On the collective network of ionic liquid/water mixtures. IV. Kinetic and rotational depolarization

Abstract

Dielectric spectroscopy is a measure of the collective Coulomb interaction in liquid systems. Adding ionic liquids to an aqueous solution results in a decrease of the static value of the generalized dielectric constant which cannot be attributed to kinetic depolarization models characterized by the static conductivity and rotational relaxation constant. However, a dipolar Poisson-Boltzmann model computing the water depolarization in the proximity of ions is not only successful for simple electrolytes but also in case of molecular ionic liquids. Moreover, our simple geometric hydration model is also capable to explain the dielectric depolarization. Both models compute the dielectric constant of water and obtain the overall dielectric constant by averaging the values of its components, water and the ionic liquid, weighted by their volume occupancies. In this sense, aqueous ionic liquid mixtures seem to behave like polar mixtures.

I. Introduction

Ionic liquids and water have not always been best friends.¹ Even if they are completely miscible with water, there are often issues of hydrolysis.^{2, 3} However, mixtures of ionic liquids and water may show physico-chemical properties which are not accessible by the pure compounds, thus enabling novel applications for many fields.¹ Quite generally, in many applications ionic liquids suffer from their high viscosities. For example, 1-butyl-3- ethylimidazolium tetrafluoroborate [C₄ mim][BF₄] has a viscosity of approximately 200 mPa s which is more than two orders of magnitude higher compared to the value of water. Actually, one of the fluidest ionic liquids is 1-ethyl-3-methylimidazolium dicyanamide with a viscosity of roughly 20 mPa s at room temperature which is still far less fluid than water. This situation changes when mixing an ionic liquid with water. The viscosity⁴ and density^{5, 6} drop significantly with increasing water content whereas the static electric conductivity σ(0) shows a parabolic behavior as a function of the mole fraction.^{5, 7} Furthermore, the physico-chemical properties of the ionic liquid/water mixtures change with mole fraction and can be roughly classified into three regions:^8–10 At low water mole fractions, x_W < 0.2, the water self-diffusion coefficient stays nearly the same with increasing water concentration. These water molecules are diluted in the ionic liquid network. Between a mole fraction of 0.2 < x_W < 0.8, water molecules seem to aggregate in small clusters, for example dimers and trimers, within the ionic liquid network. However, an increasing water content has a stronger effect on the diffusion of anions compared to that of cations indicating stronger interactions. Moreno et al. reported that the interaction between water and the ionic liquid ions is non-selective and water mainly initiates a swelling of the [C₄mim][BF₄] network.¹¹ This corresponds to our former finding that the decreased interaction between cations and anions is not due to individual water molecules penetrating the IL network. Rather, the electric field of the water molecules screens the Coulomb interaction.¹²

At high water mole fractions, x_W > 0.8, the swelling of the ionic liquid structure, i.e. the stretching of the chain-like structures, results in a breakdown of the highly branched ionic network and the mixture decomposes into water rich and ionic liquid rich regions. A turnover of the physico-chemical properties at this mole fraction was observed by Lopes et al.^{9, 10} and Voth et al.^{13, 14}, but Maroncelli et al.¹⁵ found no evidence for abrupt changes taking place at a particular composition. The formation of micelles is only observed for long alkyl chain cations, e.g. C₈ mim⁺.

In a series of papers,^12,16,17 we analyzed the collective network of [C₄ mim][BF₄]/H₂O mixtures with x_W ranging from 0.425 to 0.967: The first manuscript¹² dealt with the orientational structure. We found clear evidence of a strong anion-water network showing hydrogen bonding but weak interactions between the cations and water. Furthermore, the water molecules mainly act on the ions via their dipolar field and the anions play the role of a mediator between water and the cations. In the second publication,¹⁶ we juxtaposed the experimental and computed dielectric spectra of the mixture at water mole fractions x_W from 0.768 to 0.967. At that time we could not resolve the translational contribution in detail and were restricted to a numerical Laplace transform of the auto-correlation function of the current, 〈J(0) · J(t)〉.¹⁶ In the present work we make up the leeway and present cosine-exponential based fits of the auto-correlation function 〈J(0)·J(t)〉.¹⁸ Moreover, we give here a representation of the frequency-dependent contribution of the ro-translational coupling ϵ_DJ(ω) between the col-lective rotational dipole moment M_D(t) and the current J(t) to the generalized dielectric constant Σ₀(ω). In the third paper¹⁷ the Voronoi decomposition proved to be helpful for the interpretation of the ionic liquid/water networks: We divided the C₄ mim⁺-cations into head (methyl group), imidazolium ring and tail (butyl chain) and split the radial and orientational correlation functions into contributions from the first, second and third solvation shell. As expected, the tails turned out to be more or less hydrophobic and to prefer interactions with other tails. We also found a strong anion-water network as reported in literature.^19–23 Nevertheless, both species compete for their favorable positions in direct vicinity of the imidazolium ring hydrogens.

With the present work we want to draw the attention on the dielectric decrement of aqueous IL mixtures as a function of increasing ion content and its interpretation in terms of electrolyte theories,^24–38 as well as models of mixtures of polar solvents.^39–42 Experimental dielectric studies concerning aqueous IL mixtures^16,43–46 show a decreasing static dielectric constant Σ₀ with increasing ion content. The depolarization effect is often partially attributed to the motion of the ions,^30–34 but we will show that the kinetic depolarization cannot explain the concentration dependence of Σ₀(c_IL) in our IL mixtures at ion concentrations above 1M and rotational depolarization plays a major role.

II. Methods

In this work we used the trajectories from our former studies concerning the mixtures of [C₄ mim][BF₄] with TIP3P water, but extended the simulation to 120 ns,^12,16,17 which is more than one order of magnitude longer than other computational studies.^9–11,13,14 Although the viscosity of these mixtures is below 6 mPa s,¹⁶ we deem long simulation periods essential for good statistics. In addition, not only the coordinates of the atoms at each time step but also their velocities were stored in order to compute more reliable data on the auto-correlation function of the electric current 〈J(0) · J(t)〉 and on the cross correlation between the collective rotational dipole moment M_D(t) and the electric current J(t). These additional data improved the statistics of our investigated properties and lead to slightly changed values of the static electric conductivity σ(0) (see Table I) compared to those reported in our previous study.¹⁶

Table I.

Composition of the [C₄ mim][BF₄]/water mixtures. n_IL and n_W denote the number of the respective molecules. Even at high water mole fractions x_W, the ionic liquid ions occupy a significant volume V of the simulation box as indicated by Φ_IL = V_IL/V. The volume fraction of water Φ_W = Φ_bulk + Φ_hyd is splitted into bulk (Φ_bulk) and hy-dration (Φ_hyd) water based on the ratio of amplitudes of $<M_{\text{D}}^W(0)\cdot M_{\text{D}}^W(t)>.$

cIL =	0.00 M	1.25 M	2.52 M	3.77 M	5.31 M
n IL	0	55	111	166	234a
n W	2440b	1592	1147	548	0
x W	1.000	0.967	0.912	0.768	0.000
Φhyd	0.000	0.338	0.372	0.214	0.000
Φbulk	1.000	0.407	0.148	0.045	0.000
ΦIL	0.000	0.255	0.480	0.741	1.000
Σ0	102	55.4	38.5	24.0	12.5
σ(0) / S m−1	0.0	11.9	11.3	5.0	0.44c

^aExtrapolated number since Ref. 47 used a different box size.

^bExtrapolated number since Ref. 48 used a different box size.

^csee Ref. 47.

A detailed description of the simulation setup is given in Ref. 12 and summarized here briefly. Three independent molecular dynamics simulations at constant volume, V =(41.8 Å)³, with ionic liquid concentration c_IL =1.25, 2.52, 3.77 M (see Table I) were performed at 300 K with a time step of 2 fs using particle mesh Ewald summation with a real-space cut-off of 10 Å and a screening parameter κ = 0.41 Å⁻¹. The same cut-off was also used for dispersion forces. These mixtures are accompanied by the results of neat [C₄mim][BF₄] (Ref. 47) and pure TIP3P water⁴⁸ in Table I.

III. Theory

A. Dielectric spectrum and its components

This present manuscript extends the dielectric theory already used in Ref. 16. The generalized dielectric constant, Σ₀(ω) is composed of a dielectric permittivity, ε(ω), and dielectric conductivity, ϑ₀(ω) and their ro-translational coupling ε_DJ(ω):¹⁸

$${\text{Σ}}_0(\omega )=ϵ(\omega )-1+{\vartheta }_0(\omega )+2{ϵ}_{\text{DJ}}(\omega )$$ (1)

Usually, the latter contribution ε_DJ(ω) is equally distributed between the dielectric permittivity and dielectric conductivity. Since we are interested in the coupling between the ionic motion characterized by the electric current J(t) and the rotation of dipoles M_D(t) for the interpretation of the kinetic depolarization, this contribution is considered separately here. For the sake of simplicity, the static values of the dielectric properties in this work are denoted without the corresponding arguments, i.e.

$$\underset{\omega \to 0}{\lim }{\text{Σ}}_0(\omega )={\text{Σ}}_0(0)={\text{Σ}}_0.$$ (2)

In Ref. 16, analytical representations of the collective dipole correlation functions for the cations $<M_{\text{D}}^C(0)\cdot M_{\text{D}}^C(t)>$ and for water $\langle M_{\text{D}}^W(0)\cdot M_{\text{D}}^W(t)\rangle ,$

$$f_{\text{DD}}(t)=\sum _k{A}_k\exp (-t/{\tau }_k),$$ (3)

were used to compute the dielectric permittivity ε(ω). Since these relaxation processes are of collective nature, an assignment of the kth-contribution to a distinct physical process is complicated. However, often the process with the longest τ_k is used to correlate single particle and collective rotation via the Kivelson-Madden equation.⁴⁹

In the simulations, the dielectric permittivity ε(ω) can be decomposed in pure cationic, ε^CC(ω), and pure water, ε^WW(ω) contributions as well as their coupling, ε^WC(ω), using the collective dipole moments $M_{\text{D}}^C(0)$ and $M_{\text{D}}^W(0)$ for the auto-correlation functions, $\langle M_{\text{D}}^C(0)\cdot M_{\text{D}}^C(t)\rangle$ and $\langle M_{\text{D}}^W(0)\cdot M_{\text{D}}^W(t)\rangle ,$ or the cross-correlation function $\langle M_{\text{D}}^C(0)\cdot M_{\text{D}}^W(t)\rangle ,$ respectively. The anion ${\text{BF}}_4^{-}$ has tetrahedral symmetry and therefore no permanent dipole moment. The transient dipole of ${\text{BF}}_4^{-}$ due the vibrations is negligibly small to contribute to ε(ω).

The dielectric conductivity ϑ₀(ω) is the translational contribution of the ions to the dielectric spectrum Σ₀(ω). It is defined by the ratio of the frequency-dependent electric conductivity σ(ω) and the angular frequency ω:

$${\vartheta }_0(\omega )=4\pi i\frac{\sigma (\omega )-\sigma (0)}{\omega }$$ (4)

In Ref. 16, the electric conductivity σ(ω) was only computed by numerically Laplace transforming the current correlation function 〈J(0) · J(t)〉. Afterwards, the dielectric conductivity ϑ₀(ω) was obtained by eq. (4). This results in noisy spectra. The corresponding fit functions for 〈J(0) · J(t)〉 were first introduced in Ref. 18 and 50:

$$f_{\text{JJ}}(t)=\sum _k{A}_k\cos ({\omega }_k\cdot t+{\delta }_k)\exp (-t/{\tau }_k).$$ (5)

Slower translational relaxation processes were detected by the mean-squared displacement of the collective translational dipole moment $\langle \Delta M_J^2(t)\rangle$ and then added to the fit of 〈J(0)· J(t)〉 since $d^2\langle \Delta M_J^2(t)\rangle /d{t}^2=2\langle J(0)\cdot J(t)\rangle$ as shown in Ref. 51. In the same reference we showed that all time constants τ_k are also valid for relaxation processes of the ion cage, i.e. the first shell around a central ion. These shells contain a significant number of ions with the same charge as the central ion.⁵¹ Cationcation contacts are still very prominent in our aqueous [C₄mim][BF₄] mixtures.¹⁷

The ro-translational coupling ε_DJ(ω) can be computed by the Laplace transform of the cross-correlation function 〈M_D(0) · J(t)〉 which can be represented analytically by

$$f_{\text{DJ}}(t)=\sum _k{A}_k{t}^{{\gamma }_k-1}\exp \left(-t/{\tau }_k\right)$$ (6)

leading to a sum of Cole-Davidson peaks in the dielectric spectrum.⁵² Generally, the ro-translational contribution ε_DJ(ω) to the dielectric spectrum of pure ionic liquids is quite small due to moderate dipole densities and a small electric current J(t) of the viscous systems.^{47, 53} However, in ionic liquid/water mixtures, the strong dipolar density of water is combined with more mobile charge carriers, thus enhancing J(t). Consequently, the ro-translational contribution should be stronger than in pure ionic liquids.

B. Electrolyte theories

In a series of papers, Hubbard et al.^{30, 31} and Wolynes et al.^32–34 attributed part of the dielectric decrement to the kinetic depolarization of the ions moving through the water media which should be reflected by the ro- translational coupling ε_DJ. The general idea behind these models is the following: A moving ion enforces the rotation of the water dipoles which disturbs the normal alignment of water dipoles and results in a dielectric decrement. This is also achieved, if the rotation of the water molecules slows down the ionic motion. In case of the Hubbard model

$$\Delta {ϵ}^{\text{Hub}}=-p\frac{{\tau }_D\cdot \sigma (0)}{{ϵ}_0}\cdot \frac{{\text{Σ}}_0}{{\text{Σ}}_0+1}$$ (7)

we assume perfect slip (p = 2/3) boundary conditions between the ions and water. τ_D is the rotational relaxation constant of the auto-correlation $\langle M_{\text{D}}^W(0)\cdot M_{\text{D}}^W(t)\rangle$ of the collective rotational water dipoles, σ(0) the static conductivity and ε₀ = 8.85 · 10⁻¹²As/Vm the dielectric constant in vacuo. Wolynes et al.³³ developed a molecular picture of the ion-water interaction, that leads to a kinetic decrement

$$\Delta {ϵ}^{\text{Wol}}=-\frac{2}{3}\frac{{\tau }_D\cdot \sigma (0)}{{ϵ}_0}\sum _{i,j}\frac{R_i}{q_i}\langle \frac{{\mu }_j\cdot r_{ij}}{{\left|r_{ij}\right|}^3}\rangle \frac{1}{n_{\text{IL}}}.$$ (8)

Here, the index i runs over all ions and the index j over all water molecules with their dipolar vector μ_j. The vector r_ij points from the respective ion i to the respective water molecule j. The ion radius R_i can be easily approximated assuming a spherical shape and using the corresponding Voronoi volumes of the species.¹⁷ Nevertheless, the depolarization in both models depends linearly on the static conductivity σ(0) and the rotational relaxation constant τ_D

In contrast to the kinetic depolarization of Hubbard and Wolynes, also pure static approaches exist. For example, the dressed ion theory,⁵⁴ which is an extension to the classical Debye-Hückel theory, computes the depolarization due to the ions in Fourier space as

$$\Delta {\Sigma }_0(k)=\frac{1}{k^2}({\text{Σ}}_0{\kappa }_D^2+\frac{S_{zz}(k)}{k_{B}T{ϵ}_0}),$$ (9)

using the inversed Debye screening length κ_D and the charge-ordering function S_zz(k). For very dilute electrolyte solutions, the charge-ordering function becomes negligible and the dressed ion theory simplifies to the result of Debye-Hückel.^{37, 55} However, the solvent enters these two theories only by its static dielectric constant and the complete effect on the overall static dielectric constant is attributed to ion-ion interactions with a solvent continuum as background.

Levy et al. studied the interactions of the water dipoles in the proximity of ions using a static dipolar Poisson- Boltzmann approach. Thus, they were able to reproduce the dielectric decrement of simple electrolyte solutions up to ion concentrations of 5M.³⁸ The static value ε^DPB depends on the water concentration cW and a molecular water dipole μ_W.

$${ϵ}^{\text{DPB}}-1=\frac{c_W{\mu }_W^2}{3k_{B}T{ϵ}_0}$$ (10)

In order to yield the experimental dielectric permittivity of pure water ${ϵ}_W^{\exp }=78.2$ they used the dipole moment μ_W as a fit parameter which resulted in a value of μ_W = 4.80 D instead of the ${\mu }_W^{\exp }=1.85$ D.³⁸ This scaling was explained by missing correlation effects in the liquid phase. However, in contrast to their approach, we will use the static dielectric constant ε_W = 102 of pure TIP3P for the analysis of our computational mixtures. This corresponds to Σ₀ at c_IL = 0 M in Table I. Assuming point-shaped ions, the distance dependence of the dielectric permittivity from a central ion can be obtained by

$$ϵ(r)\simeq \frac{{ϵ}_{\text{W}}}{3h^2(l_h/r)+1}$$ (11)

with an auxillary function h(…) (see Ref.³⁸ for further details) and l_h characterizing the shell thickness of water layers

$$l_h=\sqrt{{\lambda }_B\frac{{\mu }_W}{e\sqrt{10}}}.$$ (12)

Here, μ_W =2.35 D corresponds to the molecular dipole moment of TIP3P and λ_B is the Bjerrum length

$${\lambda }_B=\frac{e^2}{4\pi {ϵ}_0{ϵ}_{\text{W}}{k}_{B}T}.$$ (13)

The dependence of the static dielectric constant 〈ε(c_IL)〉 on the concentration of ions c_IL can be computed by spherical averaging,

$$\langle ϵ(c_{\text{IL}})\rangle =\frac{{\displaystyle \underset{0}{\overset{r(c_{\text{IL}})}{\int }}4\pi {(r\prime )}^2ϵ(r\prime )dr\prime }}{\frac{4}{3}\pi r^3(c_{\text{IL}})},$$ (14)

using a typical distance between two ions $r(c_{\text{IL}})=\frac{1}{2}{(2c_{\text{IL}})}^{-1/3}.$

Furthermore, Levy et al.³⁸ proposed additional water and salt corrections to the dielectric permittivity based on the dipolar Poisson-Boltzmann approach. The first correction takes into account the fluctuation of water dipoles beyond their mean-field level. The second salt correction may be used to explain a dielectric increment with increasing ion concentrations. This phenomenon was found experimentally by Buchner et al. for [C₄mim][BF₄] mixtures with dichloromethane.⁵⁶

C. Volume occupancies of the different species

The composition of our ionic liquid/water mixtures can be characterized by various properties, e.g. the water mole fraction 0.768≤ x_W ≤0.967, the ion concentration 3.77M≥ c_IL ≥ 1.25M or the volume occupancies Φ_IL and 0.259≤ Φ_W ≤0.745. Comparing the water mole fraction x_W and the volume occupancy Φ_W reveals that even the high water mole fraction x_W =0.967 does not correspond to a dilute ion solution since only 74.5% of the volume is occupied by water molecules. This is due to the discrepancies of the molecular volumes of 270 ų and 33 ų of cations and water, respectively.¹⁷ The volume fractions of water, Φ_W, and of the ionic liquid, Φ_IL, were computed by Voronoi decomposition.^{17, 57}

In a second step, the water volume fraction Φ_W is further decomposed into bulk, Φ_bulk, and hydration water, Φ_hyd, based on a three exponential fit (k = 1 … 3) of the collective rotational dipole moment of all water molecules $\langle M_{\text{D}}^W(0)\cdot M_{\text{D}}^W(t)\rangle$ according to Eq. (3). Respective fit pa-rameters are given in the supplementary material. Since the rotation of hydration water is hindered, compared to bulk water, due to the interaction with the ionic liquid ions, the corresponding time constant τ₃ is longer than τ₂ ≃ 7.5 ps (the latter being attributed to bulk water), and is more or less independent of the IL concentration. The partitioning between hydration and bulk water is based on the relative amplitudes A_k:

$${\text{Φ}}_{\text{hyd}}=\frac{A_3}{A_1+A_2+A_3}\cdot {\text{Φ}}_W$$ (15)

$${\text{Φ}}_{\text{bulk}}=\frac{A_1+A_2}{A_1+A_2+A_3}\cdot {\text{Φ}}_W$$ (16)

Overall, the ion concentrations considered here do not correspond to a highly diluted electrolyte solution anymore as pointed out by Bowers et al.⁵⁸ as well as Singh and Kumar.⁵⁹ Both groups observed significant aggregation of ions at c_IL >0.8M and consequent changes in their physico-chemical properties. All mixtures considered here are above this threshold and overlap of hydration shells around different ions is expected.^21,58–60 Based on the coordination numbers reported in Ref. 17 it is obvious that at least the cations share some water molecules, since the number of ions, n_C = n_IL, times the coordination number of water around the cations exceeds the total number of water molecules, n_W, in each of our simulation boxes. Since cation-cation contacts are very prominent in all our mixtures, the concept of solvent- separated ion pairs, solvent-shared ion pairs and contact ion pairs is difficult to apply.⁶¹

D. Geometric hydration model

It is interesting to note that the volume fraction Φ_hyd can be also well approximated using a simplified geometrical model: Let us assume that the hydrated ions are placed randomly in a simulation box of volume V with a density n/V = c_IL. The volume of these ions and their hydration shell is V_hyd. The probability P in a Poisson process that one of the ions does not overlap with any other is

$$P=\exp (-V_{\text{hyd}}/V)$$ (17)

This is also a reasonable estimate of the volume fraction of the unbound (bulk) water,

$$P\simeq {\text{Φ}}_{\text{bulk}}=1-{\text{Φ}}_{\text{IL}}-{\text{Φ}}_{\text{hyd}}.$$ (18)

However, the ions are not exactly placed randomly since they cannot occupy the non-hydrated volume V_IL. Using the effective volume V_eff = V – V_IL in Eq. (17) instead of the total volume V yields

$$P=\exp (-\frac{V_{\text{hyd}}/V}{1-V_{\text{IL}}/V})=\exp (-\frac{{\text{Φ}}_{\text{IL}}}{1-{\text{Φ}}_{\text{IL}}}\cdot \frac{V_{\text{hyd}}}{V_{\text{IL}}})$$ (19)

using Φ_IL = V_IL/V. Altogether, the volume fraction of the hydration water Φ_hyd results in

$${\text{Φ}}_{\text{hyd}}({\text{Φ}}_{\text{IL}})=1-{\text{Φ}}_{\text{IL}}-\exp (-\frac{{\text{Φ}}_{\text{IL}}}{1-{\text{Φ}}_{\text{IL}}}\cdot \frac{V_{\text{hyd}}}{V_{\text{IL}}})$$ (20)

Of course, the volume fraction of the hydration water goes consistently to zero, both in the absence of ions (Φ_IL = 0) and in the absence of water (Φ_IL = 1). The ratio of the volumes V_hyd/V_IL can be estimated by an average computed from the space occupied of water in the first Voronoi shell of a cation or anion and the space assigned to the ions. The ratio 〈V_hyd/V_IL〉 at c_IL=1.25 M is roughly 2.12 which means that the average hydration volume is more than twice the volume of the central ion. Although simple, the geometrical model is able to reproduce reasonably well the volume fractions without any fit parameters, as shown in Fig. 2 by the respective lines. Here, Φ_bulk (blue line) corresponds to the exponential part in Eq. (20). Please note that the volume occupancy Φ_IL (red line) also depends linearly on the concentration of the ions c_IL. The amount of bulk water drops more or less exponentially and the volume occupancy of the hydration water has a parabolic behavior: At low ion concentrations, c_IL, only few possibilities for the water molecules exist to be the next neighbor of an ion. Although the number of ions increases with c_IL, the competition for the water molecules to gain a place in the first solvation shell of an ion gets correspondingly harder. We will use this model later on to describe the dielectric decrement of the mixture.

Fig. 2.

Volume fractions of the ionic liquid (Φ_IL), bulk (Φ_bulk) and hydration water (Φ_hyd) obtained by Eq. (15) and (16) (see Table I) as well as their representation by the lines based on the geometrical model (see Eq. (20)).

IV. Results And Discussion

A. Computation of the dielectric spectrum

The frequency dependent, generalized dielectric constant Σ₀(ω) emerges from the collective rotation of the dipoles within the system and the collective translational behavior of the ions. In principle, the rotational part in aqueous electrolyte solutions is dominated for dilute solutions by the auto-correlation function of the dipole moment of water molecules yielding ε^WW(ω), since atomic ions possess no dipole moment and the translational-rotational coupling ε_DJ(ω) is usually small. Increasing the concentration of the ions leads to an additional peak in the dielectric spectrum at slightly higher frequencies due to the dielectric conductivity ϑ₀(ω).^16,18,50

In case of molecular ions, e.g. ionic liquids, this situation gets a little more complicated since (1) the ions may have dipole moments as well. In the present work, each C₄mim⁺-cation has an average gas-phase dipole moment μ^C of 5.7 D which is more than double the value of TIP3P water model (μ^W =2.35 D).¹⁶ For simplicity the anion ${\text{BF}}_4^{-}$ was chosen to have no net dipole moment on average. (2) In simple electrolytes the rotational and translational contribution to Σ₀(ω) stem from different species since the ions have no dipole moment and the water molecules have no net charge. In case of ionic liquids, however, the ions contribute to both. Nevertheless, the ro-translational coupling, ε_DJ = -0.3, is small in neat ionic liquids.⁴⁷ On the one hand the viscosity is high, hampering the mobility of the charge carrier. On the other hand, the alignment of dipole moments is not very strong and the overall dipole density is low. (3) However, this cross contribution can be enhanced in ionic liquid/water mixtures since the viscosity of the system is lowered exponentially with increasing water content and the dipole density of water is significantly higher than that of common ionic liquids.

Based on the computational dielectric theory presented in Ref. 18, the auto-correlation functions of the collective rotational dipole moment M_D(t) and of the electric current J(t) yield the dielectric permittivity ε(ω) and dielectric conductivity ϑ₀(ω), respectively. For the three ionic liquid/water mixtures at c_IL =1.25, 2.52 and 3.77M the corresponding frequency dependent dielectric spectra are depicted in Fig. 3a-c). The corresponding fit parameters of the respective constituting correlation functions (see Eq. (3)-(6)) are listed in the supplementary material. The self-contributions from the cations (CC, red area) and water (WW, blue area) as well as the dielectric conductivity (orange) and the ro-translational coupling (green) are shown separately. The agreement with the experimental results (black triangles) is quite good.¹⁶ In contrast to our former work,¹⁶ the frequencydependent dielectric conductivity ϑ₀(ω) (orange area in Fig. 3a-c) is not only computed from cosine exponential fits of 〈J(0) · J(t)〉 (see Eq. (5)) but also confirmed by the fit of the initial region of the mean-squared displacement $\langle \Delta {\text{M}}_J^2\rangle$ which yields slower processes with higher accuracy.⁵¹ In particular, the major contribution to ϑ₀ stems at all ion concentrations from the slowest process (k = 4) of 〈J(0) · J(t)〉 with a time constant τ₄ of roughly 10 ps (see supplementary material). Since the amplitude of this contribution is rising with increasing ion concentration, ϑ₀ also rises. The oscillatory behavior of 〈J(0) · J(t)〉, on the other hand, modelled by k = 1, 2, contributes only marginally to the static value ϑ₀. Nevertheless, the characteristic oscillatory shape of 〈J(0) · J(t)〉 is mainly determined by the first (k = 1) cosine exponential of f_JJ(t) and the slowest contribution (k = 4) is almost invisible at first inspection.⁵¹ Interestingly, the frequency ω₁ ≃12.3 ps⁻¹ of the k = 1 oscillation does not change with increasing ion content and corresponds to a wave number $\tilde{\nu }=65{\text{cm}}^{-1}$ which is much lower than ${\tilde{\nu }}^{\exp }=94{\text{cm}}^{-1}$ as observed for the pure liquid in experiments.⁶² This finding is interpreted as an uniform weakening of the cation-anion interactions by water molecules and points towards a stable an- ion/water network persisting at all concentrations c_IL investigated here. The strength of these networks is known _{in literature.}^{12,19–23,59}

Fig. 3.

Decomposition of the frequency dependent dielectric spectra of [C₄mim][BF₄]-water mixtures at various IL concentrations. Pure water and cationic contributions to the dielectric permittivity ε(ω) are displayed as blue and red shaded areas, respectively. To some extent they overlap with the green shaded area representing the frequency-dependent ro-translational coupling ε_DJ(ω). The orange shaded areas correspond to the contribution of the dielectric conductivity ϑ₀(ω). The experimental data (black triangles) are taken from Ref. 16.

At the lowest ion concentration c_IL = 1.25M, the rotation of water molecules is responsible for more than 90% of the static dielectric constant Σ₀ (see Fig. 3a). In contrast, at c_IL = 3.77M the contributions of the water rotation, of the cationic rotation and of the dielectric conductivity ϑ₀ are of comparable size as visible in Table II. Overall, the pure dielectric permittivity contribution of cations ε^CC seems to increase linearly with Φ_IL as predicted by our geometric model and visible in Fig. 4. Both cross-terms, the rotational coupling between cations and water, ε^WC, as well as the ro-translational contribution ε_DJ go along with Φ_hyd in Eq. (20). This seems reasonable since the hydration water is the most important coupling partner for the ions. ε^WW is not displayed in Fig. 4 since the values are much higher compared to the other permittivity contributions with the exception of c_IL = 3.77M. Moreover, since hydration and bulk water are contributing, the behavior of ε^WW when changing the IL concentration cannot be described by a function like Φ_bulk solely. Nevertheless, the overall trend is roughly decaying exponentially.

Table II.

Static values of various contributions to the generalized dielectric constant Σ₀ = ε^CC + ε^WW + ε^WC + ϑ₀ + 2ε_DJ. All ε-contributions emerge from rotational motions of the cations (C) and water (W). The anion is neglected here since it does not possess a permanent dipole. ϑ₀ is the static dielectric conductivity and ε_DJ the static ro-translational coupling which can also be estimated by the predictions of Hubbard et al.³⁰ (Δε^Hub) and Wolynes et al.³⁴ (Δε^Wol)

ΦIL	ε CC	ε WW	ε WC	ϑ 0	2εDJ	ΔεHub	ΔεWol
0.000a		102.0
0.255	2.3	46.5	1.1	2.9	2.6	-9.2	-6.9
0.480	4.7	24.9	1.5	4.1	3.3	-12.7	-5.6
0.741	6.1	8.4	0.5	5.7	3.3	-10.7	-2.2
1.000b	8.2			4.6	-0.3

^aThe value of pure TIP3P water model is reported in Ref. 48

^bThe values of the pure [C₄mim][BF₄] is taken from Ref. 47.

Fig. 4.

Static dielectric contributions to Σ₀: The permittivity caused by cation-cation interactions ε^CC goes along with Φ_IL whereas contributions between water and cations, ε^WC and ε_DJ follow Φ_hyd. In contrast, Δε^Hub and Δε^Wol are negative and do not show a clear trend. All static values can be found in Table II.

B. Electrolyte models

As a consequence of a decaying ε^WW, electrolyte theories such as the Debye-Hückel model or the dressed ion theory seem not appropriate to describe the depolarization since major contributions come from the depolarization of solvent molecules and are not due to pure inter-ionic interactions.^37,54,55 Regarding the kinetic decrement, we noticed that, although the relaxation con-stant τ_D for the collective water rotation is increasing with increasing c_IL (see supplementary material), the static electric conductivity σ(0) is simultaneously decreasing. This leads to a minimum of Δε^Hub instead of the expected monotonic behavior (see Fig. 4). A monotonic behavior is recovered by Δε^Wol due to the correlation of the vector connecting the ion with a water molecule r_ij and its dipole μ_W (c.f. Eq. (8)) but the trend goes in the wrong direction yielding a dielectric increase with increasing ion concentration. Furthermore, both values should be comparable to ε_DJ which has opposite sign and is slightly smaller. This is not only the case for [C₄mim][BF₄]/water mixtures but also for [C₂mim][OTf]/water mixtures as well.¹⁸ Interestingly, the ro-translational contribution of pure ILs is negative.⁴⁷

This discussion shows that kinetic depolarization models of Hubbard and Wolynes are not compatible with the dielectric decrement in case of our IL mixtures. The most obvious reason for this failure is the high concentration of ions c_IL > 1M of the mixtures investigated here which are far from the very dilute concentrations assumed in both models. The interaction between the cations and anions with each other seems to be so strong in our case that the intuitive picture of the kinetic depolarization of water molecules around a single ion does not apply anymore. The hydration shells of the ions overlap to some extent due to the agglomeration of ions resulting in a non-homogeneous mixture which cannot be described by the continuum model of Hubbard. Moreover, ion aggregation seems common in our mixtures. We reported in Ref.⁵⁷ mean residence times of cation-anion contacts of 64.0, 147.7 and 454.8 ps for c_IL=1.25, 2.52 and 3.77 M, respectively. If charged or neutral clusters of ions move in one direction for this period of time, the rotation of neighboring water dipoles cannot be correlated with the electric current since cations and anions moving in the same direction prefer different water orientations. Consequently, the term in brackets in Eq. (8) will be oscillating around zero.

However, Levy et al.³⁸ proposed a dipolar Poisson- Boltzmann theory concerning electrolyte solutions at higher ion concentrations. Here, the dielectric constant of water increases with increasing radial distance from a central ion reaching the bulk value for very long distances as visible in the inset of Fig. 1. However, the mean separation between two ions sets the maximum distance of water from any ion. Using spherical integration in Eq. (14) an average dielectric constant of water 〈ε(c_IL)) in the proximity of ions can be computed. Since our molecular ionic liquids occupy a significant volume as pointed out in the Theory section, the overall dielectric constant Σ₀ of the mixture may be estimated by

$${\text{Σ}}_0\simeq (1-{\text{Φ}}_{\text{IL}})\langle ϵ(c_{\text{IL}})\rangle +{\text{Φ}}_{\text{IL}}{ϵ}_{\text{IL}}$$ (21)

with ε_IL = 12.5 reported in Ref.⁴⁷ The corresponding dependence of Σ₀ on the volume occupancy Φ_IL of the ionic liquid is depicted in Fig. 5 (green dots) and agrees fairly with the computed Σ₀ (black crosses). So far, we used no fit parameters. Furthermore, this theory concurs nicely with our previous finding¹² that the overall effective electric field reduces the interaction between cations and anions and not interstitial water molecules. In the previous publication we observed the effect of the water on the ions and in the present work the effect of the ions on the water. Our findings also agree with the results of Kobrak:⁶³ The solute-solvent interactions are not governed by single ions, their steric structure or specific interactions with water. It is the ion density, which enters our considerations by their occupied volume Φ_IL and by the upper limit of the spherical integration of ε(r), that yields 〈ε(c_IL)〉.

Fig. 1.

Dielectric constant 〈ε(c_IL)〉 of water in the proximity of a point-size ion according to the dipolar Poisson-Boltzmann theory. The line at r=1.1Å corresponds to 〈ε(c_IL)〉 = 40.2 = ε_hyd obtained from a spherical averaging (c.f. Eq. (14)) and indicated by the gray shaded area.

Fig. 5.

Computational static dielectric constant Σ₀ of [C₄mim][BF₄] / TIP3P mixtures and their agreement with various models discussed in the text.

C. Mixtures of polar solvents

In his review, Kaatze⁶⁴ pointed out that strong local interactions lead to a failure of kinetic depolarization models and theories concerning bound water may be more applicable. For example, strong hydrogen bonds between the water molecules and the anions are known.^{12,19–23,59} Consequently, one may think of two different water species in the electrolyte solution: hydration water which is strongly influenced by the IL ions and bulk water which is not. As mentioned in the Theory section (see Eq. (15) and (16)), the respective volume fractions Φ_hyd and Φ_bulk can be extrapolated from the amplitudes of $\langle M_{\text{D}}^W(0)\cdot M_{\text{D}}^W(t)\rangle .$

We now assume that the rotation of hydration water is different from bulk behavior due to the presence of the ions resulting in a different local permittivity ε_hyd. The overall dielectric constant of the electrolyte solution can be estimated by the average of the local dielectric constants of hydration water (ε_hyd), bulk water (ε_bulk = ε_W) and the ionic liquid (ε_IL) weighted by their volume occupancies

$${\text{Σ}}_0\simeq {\text{Φ}}_{\text{hyd}}\cdot {ϵ}_{\text{hyd}}+{\text{Φ}}_{\text{bulk}}\cdot {ϵ}_{\text{bulk}}+{\text{Φ}}_{\text{IL}}\cdot {ϵ}_{\text{IL}}.$$ (22)

The only remaining fit parameter to match the static values of the generalized dielectric constant in Fig. 5 is the dielectric constant of the hydrated water molecules ε_hyd. Using a value of ε_hyd = 40.2 results in the pentagons in Fig. 5 which fit perfectly the computed Σ₀-values. The hydration model relies on the computed Φ_hyd, Φ_bulk and Φ_IL at the discrete given concentrations. In order to predict the volume occupancies at any concentration, they may be estimated by the geometric model in Eq. (20). The volume occupancy of the ionic liquid depends linearly on c_IL, thus the exponential part of Eq. (20) represents Φ_bulk. Using this geometric model, the discrete values of the dielectric constant Σ₀ (pentagons in Fig. 5) turn into a continous prediction shown as black dashed line.

The lower value of ε_hyd compared to ε_bulk seems reasonable since hydration water is expected to relax slower^{65, 66} and mutual position and orientation of water molecules are changed.¹² This simple hydration picture assumes that only the water molecules are affected by the presence of the cations and anions, but the presence of the water molecules does not affect the rotational, ε^CC, and translational, ϑ₀, contributions of the ions, since for instance ε^CC increases linearly with increasing Φ_IL as shown by the red dashed line in Fig. 4. In principle, this simple hydration model differs not so much from the dipolar Poisson-Boltzmann approach: The hydration model varies the volume fraction as a function of the IL concentration (see Fig. 2) and keeps the local dielectric constants ε_hyd, ε_bulk and ε_IL constant. However, in the dipolar Poisson-Boltzmann model³⁸ the water is not discrimated into hydrated and bulk water but the dielectric constant 〈ε(c_IL)〉 varies continously as a function of the distance from the ion. Comparing both methods, the dielectric permittivity ε_hyd equals 〈ε(c_IL)〉 for a distance of 1.1 Å depicted as gray area in the inset of Fig. 1. This distance is slightly less than the thickness of a complete hydration shell and may be reasonably explained by overlapping hydration shells in our IL mixtures.

The interpretation in terms of rotational depolarization is also endorsed by the success of the empirical logarithmic partitioning of Pujari et al.⁴² which was applied to aqueous mixtures of organic solvents and is depicted as red dash-dotted line in Fig. 5. They also suggested an additional correction term taking into account the polarity of the organic solvent

$$\ln ({\text{Σ}}_0)=(1-{\text{Φ}}_{\text{IL}})\ln ({ϵ}_{\text{W}})+{\text{Φ}}_{\text{IL}}\ln ({ϵ}_{\text{IL}})-d(1-{\text{Φ}}_{\text{IL}}){\text{Φ}}_{\text{IL}}$$ (23)

It turns out that our data results in a d-value of -0.08 which is comparable to methanol or ethanol. Interestingly, these alcohols share almost the same static dielectric constant of the pure [C₄mim][BF₄]. Hence, the aqueous IL mixtures investigated here seems to behave like common mixtures of polar solvents. Other models^39–41 describing these mixtures of polar solvents (see supplementary material for the corresponding figure) follow the overall trend of the decreasing dielectric constant but lack strong quantitative agreement. Moreover, all these models are intrinsically decaying with increasing ion concentration. However, experimental data on [C₄mim][BF₄] mixtures in dichloromethane⁵⁶ and acetonitrile⁶⁷ show a dielectric maximum as functions of mole fraction. This finding cannot be explained with those models but may be reproduced with a hydration model with ε_hyd > ε_bulk or two different hydration shells with different properties around cations and anions, respectively.

V. Conclusion

Although kinetic depolarization theories may be successful for very dilute electrolytes,^30–34 it seems not valid anymore in case of aqueous ionic liquid mixtures with moderate to high ion concentrations of c_IL >1M. In fact, the simulated ro-translational coupling ε_DJ in this work turns out to have positive values due to strong interactions between the ions themselves and with water and consequently contradicts these models for dielectric decrement of electrolyte solutions.

Moreover, the ε_DJ contribution to the static dielectric constant shows a parabolic behavior as a function of the ion concentration which is similar to the prediction of a simple geometric model computing the volume occupancy Φ_hyd of water in the first solvation shell of the ions. However, not only the coupling between the translational motion of the ions and the rotation of the water molecules show the same behavior as Φ_hyd, but also the coupling between the rotation of the cations and the rotation of water. These findings point to a picture of rotational depolarization with strong dipolar interactions between the ions and the adjacent water molecules. Consequently, prediction models such as the hydration model or the dipolar Poisson-Boltzmann model³⁸ prove more applicable.

In principle, this change from kinetic to rotational depolarization corresponds to a shift of the point of view on aqueous ionic liquids: due to the large volume occupied by the ions and the strong interaction with water and between the ions, these mixtures do not behave like classical electrolyte solutions at high salt concentrations anymore but have to be considered as mixtures of polar solvents.