Free and Bound States of Ions in Ionic Liquids, Conductivity, and Underscreening Paradox

Using molecular dynamics simulations and theoretical analysis of velocity-autocorrelation functions, we study ion transport mechanisms in typical room-temperature ionic liquids. We show that ions may reside in two states: free and bound with an interstate exchange. We investigate quantitatively the exchange process and reveal new qualitative features of this process. To this end, we propose a dynamic criterion for free and bound ions based on the ion trajectory density and demonstrate that this criterion is consistent with a static one based on interionic distances. Analyzing the trajectories of individual cations and anions, we estimate the time that ions spend in bound “clustered states” and when they move quasifreely. Using this method, we evaluate the average portion of “free” ions as approximately 15%–25%, increasing with temperature in the range of 300–600 K. The ion diffusion coefficients and conductivities as a function of the temperature calculated from the velocity and electrical-current autocorrelation functions reproduce the reported experimental data very well. The experimental data for the direct-current conductivity (constant ionic current) is in good agreement with theoretical predictions of the Nernst-Einstein equation based on the concentrations and diffusion coefficients of free ions obtained in our simulations. In analogy with electronic semiconductors, we scrutinize an “ionic semiconductor” model for ionic liquids, with valence and conduction “bands” for ions separated by an energy gap. The obtained band gap for the ionic liquid is small, around 26 meV, allowing for easy interchange between the two dynamic states. Moreover, we discuss the underscreening paradox in the context of the amount of free charge carriers, showing that the obtained results do not yet approve its simplistic resolution.

Popular Summary

Room-temperature ionic liquids (RTILs) are nearly universal solvents whose properties are of great interest for various energy devices, such as supercapacitors, batteries, and fuel cells. To perform their electrolytic functions, RTILs must be able to efficiently transfer charge. But it is unclear how they do so because the ions in these liquids are densely packed together. One hypothesis is that the ions reside in clusters, with charge transfer occurring as ions “hop” from one cluster to another. Using molecular simulations and theoretical analysis, we show that this picture is likely correct.

Our results reveal that ions in RTILs exist in two states—free and bound—and that the ions can quickly transit between these states, justifying a concept of an ionic semiconductor, with valence and conduction bands for ions. Analyzing the trajectories of individual cations and anions, we estimate the average time that an ion spends in bound states and when it moves quasifreely. From this, we estimate that, on average, 15%–25% of ions are free, a percentage that increases in the temperature range of 300–600 K. A narrow energy gap of about 26 meV between states allows for an easy interchange between the two states. Our calculated ion diffusion coefficients and conductivities reproduce reported experimental data well. Notably, the conductivities are also in good agreement with theoretical predictions from the Nernst-Einstein equation, based on free ions in RTILs only.

It will be interesting to further explore how the balance between free and bound ions will change in nanoscale confinement, including the electrified ones, in connection with the performance of ionic liquids in porous electrodes or electrotunable friction.

Figure 1

Molecular dynamics simulation of bulk RTIL [Bmim][TFSI]. (a) Snapshot of MD simulation (red and blue colors indicate cations and anions, respectively). Molecular structures of cation ${\text{Bmim}}^{+}$ (b) and anion ${\mathrm{TFSI}}^{-}$ (c). The numbers near atoms denote the partial charges on each atom in the all-atom model of ions used for MD modeling (unit: elementary charge). The solid red circle with the sign “$+$” represents the cation center (i.e., the mass center of the ring), and the solid blue circle with the sign “$-$” indicates anion center (i.e., the mass center of the whole anion).

Figure 2

Two states of ion dynamics revealed by ion trajectories based on MD simulations of [Bmim][TFSI]. Trajectories of arbitrarily chosen cation (a) and anion (b) and the contour map of trajectory density of the cation (c) and anion (d). (e) The percentage of free ion ($\gamma$, given in percentage) as a function of the temperature (lines are guides for the eye). (f) Plot for the percentage of free ions based on Eq. (1) (lines represent the linear fit). Gray, green, and red colors in (c),(d) represent small, middle, and high densities.

Figure 3

The kinetics of exchange process for ions in [Bmim][TFSI]. (a) The survival probability functions of cations in free (top panel) and bound (bottom panel) states. Red solid lines are obtained from MD trajectories by Eq. (2); blue dashed and green dotted lines represent the exponential and biexponential fittings, respectively, and biexponential rather than exponential fitting describes $c(t)$ reasonably well. The exponential function is $c(t)=e^{-t/\tau }$, where $\tau$ is the mean residence time; the biexponential function is $c(t)=a{e}^{-t/{\tau }_f}+b{e}^{-t/{\tau }_s}$, where $a+b=1$ and ${\tau }_f$ and ${\tau }_s$ denote, respectively, the fast and slow relaxation times (see Table S1 in the Supplemental Material [108] for details). (b) The mean residence time of cations and anions in free (top panel) and bound (bottom panel) states calculated by Eq. (3).

Figure 4

Impact of the temperature on the ion kinetics in RTIL [Bmim][TFSI]. (a) VACFs of cations (top panel) and anions (bottom panel) at indicated temperatures. (b) Diffusion coefficients of cations (top panel) and anions (bottom panel) as a function of the temperature shown in Arrhenius coordinates. Red solid, green dotted, and blue dashed lines represent free ions, overall ions, and bound ions, respectively. Magenta circles in (b) represent experimental data from Ref. [76].

Figure 5

Two-state theory of VACFs of cations in RTIL [Bmim][TFSI]. (a) Comparison of the Fourier transforms ${\widehat{K}}_v(\omega )$ of VACFs obtained in MD simulations and that for the two-state model based on Eq. (7), where ${\widehat{K}}_v(\omega )=2\mathrm{Re}{\tilde{K}}_v(s=j\omega )$. (b) VACFs in the time domain for cations as a linear combination of those of free and bound cations based on Eq. (10) as obtained for the same data as in (a). The statistical coefficient of determination is 0.994, 0.979, and 0.971 for 300, 450, and 600 K, respectively.

Figure 6

The impact of the temperature on (a) the ECACF and (b) spectrum of frequency-dependent electrical conductivity. The data are obtained from MD simulations of RTIL [Bmim][TFSI].

Figure 7

The molar electrical conductivity $\mathrm{\Lambda }$ of RTIL [Bmim][TFSI] as a function of temperature. Red squares show the values calculated from the MD-simulated electric current autocorrelation function [Fig. 6]. Blue dots are experimental data from Ref. [86]. Black circles display the values computed via original Nernst-Einstein equation (see Ref. [50]) with the overall diffusion coefficients of cations and anions [shown in Fig. 4]. Purple stars are based on Eq. (12) (modified Nernst-Einstein equation), with the free-ion percentage evaluated by the trajectory-density method, while the green diamonds are the results of application of the ion-pairing method.

Figure 8

The influence of MD simulation box size on (a) the free-ion percentage, (b) diffusion coefficient of ions in the free state, (c) diffusion coefficient of ions in the overall state, (d) diffusion coefficient of ions in the bound state, and (e) RTIL conductivity.

Figure 9

For small enough cubicles, the distribution of sizes of the tumbling areas as obtained in the box-counting method weakly depends on the cubicle size $l_{\mathrm{cub}}:$ (a) 0.2 nm, (b) 0.3 nm, and (c) 0.4 nm.

Figure 10

Free-ion percentage $\gamma$ of (a) cations and (b) anions in [Bmim][TFSI] at 300, 450, and 600 K for different trajectory lengths. Dots for each temperature represent the average values for different running times of the MD simulation. The lines are a guide to the eye. One can clearly see a lack of dependence of $\gamma$ on the trajectory length for $t\ge 2\mathrm{ns}$ (very tiny deviations from a constant $\gamma$ have a statistical nature).