3.31·10^{−13} | 2.62·10^{−8} | 8.0 | 2.79·10^{−7} | 10.6 |

268 | 5.95·10^{−13} | 1.57·10^{−8} | 3.1 | 1.55·10^{−7} | 9.9 |

273 | 8.80·10^{−13} | 1.22·10^{−8} | 2.9 | 1.05·10^{−7} | 8.6 |

278 | 1.33·10^{−12} | 7.99·10^{−9} | 2.6 | 6.95·10^{−8} | 8.7 |

283 | 2.08·10^{−12} | 5.75·10^{−9} | 2.9 | 4.44·10^{−8} | 7.7 |

288 | 2.88·10^{−12} | 4.45·10^{−9} | 3.4 | 3.21·10^{−8} | 7.2 |

298 | 5.70·10^{−12} | 2.31·10^{−9} | 1.2 | 1.62·10^{−8} | 7.0 |

**Table 2.**Parameters characterising the translational and rotational dynamics of TEA-C4 anion in [TEA-C4][TFSI]; ${C}_{DD}^{HH}$ = 1.49·10

^{9}Hz

^{2}, ${d}_{CC}$ = 4.30 Å. The correlation time ${\tau}_{trans}^{C}$ has been calculated from the relationship:${\tau}_{trans}^{C}=\frac{{d}_{CC}^{2}}{2{D}_{trans}^{C}}$.

Temp. (K) | ${\mathit{D}}_{\mathit{t}\mathit{r}\mathit{a}\mathit{n}\mathit{s}}^{\mathit{A}}\text{}({\mathbf{m}}^{2}/\mathbf{s})$ | ${\mathit{D}}_{\mathit{t}\mathit{r}\mathit{a}\mathit{n}\mathit{s}}^{\mathit{C}\mathit{A}}\text{}({\mathbf{m}}^{2}/\mathbf{s})$ | Rel. Error (%) | ${\mathit{\tau}}_{\mathit{t}\mathit{r}\mathit{a}\mathit{n}\mathit{s}}^{\mathit{A}}\text{}\left(\mathbf{s}\right)$ | ${\mathit{\tau}}_{\mathit{t}\mathit{r}\mathit{a}\mathit{n}\mathit{s}}^{\mathit{C}\mathit{A}}\text{}\left(\mathbf{s}\right)$ |
---|---|---|---|---|---|

258 | 3.14·10^{−13} | 3.05·10^{−13} | 7.5 | 1.08·10^{−7} | 4.94·10^{−7} |

263 | 5.55·10^{−13} | 5.32·10^{−13} | 18.0 | 6.09·10^{−8} | 2.83·10^{−7} |

268 | 1.26·10^{−12} | 9.02·10^{−13} | 10.9 | 2.68·10^{−8} | 1.67·10^{−7} |

273 | 1.93·10^{−12} | 1.37·10^{−12} | 11.1 | 1.75·10^{−8} | 1.10·10^{−7} |

278 | 4.19·10^{−12} | 2.01·10^{−12} | 11.6 | 8.07·10^{−9} | 7.49·10^{−8} |

283 | 7.50·10^{−12} | 3.05·10^{−12} | 16.7 | 4.51·10^{−9} | 4.94·10^{−8} |

288 | 1.43·10^{−11} | 4.11·10^{−12} | 12.0 | 2.36·10^{−9} | 3.66·10^{−8} |

298 | 2.51·10^{−11} | 7.76·10^{−12} | 9.9 | 1.35·10^{−9} | 1.83·10^{−8} |

**Table 3.**Parameters characterising the translational and rotational dynamics of TEA-C4 cations in [TEA-C4][TFSI] including the ${R}_{1,F}^{inter,HF}\left({\omega}_{F}\right)$ relaxation contribution; ${C}_{DD}^{HH}$ = 1.62·10

^{9}Hz

^{2}, ${d}_{CC}^{*}$ = 5.95 Å. The correlation time ${\tau}_{trans}^{C*}$ has been calculated from the relationship:${\tau}_{trans}^{C*}=\frac{{\left({d}_{CC}^{*}\right)}^{2}}{2{D}_{trans}^{C*}}$ (“$*$ ” indicates the presence of the ${R}_{1,F}^{inter,HF}\left({\omega}_{F}\right)$ relaxation contribution). The last column includes the value of the diffusion coefficients for the cation obtained by means of NMR gradient methods [34].

Temp. (K) | ${\mathit{D}}_{\mathit{t}\mathit{r}\mathit{a}\mathit{n}\mathit{s}}^{\mathit{C}*}\text{}({\mathbf{m}}^{2}/\mathbf{s})$ | ${\mathit{\tau}}_{\mathit{r}\mathit{o}\mathit{t}}^{\mathit{C}*}\left(\mathbf{s}\right)$ | Rel. Error (%) | ${\mathit{\tau}}_{\mathit{t}\mathit{r}\mathit{a}\mathit{n}\mathit{s}}^{\mathit{C}*}\text{}\left(\mathbf{s}\right)$ | ${\mathit{\tau}}_{\mathit{t}\mathit{r}\mathit{a}\mathit{n}\mathit{s}}^{\mathit{C}*}/{\mathit{\tau}}_{\mathit{r}\mathit{o}\mathit{t}}^{\mathit{C}*}$ |
---|---|---|---|---|---|

258 | 2.16·10^{−13} | 3.56·10^{−8} | 9.7 | 8.19·10^{−7} | 23.0 |

263 | 3.79·10^{−13} | 2.55·10^{−8} | 7.8 | 4.67·10^{−7} | 18.3 |

268 | 6.75·10^{−13} | 1.65·10^{−8} | 3.1 | 2.62·10^{−7} | 15.9 |

273 | 9.55·10^{−13} | 1.20·10^{−8} | 2.8 | 1.85·10^{−7} | 15.4 |

278 | 1.45·10^{−12} | 7.83·10^{−9} | 2.2 | 1.22·10^{−7} | 15.5 |

283 | 2.34·10^{−12} | 5.61·10^{−9} | 2.3 | 7.56·10^{−8} | 13.5 |

288 | 3.64·10^{−12} | 4.29·10^{−9} | 2.8 | 4.86·10^{−8} | 11.3 |

298 | 7.60·10^{−12} | 2.24·10^{−9} | 1.2 | 2.33·10^{−8} | 10.4 |

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**MDPI and ACS Style**

Kruk, D.; Masiewicz, E.; Lotarska, S.; Markiewicz, R.; Jurga, S. Correlated Dynamics in Ionic Liquids by Means of NMR Relaxometry: Butyltriethylammonium bis(Trifluoromethanesulfonyl)imide as an Example. *Int. J. Mol. Sci.* **2021**, *22*, 9117.
https://doi.org/10.3390/ijms22179117

**AMA Style**

Kruk D, Masiewicz E, Lotarska S, Markiewicz R, Jurga S. Correlated Dynamics in Ionic Liquids by Means of NMR Relaxometry: Butyltriethylammonium bis(Trifluoromethanesulfonyl)imide as an Example. *International Journal of Molecular Sciences*. 2021; 22(17):9117.
https://doi.org/10.3390/ijms22179117

**Chicago/Turabian Style**

Kruk, Danuta, Elzbieta Masiewicz, Sylwia Lotarska, Roksana Markiewicz, and Stefan Jurga. 2021. "Correlated Dynamics in Ionic Liquids by Means of NMR Relaxometry: Butyltriethylammonium bis(Trifluoromethanesulfonyl)imide as an Example" *International Journal of Molecular Sciences* 22, no. 17: 9117.
https://doi.org/10.3390/ijms22179117