Quantification of cation–cation, anion–anion and cation–anion correlations in Li salt/glyme mixtures by combining very-low-frequency impedance spectroscopy with diffusion and electrophoretic NMR

Abstract

Directional correlations between the movements of cations and anions exert a strong influence on the charge and mass transport properties of concentrated battery electrolytes. Here, we combine, for the first time, very-low-frequency impedance spectroscopy on symmetrical Li|electrolyte|Li cells with diffusion and electrophoretic NMR in order to quantify cation–cation, anion–anion and cation–anion correlations in Li salt/tetraglyme (G4) mixtures with Li salt to G4 ratios between 1 : 1 and 1 : 2. We find that all correlations are negative, with like-ion anticorrelations (cation–cation and anion–anion) being generally stronger than cation–anion anticorrelations. In addition, we observe that like-ion anticorrelations are stronger for the heavier type of ion and that all anticorrelations become weaker with decreasing Li salt to G4 ratio. These findings are in contrast to theories considering exclusively anion–cation correlations in form of ion pairs, as the latter imply positive cation–anion correlations. We analyze in detail the influence of anticorrelations on Li⁺ transference numbers and on the Haven ratio. In order to rationalize our results, we derive linear response theory expressions for all ion correlations. These expressions show that the Li⁺ ion transport under anion-blocking conditions in a battery is governed by equilibrium center-of-mass fluctuations in the electrolytes. This suggests that in future electrolyte theories and computer simulations, more attention should be paid to equilibrium center-of-mass fluctuations.

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Introduction

The liquid electrolytes used in state-of-the art lithium-ion batteries are based on volatile organic carbonates. Due to their volatility, the electrolytes are flammable, leading to serious safety concerns. Consequently, there is strong interest in using alternative classes of highly concentrated electrolytes with low vapor pressure, such as solvent-in-salt electrolytes and electrolytes based on ionic liquids (ILs).^1–10 ILs can be separated into four groups, namely aprotic, protic, inorganic and solvate (or chelate) ILs.¹¹ For battery applications, most studies have focused on aprotic and solvate ILs. In the case of aprotic ILs, lithium salts have to be added in order to generate Li⁺ ion conductivity. However, the addition of lithium salts often leads to high viscosities and to low Li⁺ transport numbers.^12,13 Furthermore, lithium transference numbers are often low and have even been reported to exhibit a negative sign.^14–16 This was found in electrophoretic NMR (eNMR) studies and attributed to a vehicular transport mechanism of lithium in negatively charged lithium–anion complexes.

On the other hand, lower viscosities and higher Li⁺ transport numbers have been achieved for solvate ionic liquids (SILs), which consist of equimolar mixtures of a lithium salt with glymes. The glyme molecules form strong complexes with the Li⁺ ions.^17,18 Examples are equimolar mixtures of triglyme (G3) or tetraglyme (G4) with lithium bis(trifluoromethanesulfonyl)imide (LiTFSI)^19,20 or lithium bis(fluoro-sulfonyl)imide (LiFSI).^13,21 These mixtures exhibit ionic conductivities in the range of 10⁻³ S cm⁻¹ and Li⁺ transport numbers, derived from the diffusion coefficients of the ions, around 0.5.^13,22 Recent eNMR measurements showed virtually identical mobilities of Li⁺ ions and G4 molecules, pointing to the migration of complex [Li(G4)]⁺ cations instead of single Li⁺ ions.²³ SILs are thus prototypic highly concentrated electrolytes with properties similar to conventional ionic liquids.¹⁷ Thus they also exhibit strong interionic interactions, which lead to strongly correlated movements of the ions.^22–25 This implies directional correlations between the movements of distinct cations, between the movements of distinct anions, and also between the movements of cations and anions. Correlations can be positive, if distinct ions move preferentially into the same direction, or negative, if distinct ions move preferentially into opposite directions.²⁶ These ion correlations exert a strong influence on the transport properties of the electrolytes in batteries. One important implication of ion correlations is that diffusion-based Li⁺ transport numbers measured by pulsed-field gradient NMR (PFG-NMR)¹³ are distinct from mobility-based Li⁺ transference numbers measured by electrophoretic NMR (eNMR)^14,15 and also distinct from Li⁺ transference numbers under anion-blocking conditions measured by potentiostatic polarization or very-low-frequency impedance spectroscopy (VLF-IS).^24,27 By independently employing either eNMR or VLF-IS, we have recently demonstrated the role of ion correlations. For example, we have shown that the application of SILs in lithium-ion batteries suffers from very low Li⁺ transference numbers under anion-blocking conditions caused by anticorrelated movements of cations and anions.^22,24 The large role of cation–anion anticorrelations is also shown by the decrease of the Haven ratio with increasing Li salt to glyme ratio. Interestingly, when comparing Li salt/G4 mixtures containing either lithium bis(trifluoromethanesulfonyl)imide (LiTFSI) or lithium tetrafluoroborate (LiBF₄), anticorrelations between the complex cation and the anion are most dominant at high salt concentrations. This fact clearly points at a dominant role of momentum conservation.²³ MD simulations predicted that the addition of glyme to SILs weakens the cation–anion anticorrelations leading to higher Li⁺ transference numbers under anion-blocking conditions.²⁴ So far, while techniques such as VLF-IS and eNMR have been helpful to qualitatively elucidate the role of ion correlations, either of them separately does not deliver sufficient information for a full description of ion correlations, for example in the framework of Onsager coefficients.

In this framework, the ionic conductivity of a binary electrolyte containing a monovalent salt can be written as:

σ_ion = σ^self₊ + σ^distinct₊₊ + σ^self₋ + σ^distinct₋₋ − 2σ₊₋ (1)

The self-parts σ^self₊ and σ^self₋ are related to the self-diffusion coefficients of cations, , and anions, , respectively:Here, c_salt denotes the salt concentration, while F, R and T are the Faraday constant, the ideal gas constant and the temperature, respectively. The transport coefficients σ^distinct₊₊, σ^distinct₋₋ and σ₊₋ account for cation–cation, anion–anion, and cation–anion correlations.²⁶ For a complete description of the ion transport properties of a binary electrolyte, the five transport coefficients on the right-hand side of eqn (1) and the thermodynamic factor , with a_± denoting the cations’ and anions’ mean activity, are needed. Consequently, six measurable quantities are required in order to derive the five coefficients and the thermodynamic factor. In this way, information about cation–cation, anion–anion and cation–anion correlations and their impact on the transport properties can be obtained. To our knowledge, there is only a single example in the literature with such a complete experimental information.²⁵ On the other hand, a small number of recent MD studies on ion correlations in concentrated electrolytes has been published.^28–30 Harris showed that under the assumption of strict ion momentum conservation in a solvent-free electrolyte with only a single type of cations and anions, measurements of the ionic conductivity and of the self-diffusion coefficients of the ions are sufficient for elucidating all ion correlations.^31,32 However, this approach cannot be applied to electrolytes with more than two ionic species or to solvent-containing electrolytes, such as solvent-in-salt electrolytes and solvate ionic liquids.

In this paper, we combine, for the first time, very-low-frequency impedance spectroscopy on symmetrical Li|electrolyte|Li cells with pulsed-field gradient NMR and electrophoretic NMR, and we show that this is a powerful combination for the quantification of the ion transport coefficients in eqn (1), fully based on experimental data. We use the ion correlations in SILs and in diluted Li salt/glyme mixtures as a showcase to demonstrate the potential of this methodological combination. Complete experimental information is obtained for LiFSI/glyme mixtures with molar ratios of 1 : 1, 1 : 1.5, and 1 : 2 in VLF-IS, PFG-NMR and eNMR, while using our previous VLF-IS results for the LiFSI/G4 1 : 1 mixture.²² In addition, for comparison, we use previously published eNMR and VLF-IS data of a LiTFSI/G4 1 : 1 mixture.^23,27 In this way, we analyse both the influence of salt concentration and anion type on the ion correlations. We discuss the impact of different correlations (cation–cation, anion–anion, and cation–anion) on the Li⁺ transference number under anion-blocking conditions, on the mobility-based Li⁺ transference number, and on the Haven ratio. In order to rationalize the experimental findings, we derive and discuss linear response theory expressions for a binary electrolyte with dipole fluctuations and center-of-mass fluctuations.

Experimental section

Electrolyte preparation

The chemical structures of the different electrolytes studied in this work are shown in Fig. 1. They were prepared in an argon inert gas atmosphere with water and oxygen content below 1 ppm. The electrolyte compositions listed in Table 1 were prepared by solving lithium bis(fluorosulfonyl)imide (LiFSI) (Fluorochem. Ltd, battery grade) and lithium bis(trifluoromethanesulfonyl)imide (LiTFSI) (Sigma-Aldrich, 99.95% trace metals basis), respectively, in tetraglyme (G4) (Sigma-Aldrich, >99%) while stirring overnight, until a homogeneous mixture was obtained. Prior to use, the G4 was flushed with dry argon, degassed carefully in vacuum and dried over molecular sieve (3 Å) for at least 24 h.

Fig. 1 Chemical structures of the different electrolyte components studied in this work.

Table 1 Composition of the different electrolytes studied in this work

LiFSI/G4	Ratio	x G4	x salt	c salt/mol l^−1
1 : 1	1	0.50	0.50	3.23
1 : 1.5	0.667	0.60	0.40	2.38
1 : 2	0.5	0.67	0.33	1.72

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LiTFSI/G4	Ratio	x G4	x salt	c salt/mol l^−1
1 : 1	1	0.50	0.50	2.75

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Total ionic conductivity

Ionic conductivity measurements were carried out using a closed TSC70 liquid sample cell with a Microcell HC cell stand (rhd instruments). The cell consists of a PEEK crucible, which was filled with 70 μl of the electrolyte, and of two polished platinum electrodes. The impedance measurements were conducted in a frequency range from 100 kHz to 1 Hz with an ac voltage of 10 mV_rms by means of an Alpha-A impedance analyzer (Novocontrol Technologies) equipped with a ZG2 interface. The temperature was varied from 10 °C to 80 °C by means of an Eurotherm 2416 temperature controller (accuracy ±0.1 K). The cell constant of the liquid sample cell was determined using a 0.1 mol l⁻¹ KCl standard solution (HANNA instruments). The software RelaxIS (rhd instruments) was used for fitting the impedance spectra with an equivalent circuit consisting of a resistance in series with a constant phase element (CPE). The resistance describes the bulk ion transport in the electrolyte, while the CPE accounts for electrode polarization.

Very-low-frequency impedance spectroscopy

In order to study Li⁺ ion transport under anion-blocking conditions, very-low-frequency impedance spectroscopic (VLF-IS) measurements on symmetrical Li|electrolyte|Li cells²⁷ were carried out. The distance between the two parallel lithium electrodes with 12 mm diameter was varied between 100 μm and 350 μm. The cells were assembled in a glovebox under inert gas atmosphere and were subsequently closed airtight. Then the cells were connected to an Alpha-A impedance analyzer (Novocontrol Technologies) equipped with a ZG2 interface and a Quatro Cryosystem.

Before carrying out the very-low-frequency impedance measurements, the stability of the Li|electrolyte interface was monitored by continuous impedance measurements at 30 °C in a frequency range from 1 MHz to 0.1 Hz over several hours. After achieving a stable interfacial impedance, the very-low-frequency impedance measurements were carried out at 30 °C in a frequency range from 1 MHz to 100 μHz using an ac voltage of 2 mV_rms. The measurements were done at different distances between the Li electrodes. The software RelaxIS (rhd instruments) was used for the fitting of the impedance data.

Diffusion NMR and electrophoretic NMR

All NMR investigations were performed on a 400 MHz AVANCE III HD NMR spectrometer (Bruker) equipped a gradient probe head (Bruker, Diff50), providing a maximum gradient strength of 30 T m⁻¹. Selective radiofrequency inserts for ¹H, ⁷Li and ¹⁹F were used with 5 mm sample tubes. A power source designated for eNMR experiments was employed to provide electric field pulses (P&L Scientific, Sweden). An in-house designed eNMR probe³³ was equipped with palladium wire electrodes at a distance of 22 mm. For each measurement 19 mm long gas chromatography capillaries (fused silica capillary tubing, Polymicro Technologies, ID = 250 μm, OD = 350 μm), dried at 100 °C under high vacuum overnight, were placed between the electrodes before filling the probe with the liquid sample under argon atmosphere. They serve to prevent convection during the experiment.

Diffusion experiments were conducted at 30 °C using a stimulated echo pulse sequence with magnetic field gradient pulses.³⁴ The observation time Δ was 100 ms, and the applied gradient duration δ was set depending on the observed nucleus (1 ms for ¹H and ¹⁹F, and 3 ms for ⁷Li). The gradient pulse strength was increased, until the observed NMR signal was attenuated by at least two orders of magnitude. The self-diffusion coefficients of G4, Li⁺, and FSI⁻ were determined from ¹H, ⁷Li, and ¹⁹F PFG-NMR measurements via the Stejskal–Tanner equation,^34,35 with the gyromagnetic ratio γ, the applied magnetic field gradient pulse strength G, the magnetic field gradient pulse duration δ, the observation time Δ, and the self-diffusion coefficient , see eqn (4).

The eNMR experiments were conducted at 30 °C using a double stimulated echo pulse sequence in combination with two electric field pulses of opposite polarity and four magnetic field gradient pulses.³⁶ The voltage was incremented from 0 to 100 V, and the magnetic field gradient strength set to a constant value depending on the investigated sample and nucleus. The observation time Δ was 100 ms. The gradient duration δ was set depending on the observed nucleus (1 ms for ¹H and ¹⁹F, 3 ms for ⁷Li). eNMR spectra were recorded at distinct voltages and the phase shifts analyzed via spectral deconvolution using single or multiple phase-sensitive Lorentz profiles, as described earlier.¹⁴ The electrophoretic mobility was extracted from a linear fit of the resulting phase angles against the voltage. In the following, mobility values are given as the average of the slopes of at least 3 independent experiments per sample and nucleus.

Results and discussion

Total ionic conductivities, Li⁺ transference numbers under anion-blocking conditions and salt diffusion coefficients

In Fig. 2a, we show an Arrhenius plot of the total ionic conductivity σ_ion of all samples. The ionic conductivity is lowest for the LiTFSI/G4 1 : 1 mixture and slightly higher for the LiFSI/G4 1 : 1 mixture. With increasing G4 content of the LiFSI/G4 mixtures, the total ionic conductivity increases.

Fig. 2 (a) Arrhenius plots for the different Li salt/G4 mixtures. (b) Very-low-frequency impedance spectrum and fit for the LiFSI/G4 1 : 1.5 mixture at 30 °C and at an electrode distance d of 0.192 mm. The inset shows the equivalent circuit used for fitting. (c) Diffusion resistance R_diffvs. electrode distance d and (d) time constant τ vs. squared electrode distance d² for the LiFSI/G4 1 : 1.5 mixture shown in (b).

In Fig. 2b, very-low-frequency (VLF) impedance spectra of a LiFSI/G4 1 : 1.5 mixture between two Li electrodes with a distance of 192 μm are shown. The equivalent circuit for fitting is shown in the inset. The circuit consists of a bulk resistance R_bulk caused by the total ionic conductivity of the electrolyte, in series with two R-CPE elements accounting for the Li|electrolyte interface (impedance of solid electrolyte interphase and charge transfer impedance), in series with a Warburg-short (Ws) element describing the formation of salt diffusion layers and the stationary Li⁺ current under anion-blocking conditions. The Warburg-short impedance is given by:

Here, R_diff denotes a diffusion resistance, ω the angular frequency, j the imaginary unit, τ a characteristic time for establishing stationary salt diffusion profiles across the electrolyte, and α is a characteristic exponent. τ can be written as:with D_salt and d denoting the salt diffusion coefficient and the distance between the Li electrodes. In Fig. 2c and d, we plot R_diffversus d and τ versus d², respectively. The plots verify the expected linear dependence of R_diff on d and the expected quadratic dependence of τ on d, respectively. From the slope of the data in Fig. 2d, the salt diffusion coefficient D_salt is obtained.

The Li⁺ transference number under anion-blocking conditions is given by:²²

In Table 2, we summarize the results obtained for the total ionic conductivity σ_ion, the Li⁺ transference number under anion-blocking conditions t^abc₊ and the salt diffusion coefficient D_salt of all mixtures. The data for LiTFSI/G4 1 : 1 were taken from ref. 27 and the data for LiFSI/G4 1 : 1 from ref. 22, respectively.

Table 2 Ionic conductivities σ_ion, lithium ion transference numbers under anion-blocking conditions t^abc₊ and salt diffusion coefficients D_salt obtained at 30 °C for the different Li salt/G4 mixtures

LiFSI/G4	σ ion/mS cm^−1	t ^abc+	D salt/cm^2 s^−1
1 : 1^22	1.58	0.015 ± 0.003	(5.3 ± 0.5) × 10^−8
1 : 1.5	1.95	0.082 ± 0.015	(1.8 ± 0.5) × 10^−7
1 : 2	2.45	0.091 ± 0.015	(4.4 ± 3.1) × 10^−7

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LiTFSI/G4	σ ion/mS cm^−1	t ^abc+	D salt/cm^2 s^−1
1 : 1^27	1.31	0.025 ± 0.005	(7.0 ± 0.9) × 10^−8

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Li⁺ transport numbers and mobility-based Li⁺ transference numbers

The self-diffusion coefficients of G4, Li⁺, and FSI⁻ were determined from ¹H, ⁷Li, and ¹⁹F PFG-NMR measurements. The signal attenuation I/I₀ for each NMR nucleus of LiFSI/G4 1 : 1.5 as an exemplary sample is shown in Fig. 3a in dependence of b. In this depiction, the slope is a direct representation of the self-diffusion coefficient , which is evaluated according to eqn (4). As expected, we observe a monoexponential decay of the investigated NMR signals with increasing b values. This behavior indicates the fast exchange averaging of the diffusion coefficient in case of the existence of different species, such as single ions or complexes containing the observed nucleus during the observation time of 100 ms. In the case of LiFSI/G4 1 : 1 mixture, ⁷Li and ¹H exhibit the same slope (see Fig. S1 in ESI†), and therefore the same diffusion coefficient. In the past, this behavior was attributed to the formation of a complex [Li(G4)]⁺ solvate cation, and is the main characteristic of solvate ionic liquids.^13,17,37 With decreasing lithium to glyme ratio, G4 diffuses faster than Li⁺, as indicated by a steeper slope for ¹H, see Fig. 3a.

Fig. 3 (a) Signal attenuation I/I₀ observed in ¹H, ⁷Li and ¹⁹F PFG-NMR experiments and (b) voltage U dependent reduced phase shift values (ϕ − ϕ₀)dγ⁻¹G⁻¹δ⁻¹Δ⁻¹ observed via¹H, ⁷Li and ¹⁹F eNMR for the LiFSI/G4 1 : 1.5 mixture.

The electrophoretic mobilities μ_i were determined from the voltage-dependent phase shift ϕ − ϕ₀ observed via¹H, ⁷Li and ¹⁹F eNMR, as shown in the example in Fig. 3b. According to eqn (8), the signal phase shift is linearly depending on the applied voltage U divided by the electrode distance d, the gyromagnetic ratio γ, the applied magnetic field gradient pulse strength G, its duration δ, the observation time Δ and the mobility μ_LF.^38,39

In order to compare eNMR measurements conducted on distinct nuclei using different experimental parameters, reduced phase shift values (ϕ − ϕ₀)dγ⁻¹G⁻¹δ⁻¹Δ⁻¹ are depicted in Fig. 3b for a measurement series for an exemplary LiFSI/G4 1 : 1.5 mixture. In this depiction, the slope represents the mobility μ_LF of the observed nucleus in the laboratory reference frame (LF). As expected, the depicted eNMR results exhibit a phase shift linearly depending on the applied voltage. While the scattering of the phase shift is rather low for ¹H and ¹⁹F, it is somewhat larger for ⁷Li, due to the lower signal-to-noise ratio of ⁷Li spectra, resulting from a lower value of γ compared to ¹H and ¹⁹F.

The opposite sign of the slopes of the ⁷Li and ¹⁹F phase shifts signifies their migration in opposite directions as a consequence of the opposite sign of charge of Li⁺ and FSI⁻. Despite G4 lacking an own nominal charge, the observed ¹H phase shift values exhibit the same slope, and therefore mobility, as ⁷Li for the LiFSI/G4 1 : 1 mixture (see Fig. S2 in ESI†). This observation shows the migration of a complex cation [Li(G4)]⁺ in an electric field in the LiFSI/G4 1 : 1 mixture, similar to the previously studied SIL systems Li(G4)TFSI and Li(G4)BF₄,^14,23 underlining the ionic liquid-like character of Li(G4)FSI. All resulting mobility values obtained from ¹H, ⁷Li and ¹⁹F eNMR measurements on the LiFSI/G4 samples are given in Fig. 4 as open symbols. The overall trend is a reduction of the modulus of mobility with increasing salt concentration related to a viscosity increase. With decreasing lithium salt to glyme ratio, the fast exchange averaging over free glyme molecules exhibiting no nominal charge and glyme molecules bound to lithium leads to an average mobility of G4, which is lower than that of lithium, compare open squares and diamonds in Fig. 4. This hints at the coexistence of free glyme molecules and Li/glyme complexes in the dilute samples, which will be discussed in more detail below.

Fig. 4 Electrophoretic mobilities μ obtained from ¹H, ⁷Li, and ¹⁹F eNMR measurements in the laboratory (LF, open symbols) and center-of-mass (CM, filled symbols) reference frames. Salt concentrations correspond to LiFSI/G4 molar ratios of 1 : 2, 1 : 1.5, and 1 : 1 (from left to right).

Mobilities μ_LF extracted from eNMR are based on drift velocities determined from the particles’ displacements in space in the laboratory reference frame (LF). During the two electric field pulses with opposite sign, the pulse sequence ensures constructive addition of the phase shifts. On the other hand, in the ideal case, the ion displacements in space during either pulse should exactly compensate, which implies a net zero drift of the center-of-mass of the whole electrolyte. In practice, the ideal compensation is often not given, possible sources of experimental imperfections could be Joule heating, gravitational effects causing differences between upper and lower electrode, or irreversible electrochemical decomposition processes at the electrodes. As a consequence, in eNMR experiments, the net drift of the center-of-mass may deviate from zero. Several methodological papers have dealt with describing and partially improving the effect of experimental imperfections.^40–42 While several improvements have been achieved concerning the effects of electroosmotic flow and convection, at the current state there is still a net drift of the center-of-mass observed. Consequently, in order to minimize the influence of the experimental imperfections on our present results, we chose to impose a first order correction by transforming the mobilities to the center-of-mass reference frame (CM) according to eqn (9a) and (9b). The net drift of the center-of-mass is calculated by summing over all molecular species i in the electrolyte with c_i as the molar concentration, the molar mass M_i, the respective mobility μ^LF_i obtained from eNMR experiments, and the electrolyte density ρ.

μ^CM_i = μ^LF_i − μ_{mass center} (9b)

The mobilities in the center-of-mass reference frame μ^CM_i are depicted as full symbols in Fig. 4. They differ slightly from the lab frame values, visualizing the effect of experimental imperfections, while the qualitative behavior of the different species and their concentration dependence remains the same. We note that this transformation also relates to an earlier discussion of reference frames in eNMR experiments on concentrated electrolytes, where the CM frame was suggested as an internal reference frame with the benefit of allowing comparison to simulations or other theoretical approaches which presume momentum conservation.⁴³ Similar to the diffusion data, upon enhancing salt concentration, the mobilities also reflect the viscosity influence, and in addition the merging of Li and glyme mobilities due to complex cation formation.

The mobilities μ^CM_i and mobility-based lithium ion transference numbers t^μ₊ in the center-of-mass reference frame (see eqn (10)) as well as self-diffusion coefficients and diffusion-based lithium ion transport numbers t^PFG₊ (see eqn (11)) for the different electrolyte compositions compared in this work are summarized in Tables 3 and 4.

In the 1 : 1 samples, the respective cation and glyme mobilities agree within error (see Table 3), consistent with the formation of a complex cation [Li(G4)]⁺ as the only cationic species. In the 1 : 2 sample, however, cation and glyme mobilities do not agree within error, reflecting an averaging over different glyme-containing species, possibly free glyme and complexes [Li(G4)_n]⁺, with different integer values of the coordination number n. Assuming that the species in this sample would be [Li(G4)₁]⁺ complexes and free G4 with zero mobility, the resulting average glyme mobility would be μ_G4 = 0.5μ₊. If on the other hand exclusively [Li(G4)₂]⁺ complexes existed, the G4 mobility would be μ_G4 = μ₊. Experimentally, we find μ_G4 = 0.73μ₊. This points to an average coordination number n̄ between 1 and 2. We will discuss the complex compositions in more detail at the end of the discussion and provide an estimate of n̄ for the dilute samples.

Table 3 Center-of-mass frame based mobilities μ^CM_i and mobility-based Li⁺ transference numbers t^μ₊ obtained at 30 °C for the different electrolyte mixtures

LiFSI/G4	μ ^CM+/cm^2 (V s)^−1	μ ^CM−/cm^2 (V s)^−1	μ ^CMG4/cm^2 (V s)^−1	t ^μ+
1 : 1	(2.7 ± 0.4) × 10^−6	(−3.6 ± 0.5) × 10^−6	(2.8 ± 0.2) × 10^−6	0.43 ± 0.05
1 : 1.5	(3.4 ± 0.6) × 10^−6	(−5.8 ± 0.6) × 10^−6	(3.1 ± 0.4) × 10^−6	0.37 ± 0.05
1 : 2	(5.1 ± 0.5) × 10^−6	(−9.3 ± 0.3) × 10^−6	(3.7 ± 0.6) × 10^−6	0.36 ± 0.02

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LiTFSI/G4	μ ^CM+/cm^2 (V s)^−1	μ ^CM−/cm^2 (V s)^−1	μ ^CMG4/cm^2 (V s)^−1	t ^μ+
1 : 1^23	(3.3 ± 0.7) × 10^−6	(−2.4 ± 1.0) × 10^−6	(3.0 ± 0.7) × 10^−6	0.58 ± 0.11

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Table 4 Self-diffusion coefficients and diffusion-based Li⁺ transport numbers t^PFG₊ obtained at 30 °C for the different electrolytes

LiFSI/G4	/cm^2 s^−1	/cm^2 s^−1	t ^PFG+
1 : 1	(1.2 ± 0.1) × 10^−7	(1.4 ± 0.1) × 10^−7	0.46 ± 0.02
1 : 1.5	(1.7 ± 0.1) × 10^−7	(2.2 ± 0.1) × 10^−7	0.44 ± 0.02
1 : 2	(2.7 ± 0.1) × 10^−7	(3.6 ± 0.2) × 10^−7	0.43 ± 0.02

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LiTFSI/G4	/cm^2 s^−1	/cm^2 s^−1	t ^PFG+
1 : 1^23	(1.3 ± 0.1) × 10^−7	(1.2 ± 0.1) × 10^−7	0.51 ± 0.02

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Calculation of transport coefficients from the experimental data

In a first step, we split the ionic conductivity into three Onsager coefficients:^26,27

σ_ion = σ₊₊ + σ₋₋ − 2σ₊₋ (12)

In the framework of the Onsager formalism, the Li⁺ transference number under anion-blocking conditions and the mobility-based Li⁺ transference number are given by:^26,27Solving eqn (12)–(14) for the Onsager coefficients results in:Furthermore, the salt diffusion coefficient can be written as:²⁶Thus, the thermodynamic factor is given by:Finally, we split the Onsager coefficients σ₊₊ and σ₋₋ into their self- and distinct-parts:

σ₊₊ = σ^distinct₊₊ + σ^self₊ (20)

σ₋₋ = σ^distinct₋₋ + σ^self₋ (21)

with the self-parts obtained from eqn (2) and (3).

The obtained results are summarized in Tables 5 and 6 and are illustrated in Fig. 5. We start by comparing the diffusion-based Li⁺ transport numbers and the mobility-based Li⁺ transference numbers (Fig. 5a). In the case of LiTFSI/G4 1 : 1, we find t^μ₊ > t^PFG₊, while for LiFSI/G4 1 : 1, we observe t^μ₊ ≲ t^PFG₊. In Fig. 5b, the strength of the cation–cation and anion–anion correlations is illustrated by plotting the distinct-parts of σ₊₊ and σ₋₋ normalized by their self-part. In the case of LiTFSI/G4 1 : 1, the cation dynamics is characterized by cation–cation anticorrelations with a ratio of σ^distinct₊₊/σ^self₊ = −0.66, see open black square. This number implies that cation–cation anticorrelations reduce the contribution of diffusive cationic displacements to the ionic conductivity by 66%. The anion dynamics shows even stronger anticorrelations with a ratio of σ^distinct₋₋/σ^self₋ = −0.80, see open red circle. The stronger anion–anion anticorrelations reduce the anionic contribution to the conductivity and lead to t^μ₊ > t^PFG₊. In contrast, in the case of LiFSI/G4 1 : 1, the cation–cation anticorrelations are slightly stronger (σ^distinct₊₊/σ^self₊ = −0.78) than the anion–anion anticorrelations (σ^distinct₋₋/σ^self₋ = −0.68). This leads to t^μ₊ ≲ t^PFG₊. It is thus evident that like-ion anticorrelations are responsible for the deviations of transference numbers from transport numbers. In this context, it should be noted that the molar mass of FSI anions (M = 178 g mol⁻¹) is lower than the molar mass of [Li(G4)]⁺ complex cations (M = 229 g mol⁻¹), while the molar mass of TFSI anions (M = 280 g mol⁻¹) is higher. Consequently, our results suggest that the respective heavier ion type shows stronger anticorrelations.

Table 5 Onsager conductivity coefficients σ₊₊, σ₋₋ and σ₊₋ obtained at 30 °C for the different electrolyte mixtures

LiFSI/G4	σ ++/mS cm^−1	σ −−/mS cm^−1	σ +−/mS cm^−1
1 : 1	0.30 ± 0.07	0.52 ± 0.16	−0.38 ± 0.01
1 : 1.5	0.34 ± 0.06	0.84 ± 0.41	−0.39 ± 0.04
1 : 2	0.40 ± 0.04	1.14 ± 0.32	−0.45 ± 0.03

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LiTFSI/G4	σ ++/mS cm^−1	σ −−/mS cm^−1	σ +−/mS cm^−1
1 : 1	0.44 ± 0.16	0.24 ± 0.07	−0.31 ± 0.02

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Table 6 Self and distinct parts of the Onsager conductivity coefficients σ^self₊, σ^self₋, σ^distinct₊₊ and σ^distinct₋₋, obtained at 30 °C for the different electrolyte mixtures

LiFSI/G4	σ ^self+/mS cm^−1	σ ^self−/mS cm^−1	σ ^distinct++/mS cm^−1	σ ^distinct−−/mS cm^−1
1 : 1	1.37 ± 0.07	1.65 ± 0.08	−1.07 ± 0.10	−1.13 ± 0.18
1 : 1.5	1.52 ± 0.08	1.95 ± 0.10	−1.19 ± 0.10	−1.11 ± 0.42
1 : 2	1.74 ± 0.09	2.29 ± 0.11	−1.33 ± 0.10	−1.15 ± 0.34

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LiTFSI/G4	σ ^self+/mS cm^−1	σ ^self−/mS cm^−1	σ ^distinct++/mS cm^−1	σ ^distinct−−/mS cm^−1
1 : 1	1.28 ± 0.06	1.23 ± 0.06	−0.84 ± 0.18	−0.99 ± 0.09

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Fig. 5 (a) Lithium ion transference numbers t^abc₊ and t^μ₊ as well as transport numbers t^PFG₊ and (b) ion correlations manifesting in the ratios σ^distinct₊₊/σ^self₊, σ^distinct₋₋/σ^self₋ and σ₊₋/σ_ion for the different electrolyte mixtures, see text for further explanation.

The same is true for the more dilute (1 : 1.5 and 1 : 2) LiFSI/G4 mixtures, where t^μ₊ < t^PFG₊ is valid for both concentrations. Thus, in all FSI-containing mixtures, due to cation–cation anticorrelations being stronger than anion–anion anticorrelations, the cationic contribution to the conductivity is lower than expected from the diffusion coefficients. While the cation–cation anticorrelation is similar for all investigated LiFSI/G4 mixtures, both anion–anion and cation–anion anticorrelations decrease with decreasing LiFSI/G4 ratio.

We note that we have also calculated the thermodynamic factor from eqn (19). The obtained values are given in Fig. S3 (ESI†).

The Li⁺ transference numbers under anion-blocking conditions t^abc₊ are much lower than t^μ₊ and t^PFG₊, see black squares in Fig. 5a. For the 1 : 1 mixtures, the very small numbers obtained previously, i.e. t^abc₊ = 0.025 for LiTFSI/G4²⁷ and t^abc₊ = 0.015 for LiFSI/G4,²² are caused by cation–anion anticorrelations due to ion momentum conservation. Since the exchange of the G4 molecules between [Li(G4)]⁺ complexes is slower than the elementary displacement steps of the ions, there is virtually no momentum exchange between G4 molecules and ions, so that ion momentum conservation plays an important role.^24,27 Ion momentum conservation implies that under anion-blocking conditions in a battery, the cation momentum becomes very small.

With decreasing ratio of LiFSI/G4, the strength of the cation–anion anticorrelations, manifesting in the ratio σ₊₋/σ_ion, decreases, namely from −0.24 for the 1 : 1 mixture to −0.19 for the 1 : 2 mixture, see Fig. 5b. According to eqn (13), this leads to a considerably larger Li⁺ transference number under anion-blocking conditions for the 1 : 2 mixture (t^abc₊ = 0.091).

In Fig. 6a, we illustrate the influence of cation–cation, anion–anion, and cation–anion correlations on the two transference numbers t^μ₊ and t^abc₊ of the two 1 : 1 mixtures. These illustrations are based on rewriting eqn (13) and (14) as:

Fig. 6 Influence of the cation–cation (C–C), anion–anion (A–A) and cation–anion (C–A) correlations on (a) the two transference numbers t^abc₊ and t^μ₊ (eqn (22) and (23)) and (b) the inverse Haven ratio H_R⁻¹ (eqn (25)) for the two Li salt/G4 1 : 1 mixtures.

In the absence of ion correlations (σ^distinct₊₊ = 0, σ^distinct₋₋ = 0, σ₊₋ = 0), both transference numbers are identical to the transport number t^PFG₊. In the illustration in Fig. 6a, we start from t^PFG₊ and then consecutively insert the non-zero values for σ^distinct₊₊, σ^distinct₋₋ and σ₊₋ into eqn (22) and (23). The plot illustrates the implications of the different strength of the anticorrelations in the two electrolytes for t^μ₊ and t^abc₊. While the cation–cation anticorrelations reduce t^μ₊, the anion–anion anticorrelations have the opposite effect. The influence of the cation–anion anticorrelations on t^μ₊ depends on the mass ratio of the [Li(G4)]⁺ cation to the anion. The cation–anion anticorrelations enhance the conductivity contribution of the heavier ion more strongly than the contribution of the lighter ion. Consequently, t^μ₊ is enhanced in the FSI system, while it is reduced in the TFSI system.

The most important ion correlation for t^abc₊ is the cation–anion anticorrelation. This anticorrelation decreases t^abc₊ severely, since the strong cation–anion correlations imply that under anion-blocking conditions, the cation current is strongly hindered. Thus, the large difference between t^abc₊ and t^μ₊ for the 1 : 1 mixtures is clearly a consequence of the cation–anion anticorrelation.

Next we discuss the influence of the ion correlations on the Haven ratio H_R, which is defined as:

The calculated values for H_R of the different electrolyte mixtures are summarized in Table 7.

Table 7 Haven ratios H_R for the different electrolyte mixtures

LiFSI/G4	H R
1 : 1	1.9
1 : 1.5	1.8
1 : 2	1.6

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LiTFSI/G4	H R
1 : 1	1.9

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The inverse Haven ratio H_R⁻¹

is sometimes termed as “ionicity”. In the absence of ion correlations, the inverse Haven ratio would be unity. As illustrated in Fig. 6b, the cation–cation and anion–anion anticorrelations lower the inverse Haven ratio, while the cation–anion anticorrelations increase the inverse Haven ratio. Overall, the influence of the like-ion anticorrelations (cation–cation and anion–anion) dominates, so that the inverse Haven ratio is <1 and the Haven ratio is >1. This shows clearly that the term “ionicity” for the inverse Haven ratio does not properly describe the physical origin of H_R > 1. The Haven ratio might partly be enhanced by the formation of ion pairs, however, this contribution would manifest in positive cation–anion correlations (movements of the cations and anions bound in pairs into the same direction). Since the latter are negative, our analysis clearly demonstrates the dominating relevance of anticorrelation contributions. These occur between all types of ions and depend strongly on the relative mass of cationic and anionic species. Thus, the molecular picture of ion correlations is far less than anticipated influenced by Coulombic and dispersive interactions and potential pair formation, but rather by momentum exchange between the ions.

Relation to center-of-mass and dipole fluctuations by linear response theory

In order to rationalize our results for the cation–cation, anion–anion, and cation–anion correlations, we start with a general linear response theory expression for the ionic conductivity of an isotropic liquid electrolyte:²⁶Here, N is the total number of ions in the electrolyte, z_i is the charge number of ion i, and V is the sample volume. k_B and T denote Boltzmann's constant and the temperature, respectively. The brackets 〈…〉 stand for the ensemble average. The ion displacement vectors ΔR⃑_i(t) refer to the self-diffusive displacements in thermal equilibrium in the absence of an electric field. ΔR⃑_i(t) consists of three components in x, y, and z direction:For isotropic materials, we can write:and we can express the ionic conductivity via the ionic displacements in x direction:We assume that the ion dynamics is diffusive, i.e.. In this case, we can rewrite eqn (29) as:with x_i(Δt) denoting the ionic displacements during a time interval Δt.

Now we consider a 1 : 1 electrolyte with ionic charge numbers z₊ = 1 and z₋ = −1 and with identical numbers of cations and anions, N₊ = N₋ = N/2. We note that the ions might be single ions or complex ions, such as [Li(G4)]⁺. From the displacements of the cations, x⁺_i, and of the anions, x⁻_i, during a time interval Δt, the Onsager coefficients σ₊₊, σ₋₋, and σ₊₋ can be calculated:

The Onsager coefficients σ₊₊ and σ₋₋ can be split into self- and distinct-parts:The ionic displacements during the time interval Δt lead to a center-of-mass fluctuation and to a dipole fluctuation, which can be expressed as:Here, denotes the mass ratio of the ions. A and B characterize center-of-mass and dipole fluctuations in the electrolyte, respectively, and exhibit ensemble averages, 〈A〉 = 0 and 〈B〉 = 0, and variances 〈A²〉 and 〈B²〉. These variances are limited by intermolecular forces, which prevent the formation of large local center-of-mass and large local dipole fluctuations. x₀ is a characteristic length scale, which is explained below, and the factor accounts for the dependence of fluctuations on the number of particles in a system.

Now we express the ionic displacement as follows:

Δx⁺_i = x₀(g_i + a⁺_i·A + b⁺_i·B) (34a)

Δx⁻_i = x₀(g_i + a⁻_i·A + b⁻_i·B) (34b)

with g_i denoting a displacement distribution function with zero particle mean, , and variance . Eqn (34a) and (34b) imply that the characteristic length scale x₀ is identical to the root mean square displacement of both types of ions in the absence of center-of-mass and dipole fluctuations. The quantities a^±_i and b^±_i characterize the individual contributions of cations and anions to the center-of-mass and to the dipole fluctuation, respectively.

Inserting eqn (34a) and (34b) into (33a) and (33b) results in (see ESI† for details):

Now, we insert eqn (34a), (34b) and (35a)–(35c) into (31a)–(31c), (32a) and (32b) in order to obtain expressions for the different transport coefficients (see ESI† for details):

σ_ion = σ₀·2·〈B²〉 (36h)

with . Here, denotes the number density of each type of ion.

The cationic and anionic contributions to the ionic conductivity can then be expressed as (see ESI† for details):

The cation transport coefficient under anion-blocking conditions, t^abc₊·σ_ion, can be written as:If center-of-mass fluctuations are much weaker than dipole fluctuations, such that and , the cation transport coefficient under anion-blocking conditions can be approximated by (see ESI† for details):

t^abc₊·σ_ion ≈ σ₀·2·(1 + k)²·〈A²〉 (39)

Finally, we calculate the ratio of the distinct-parts of σ₊₊ and σ₋₋ to their self-parts in order to interpret the data in Fig. 5b in terms of fluctuations. If the fluctuation terms in the self-parts and in the distinct-parts are much smaller than unity, i.e., , , and , the ratios can be approximated by (see ESI† for details):

From eqn (36)–(40), we draw the following conclusions: (i) While the self- and distinct-parts σ^self₊, σ^distinct₊₊, σ^self₋, and σ^distinct₋₋ depend on the distribution function of the prefactors a_i and b_i, the Onsager coefficients σ₊₊, σ₋₋, and σ₊₋ are independent of this distribution function (eqn (36a)–(36g)). (ii) In the case of weak center-of-mass and dipole fluctuations, such that (eqn (40a)) and (eqn (40b)), the distinct-parts σ^distinct₊₊ and σ^distinct₋₋ are negative, i.e. cation–cation and anion–anion anticorrelations prevail. The physical reason is that anticorrelated displacements of the same type of ions create neither a large dipole nor a large center-of-mass shift. This is illustrated schematically in Fig. 7a and b. (iii) According to eqn (36g), the Onsager coefficient σ₊₋ is negative, if the center-of-mass fluctuations are weaker than the dipole fluctuations, such that . The reason is that a dipole fluctuation without a center-of-mass fluctuation is only possible, if the center-of-mass of the cations moves into the opposite direction as the center-of-mass of the anions, as illustrated schematically in Fig. 7c. (iv) While the Onsager coefficients σ₊₊, σ₋₋ and σ₊₋ depend on the mass ratio k as well as on the center-of-mass and dipole fluctuations 〈A²〉 and 〈B²〉, the ionic conductivity σ_ion (eqn (36h)) depends exclusively on the dipole fluctuations 〈B²〉. (v) If center-of-mass fluctuations are absent, i.e. 〈A²〉 = 0 in eqn (36b), (36e), and (36g), the Onsager coefficients exhibit the following mass dependence: ; ; and , in agreement with the results obtained by Kashyap et al.⁴⁴ (vi) The mobility-based cation transference number, according to eqn (37a) and (37b), exhibits the following mass dependence: , which does not depend on the center-of-mass fluctuations 〈A²〉. While the relation is evident in solvent-free systems,^32,45 since it follows from ion momentum conservation, our derivation implies that even in case of diluted systems containing free solvent molecules, t^μ₊ does not depend on center-of-mass fluctuations. (vii) Following eqn (39), the center-of-mass fluctuations 〈A²〉 govern the cation transport under anion-blocking conditions. In the absence of center-of-mass fluctuations, t^abc₊·σ_ion = 0. For 〈A²〉 > 0 it follows that t^abc₊·σ_ion > 0.

Fig. 7 (a and b) Illustration of anticorrelated displacements of the same type of ion, yielding negative contributions to σ^distinct₊₊ and σ^distinct₋₋, respectively. These anticorrelated displacements create neither a large dipole nor a large center-of-mass shift. (c) Illustration of anticorrelated cation–anion displacements, resulting in a negative contribution to σ₊₋. These displacements create a large dipole, but not necessarily a center-of-mass shift. Thus, cation–anion anticorrelations prevail in electrolytes, in which dipole fluctuations are much stronger than center-of-mass fluctuations.

Now, we use these equations to derive values for the mass ratio k and for fluctuation parameters 〈A²〉 and 〈B²〉, presented in Table 8. The mass ratio k can be directly determined from the mobility-based transference number t^μ₊, see eqn (36h) and (37a).

Table 8 Center-of-mass and dipole fluctuations 〈A²〉 and 〈B²〉 as well as ion mass ratio k and average coordination number n̄ for the different electrolytes

LiFSI/G4	〈A^2〉	〈B^2〉	k	Average coordination number n̄
1 : 1	0.003 ± 0.0015	0.57 ± 0.25	0.76 ± 0.15	1.0 ± 0.3
1 : 1.5	0.021 ± 0.010	0.65 ± 0.25	0.59 ± 0.11	1.3 ± 0.4
1 : 2	0.027 ± 0.008	0.71 ± 0.22	0.55 ± 0.055	1.4 ± 0.2

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LiTFSI/G4	〈A^2〉	〈B^2〉	k	Average coordination number n̄
1 : 1	0.002 ± 0.001	0.54 ± 0.28	1.4 ± 0.62	0.90 ± 0.25

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We note that with this experimental determination of the mass ratio k, no assumption about the nature or stoichiometry of the ionic species, in particular of the complex cations, needs to be made. For a known value of k, then a combination of either eqn (40a) or eqn (40b) with equation

can be used to determine 〈A²〉 and 〈B²〉. The choice of eqn (40a) or eqn (40b) depends on whether the approximations and or the approximations and are better fulfilled, which is the case for the ion type with the stronger like-ion anticorrelations, namely the cations in the case of FSI-containing mixtures and the anions in the case of TFSI-containing mixtures.

The obtained values for 〈A²〉, 〈B²〉, and k are given in Table 8. The 〈A²〉 values obtained for the 1 : 1 mixtures are very small compared to 〈B²〉 implying that center-of-mass fluctuations of the ions are much weaker than their dipole fluctuations. This implies quite strict ion momentum conservation due to the slow exchange of the G4 molecules between [Li(G4)]⁺ complexes in comparison to the elementary displacement steps of the ions, as discussed above. With increasing dilution of the Li salt (1 : 1.5 and 1 : 2 mixtures), the center-of-mass fluctuations of the ions increase significantly, since excess G4 molecules exist, which are not strongly bound to Li⁺ ions. These excess G4 molecules are able to exchange momentum with the ions, so that ion momentum conservation is much less strictly fulfilled.

For all electrolytes, dipole fluctuations 〈B²〉 are significantly larger than center-of-mass fluctuations 〈A²〉, and the dipole fluctuations show only a slight relative increase with increasing dilution, see Table 8. This suggests that the microscopic mechanisms limiting the equilibrium center-of-mass and dipole fluctuations in these electrolytes are distinct and should be considered separately in computer simulations of electrolytes. In particular, the mechanisms limiting center-of-mass fluctuations have not been analysed in detail, but are highly relevant, since Li⁺ transport under anion-blocking conditions is governed by center-of-mass fluctuations, see eqn (39).

For the interpretation of the obtained k values, we assume that the Li⁺ ions form complex cations [Li(G4)_n]⁺, where different integer values of n may coexist. With n̄ denoting the average number of G4 molecules bound to a single Li⁺ ion, the mass ratio k is given by:

The obtained values for n̄ are also given in Table 8. As expected, for the 1 : 1 mixtures, n̄ is close to unity within the experimental error. However, with increasing dilution, n̄ becomes significantly larger than unity, suggesting that, on average, more than one G4 molecule is bound to a Li⁺ ion. Evidence for this is also directly seen from the similar mobilities of Li⁺ and G4 in the sample with Li salt/G4 of 1 : 2 in Fig. 4. If only one G4 molecule was bound to a Li⁺ ion, a mobility of μ_G4 = 1/2μ_Li would be expected, since the G4 mobility results as a fast exchange average of the bound and the free glyme molecules. Instead, the experimental value of μ_G4 is clearly larger than 1/2μ_Li. However, since n̄ is lower than the molar mixing ratio G4/Li salt, a significant amount of free G4 molecules exist in the 1 : 1.5 and 1 : 2 mixtures, which are not strongly bound to Li⁺ ions. This is consistent with the large center-of-mass fluctuations of the ionic species observed in these mixtures.

Conclusions

We have combined, for the first time, very-low-frequency impedance spectroscopy on symmetrical Li|electrolyte|Li cells with diffusion and electrophoretic NMR in order to obtain quantitative information about ion correlations in Li salt/G4 mixtures with molar ratios between 1 : 1 and 1 : 2. Our results show that cation–cation, anion–anion, and cation–anion movements are anticorrelated. The anticorrelations are particularly strong in the 1 : 1 mixture and decrease with decreasing molar ratio of Li salt to G4. Like-ion anticorrelations are generally stronger than cation–anion anticorrelations and are particularly strong for the heavier type of ion. We have analyzed in detail the influence of these ion correlations on two types of Li⁺ transference numbers, namely the mobility-based Li⁺ transference numbers and the Li⁺ transference numbers under anion-blocking conditions, as well as on the Haven ratio. Our results are at variance with concepts explaining transference numbers and Haven ratio by ion pair formation and ionicity exclusively, since these imply positive cation–anion correlations. We conclude that in the present systems, momentum conservation and the resulting ion anticorrelations are far more important than ion pair formation.

In order to rationalize our findings, we have derived linear response theory expressions for all correlations and for the resulting charge and mass transport coefficients of the electrolytes. These expressions show that the Li⁺ ion transport under anion-blocking conditions in a battery is governed by equilibrium center-of-mass fluctuations in the electrolytes. Therefore, we suggest that in future computer simulations of battery electrolytes, not only dipole fluctuations should be considered, which govern the overall ionic conductivity, but that particular attention should be paid to the analysis of center-of-mass fluctuations.

Conflicts of interest

There are no conflicts to declare.

Acknowledgements

Financial support of this work by the Federal State of Hesse is gratefully acknowledged. Furthermore, we thank Albemarle (Rockwood Lithium GmbH) for providing the lithium foil.

Notes and references

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Footnote

† Electronic supplementary information (ESI) available: Plots of PFG-NMR and eNMR data for the different LiFSI/G4 mixtures, plot of the thermodynamic factor for the LiTFSI/G4 and LiFSI/G4 1 : 1 mixtures, and derivations of linear response theory expressions. See DOI: 10.1039/d0cp06147f

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