Conductometric Analysis of Ion Equilibrium in Li⁺/F⁻ and Mg²⁺/F⁻ Hybrid Electrolyte Solutions

Abstract

Fluoride shuttle batteries are expected to be innovative with high energy density superior to lithium-ion ones as on-board power-source for electric vehicles. However, the low solubility of fluoride ion in organic solvents makes it difficult to choose electrolytes. Kawasaki et al. have reported that no precipitation appears in γ-butyrolactone solution of CsF at high concentrations in the presence of excess amounts of Li⁺ or Mg²⁺ cation [M. Kawasaki, et al., J. Electrochem. Soc., 169, 110508 (2022).]. Such solutions are called hybrid electrolyte ones. In this study, we performed conductometric titration of Li⁺- or Mg²⁺-containing solutions with CsF solutions to elucidate the ion equilibrium. The titration curves were analyzed on models including precipitation of LiF or MgF₂ and formation of Li₂F⁺ and LiF₂⁻ triple ion or MgF⁺ associated one to evaluate the corresponding equilibrium constants. The results indicate that no precipitation appears in certain ranges of the F⁻/Li⁺ or F⁻/Mg²⁺ molar ratio by the triple/associated ion formation. Details of the conductometric analysis are discussed together with some problems involved in the proposed method.

1. Introduction

With the global requirement to reduce greenhouse gases such as carbon dioxide, there is a growing demand for the replacement of gasoline-powered vehicles with electric vehicles (EVs). Lithium-ion batteries (LIBs) have high energy density and are suitable for use in small devices. However, as an on-board power-source for EVs, the energy density is not sufficient to achieve a cruising range equivalent to that of gasoline-powered vehicles. One candidate for innovative batteries with high energy density beyond LIBs is fluoride shuttle batteries (FSBs).¹ FSB is a battery that operates by metal fluorination/defluorination with fluoride ion shuttling between the electrodes;   

\begin{equation} \text{M$_{1}$F$_{x}$} + \text{$x$e$^{-}$} \rightleftarrows \text{M$_{1}$} + \text{$x$F$^{-}$}, \end{equation} (1)

  

\begin{equation} \text{M$_{2}$} + \text{$y$F$^{-}$} \rightleftarrows \text{M$_{2}$F$_{y}$} + \text{$y$e$^{-}$}, \end{equation} (2)

where M₁ and M₂ represent metals, respectively. These metals multivalently react with fluoride ion, resulting in a large theoretical capacity. In addition, since fluoride ion shows high oxidation resistance, high energy density can be achieved using appropriate metals and electrolytes.

However, many fluoride salts are poorly soluble in organic solvents, which makes it difficult to select electrolytes. Several strategies have been considered to overcome this problem. For example, borane,^2,3 borate,⁴ or boroxine,^5–7 was added as an anion acceptor (AA) to increase the solubility by binding AA to F⁻; bulky cations were used to increase the salt solubility;⁸ or ionic liquids were used instead of organic solvents.^9–11 However, these additives and ionic liquids are reductively decomposable, which limits the choice of electrodes. In addition, the presence of β-hydrogen in quaternary ammonium cations that are often used as bulky cations or ionic liquids limits the choice of cations, because F⁻ acts as a base and causes Hofmann elimination.¹²

In exploring suitable electrolytes for FSBs, Kawasaki et al. have successfully increased the total concentration of CsF and KF in γ-butyrolactone (GBL) up to ca. 0.05 M (M ≡ mol dm⁻³) by solvent substitution method.¹³ They have also found that the addition of excess amounts of lithium bis(fluorosulfonyl)amide (LiFSA) or magnesium bis(trifluoromethanesulfonyl)amide (Mg(TFSA)₂) to CsF/GBL solutions solubilizes the low solubility product of LiF or MgF₂, respectively.¹⁴ These electrolytes, named Li⁺/F⁻ and Mg²⁺/F⁻ hybrid ones, were confirmed to be applicable to FSBs. Since the hybrid electrolytes are composed of metal cations, they are free from the Hofmann elimination and show relatively low reducibility, which would allow the use of a variety of electrodes.

The hybrid electrolyte solution is a mixture of 1 M LiTFSA/GBL or Mg(TFSA)₂/GBL and 50 mM CsF/GBL at various volume ratios. No precipitates were observed at molar ratios of Li⁺/F⁻ above 7 or of Mg²⁺/F⁻ above 1.3. Kawasaki et al.¹⁴ explained that the reason for the absence of the precipitation was attributed to the Li₂F⁺ triple ion formation or MgF⁺ associated one due to excess amounts of alkali or alkali metal cations. The presence of MgF⁺ ion in the Mg²⁺/F⁻ hybrid electrolyte solution was confirmed by ¹⁹F NMR, while it was difficult to directly observe Li₂F⁺. Conductometric titration was also performed, and the triple/associated ion formation equilibria were discussed under approximated conditions. In this study, we improved the analysis of the conductometric titration data, and successfully verified the solubilization of the precipitates by the triple/associated ion formation. The range in which the precipitates do not appear and the ionic species at any given molar ratios are discussed based on the evaluated equilibrium constants. Some problems in this method are also discussed.

2. Experimental

Details of the experimental procedure were reported in the literature.¹⁴ Briefly, conductometric titration was performed for ca. 100× diluted-Li⁺/F⁻ hybrid electrolyte solution or a dilute (5 mM) Mg(TFSA)₂ electrolyte (Kishida Chemical Co., Ltd., magnesium battery grade) in GBL (Kishida Chemical Co., Ltd., 99.5 %, lithium battery grade) as an analyte with 50 mM CsF (Tokyo Chemical Industry Co., Ltd., 99.0 %) in GBL as a titrant. The Li⁺/F⁻ hybrid electrolyte solution before the dilution was a mixture of 0.75 M LiFSA (Nippon Shokubai Co., Ltd., 99.9 %) + 12.5 mM CsF/GBL. The CsF/GBL was prepared by the method of Kawasaki et al.¹³ The dilute analyte solutions containing respective analytes at initial concentrations less than 10 mM were used to simplify the concentration dependence of the molar conductivity, as described in detail in the next section. We also reason in the next section why we used the 100× diluted-Li⁺/F⁻ hybrid electrolyte solution, instead of a dilute solution of LiFSA alone. The test solution was placed in a Pt–Pt two-electrode cell (SB1400, EC Frontier Co., Ltd.) with a cell constant of 1.4 cm⁻¹. The ionic conductivity was measured by A. C. impedance spectroscopy at 0 ± 50 mV in a frequency range of 1 × 10⁶ to 10 Hz (HZ-Pro, Hokuto Denko Corp.). All measurements were performed in an Ar-filled glove box. The inside of the glove-box was maintained at 30 °C throughout the experiments by an infrared lamp connected to a digital high accuracy temperature controller (AS ONE Corporation, TJA-550P).

The molar ionic conductivity was determined by the conductivity measurements for each salt at different concentrations and the transport number measurement of LiFSA/GBL. The conductivity of the GBL solutions of LiFSA, LiTFSA (Kishida Chemical Co., Ltd., 99.9 %, lithium battery grade), CsTFSA (Tokyo Chemical Industry Co., Ltd., 98.0 %), CsF, and Mg(TFSA)₂ was measured at concentrations of 0.9, 2.5, 4.9, and 8.1 mM. The cation transport number of LiFSA, t_+,LiFSA, was measured to be 0.36 for 1.0 M LiFSA/GBL by Bruce and Vincent method.¹⁵

3. Analytical Models

The conductivity, κ, can be calculated from the sum of the products of the ion concentration, c_i, and the molar ionic conductivity, of each ion i, λ_i:   

\begin{equation} \kappa = \sum\nolimits_{\text{i}}c_{\text{i}}\lambda_{\text{i}}. \end{equation} (3)

In this section, we describe the evaluation methods for λ_i and c_i. First, the c_i dependence of λ_i is discussed. Next, the equilibrium values of c_i are discussed for the Li⁺/F⁻ and Mg²⁺/F⁻ hybrid electrolyte solutions.

3.1 Concentration dependence of the molar ionic conductivity

We can reasonably assume that the molar conductivity of salt solution I, Λ_I, given by Λ_I = κ_I/c_I (κ_I and c_I being the conductivity and concentration of salt solution I, respectively), is expressed as the sum of the molar ionic conductivities of cation, λ_+,i, and anion, λ_−,i at relatively low c_I:   

\begin{equation} \varLambda_{\text{I}} = \lambda_{+,\text{i}} + \lambda_{-,\text{i}}. \end{equation} (4)

Λ_I depends on the ionic strength of salt solution I, I_I, according to Kohlrausch’s square root law as follows.   

\begin{equation} \varLambda_{\text{I}} = \varLambda_{\text{I}}^{\infty} - S_{\text{I}}\sqrt{I_{\text{I}}}, \end{equation} (5)

where $\varLambda_{\text{I}}^{\infty }$ and S_I is the molar conductivity of infinite dilution and the concentration parameter of salt I, respectively. This Kohlrausch’s square root law holds only for dilute solutions. Therefore, the conductivity titration was performed, restricting the total ionic strength of the analyte solution to be as small as possible. I_I for solution I composed of univalent ions (I = LiFSA, LiTFSA, CsTFSA, and CsF) is given by:   

\begin{equation} I = \frac{c_{+,\text{i}}(z_{+,\text{i}})^{2} + c_{-,\text{i}}(z_{-,\text{i}})^{2}}{2} = \frac{c_{\text{I}}(1)^{2} + c_{\text{I}}(-1)^{2}}{2} = c_{\text{I}}, \end{equation} (6)

while for Mg(TFSA)₂:   

\begin{equation} I = \frac{c_{+,\text{i}}(z_{+,\text{i}})^{2} + c_{-,\text{i}}(z_{-,\text{i}})^{2}}{2} = \frac{c_{\text{I}}(2)^{2} + 2c_{\text{I}}(-1)^{2}}{2} = 3c_{\text{I}}, \end{equation} (7)

where c_+,i, c_−,i, z_+,i, and z_−,i are the concentration and valence of cation and anion, respectively. Good linearity in Λ_I vs. $\sqrt{c_{\text{I}}} $ (or $\sqrt{3c_{\text{I}}} $) plots given in Fig. S1 confirmed the Kohlrausch’s square root law over the concentration range measured for each salt solution. Thus, we can calculate Λ_I at any c_I using $\varLambda_{\text{I}}^{\infty }$ and S_I evaluated from the plots in Fig. S1. Each λ_i value at a given c_i was evaluated with Λ_I and t_+,LiFSA (= 0.36), as detailed in SI, Section 1. Thus, we may predict κ values of mixed salt tsolutions with Eq. 3 if we could only know c_i of the solute ions. Note here that S_I reflects the interaction only between the cation and anion of salt I, ignoring the possible interaction among the other ions in mixed salt solutions. Essentially, it should take into account interactions with other ions, however since there is no way to determine S_I of them, there is no other way to ignore the possible interaction among the other ions in mixed salt solutions.

3.2 Equilibrium concentration

3.2.1 Li⁺/F⁻ hybrid electrolyte solution

The following equilibria are considered in mixed solutions of LiFSA and CsF: the precipitation and the triple ion formation equilibria;   

\begin{equation} \text{Li$^{+}$} + \text{F$^{-}$} \rightleftharpoons \text{LiF$\downarrow$},\quad K_{\text{SP,LiF}} = \frac{c_{\text{Li${^{+}}$}}c_{\text{F${^{-}}$}}}{(c^{\circ})^{2}}, \end{equation} (8)

  

\begin{equation} \text{2Li$^{+}$} + \text{F$^{-}$} \rightleftharpoons \text{Li$_{2}$F$^{+}$},\quad K_{\text{T},+} = \frac{c_{\text{Li${_{2}}$F${^{+}}$}}(c^{\circ})^{2}}{(c_{\text{Li${^{+}}$}})^{2}c_{\text{F${^{-}}$}}}, \end{equation} (9)

  

\begin{equation} \text{Li$^{+}$} + \text{2F$^{-}$} \rightleftharpoons \text{LiF$_{2}{}^{-}$},\quad K_{\text{T},-} = \frac{c_{\text{LiF${_{2}}{^{-}}$}}(c^{\circ})^{2}}{c_{\text{Li${^{+}}$}}(c_{\text{F${^{-}}$}})^{2}}, \end{equation} (10)

where K_SP,LiF is the equilibrium constant for the LiF precipitation, while K_T,+ and K_T,− are the equilibrium constants for the formation of Li₂F⁺ and LiF₂⁻ triple ions, respectively. c° (= 1 M) is the standard concentration. Since the titration was performed at sufficiently dilute concentration, the activity coefficient was assumed to be one. Here, Cs⁺ and FSA⁻ in the solution were assumed to be inert and hard to bind with the counterion, since Cs⁺ is softer acid than Li⁺ and FSA⁻ is bulky. For simplicity, the equilibrium constants of the triple ion formations are approximated as follows, as often used in ion equilibrium analysis:¹⁶   

\begin{equation} K_{\text{T}} \equiv K_{\text{T},+} = K_{\text{T},-}. \end{equation} (11)

Our present analyses deal with only equilibrium systems where all the relevant reactions are in equilibrium at arbitrary titration points. This means that the following reaction: Li⁺ + LiF↓ $ \rightleftharpoons $ Li₂F⁺ given by the combination of reactions (8) and (9), in which an extra Li⁺ ion solubilizes LiF↓ by the formation of a triple ion Li₂F⁺, should also be in equilibrium at all times. However, this assumption breaks when we try to titrate a dilute (e.g., 7 mM) LiFSA/GBL with 50 mM CsF/GBL, where the LiF precipitation dominates at every titration point. This is partly due to the considerably small concentration of extra Li⁺ ions and partly because of the relatively small rate constant for the LiF↓ solubilization (Li⁺ + LiF↓ → Li₂F⁺).

We, therefore, chose the 100× diluted-Li⁺/F⁻ hybrid electrolyte solution as an alternative analyte solution. The sufficiently high concentration (∼0.7 M) of extra Li⁺ ions in the original Li⁺/F⁻ hybrid electrolyte enables facile formation of Li₂F⁺ triple ions. As a result, a substantial number of Li₂F⁺ triple ions exist even in the 100-fold diluted Li⁺/F⁻ hybrid electrolyte solution. Furthermore, as we emphasized in the previous paper,¹⁴ Li₂F⁺ triple ions are most likely able to self-catalyze the LiF↓ solubilization into another Li₂F⁺. Thus, the 100× diluted-Li⁺/F⁻ hybrid electrolyte solution affords an equilibrium (at least pseudo-equilibrium) system that warrants our present quantitative analyses as detailed below.

The total amount of Li, n_Li,t, in the diluted Li⁺/F⁻ hybrid electrolyte can be given with its initial volume, V_hybrid, and the initial concentration of Li⁺ plus twice that of Li₂F⁺, denoted by c_0,Li.   

\begin{equation} n_{\text{Li,t}} = c_{\text{0,Li}}V_{\text{hybrid}}. \end{equation} (12)

The total amount of F, n_F,t, at an arbitrary titration point can be expressed in the same way with the titration volume, V_CsF, and the concentration, c_0,CsF, of the CsF solution as a titrant, and the initial total concentration of F (present as either F⁻ or Li₂F⁺) in the analyte, c_0,F,hybrid, as follows:   

\begin{equation} n_{\text{F,t}} = \text{c}_{\text{0,CsF}}V_{\text{CsF}} + c_{\text{0,F,hybrid}}V_{\text{hybrid}}. \end{equation} (13)

The total volume of the analyte at an arbitrary titration point is:   

\begin{equation} V_{\text{t,Li/F}} = V_{\text{CsF}} + V_{\text{hybrid}}. \end{equation} (14)

The equilibrium constants of Eqs. 8–10 can be rewritten with the amounts of ions, n_i (i = Li⁺, F⁻, Li₂F⁺, $\text{LiF}_{2}^{ - }$):   

\begin{align} & K_{\text{SP,LiF}} = \frac{n_{\text{Li${^{+}}$}}n_{\text{F${^{-}}$}}}{(c^{\circ}V_{\text{t,Li/F}})^{2}},\\ & K_{1} = n_{\text{Li${^{+}}$}}n_{\text{F${^{-}}$}}\quad (K_{1} \equiv K_{\text{SP,LiF}}(c^{\circ} V_{\text{t,Li/F}})^{2}). \end{align} (15)

  

\begin{align} & K_{\text{T}} = \frac{n_{\text{Li${_{2}}$F${^{+}}$}}(c^{\circ} V_{\text{t,Li/F}})^{2}}{(n_{\text{Li${^{+}}$}})^{2}n_{\text{F${^{-}}$}}},\\ & K_{2} = \frac{n_{\text{Li${_{2}}$F${^{+}}$}}}{(n_{\text{Li${^{+}}$}})^{2}n_{\text{F${^{-}}$}}}\quad \left(K_{2} \equiv \frac{K_{\text{T}}}{(c^{\circ} V_{\text{t,Li/F}})^{2}}\right). \end{align} (16)

  

\begin{align} & K_{\text{T}} = \frac{n_{\text{LiF${_{2}}{^{-}}$}}(c^{\circ} V_{\text{t,Li/F}})^{2}}{n_{\text{Li${^{+}}$}}(n_{\text{F${^{-}}$}})^{2}},\\ & K_{2} = \frac{n_{\text{LiF${_{2}}{^{-}}$}}}{n_{\text{Li${^{+}}$}}(n_{\text{F${^{-}}$}})^{2}}\quad \left(K_{2} \equiv \frac{K_{\text{T}}}{(c^{\circ} V_{\text{t,Li/F}})^{2}}\right). \end{align} (17)

3.2.1.1 Precipitation equilibrium model without the triple ion formation

We consider here the equilibrium values of c_i during the titration by assuming that only precipitation equilibrium occurs and no triple ions are formed by considering K₁. The mass balance of the titration solution is represented by:   

\begin{equation} n_{\text{Li,t}} = n_{\text{Li${^{+}}$}} + n_{\text{LiF}}, \end{equation} (18)

for Li, and   

\begin{equation} n_{\text{F,t}} = n_{\text{F${^{-}}$}} + n_{\text{LiF}}, \end{equation} (19)

for F, where n_LiF is the amount of LiF precipitate. The total amounts are related to the concentration and volume of the analyte and titrant (Eqs. 12 and 13). Considering the K₁ (Eq. 15), $n_{\text{Li}^{ + }}$ is given as follows (details in SI, Section 2.1.1):   

\begin{equation} n_{\text{Li${^{+}}$}} = \frac{(n_{\text{Li,t}} - n_{\text{F,t}}) + \sqrt{(n_{\text{Li,t}} - n_{\text{F,t}})^{2} + 4K_{1}}}{2}, \end{equation} (20)

  

\begin{equation} n_{\text{LiF}} = n_{\text{Li,t}} - n_{\text{Li${^{+}}$}}. \end{equation} (21)

Consequently, we can evaluate c_i (i = Li⁺, F⁻, FSA⁻, and Cs⁺) at any arbitrary titration points:   

\begin{align} &c_{\text{Li${^{+}}$}} = \frac{n_{\text{Li${^{+}}$}}}{V_{\text{t,Li/F}}},\quad c_{\text{F${^{-}}$}} = \frac{n_{\text{F${^{-}}$}}}{V_{\text{t,Li/F}}},\quad \\ &c_{\text{FSA${^{-}}$}} = \frac{n_{\text{Li,t}}}{V_{\text{t,Li/F}}},\quad \text{and}\quad c_{\text{Cs${^{+}}$}} = \frac{n_{\text{F,t}}}{V_{\text{t,Li/F}}}. \end{align} (22)

3.2.1.2 Model with both of the precipitation and triple ion formation equilibria

We calculate here c_i during the titration by assuming both of the precipitation and the triple ion formation equilibria by considering K₁ and K₂. The mass balance of the titration solution is given by:   

\begin{equation} n_{\text{Li,t}} = n_{\text{Li${^{+}}$}} + n_{\text{LiF}} + 2n_{\text{Li${_{2}}$F${^{+}}$}} + n_{\text{LiF${_{2}}{^{-}}$}}, \end{equation} (23)

for Li, and   

\begin{equation} n_{\text{F,t}} = n_{\text{F${^{-}}$}} + n_{\text{LiF}} + n_{\text{Li${_{2}}$F${^{+}}$}} + 2n_{\text{LiF${_{2}}{^{-}}$}}, \end{equation} (24)

for F, where $n_{\text{Li}_{2}\text{F}^{ + }}$ and $n_{{\text{LiF}_{2}}^{ - }}$ are the amounts of Li₂F⁺ and LiF₂⁻ triple ions, respectively. The equilibrium constants (Eqs. 15–17) and the mass balance concept give (details in SI, Section 2.1.2):   

\begin{equation} n_{\text{Li${^{+}}$}} = \frac{(n_{\text{Li,t}} - n_{\text{F,t}}) + \sqrt{(n_{\text{Li,t}} - n_{\text{F,t}})^{2} + 4K_{1}(1 + K_{1}K_{2})^{2}}}{2(1 + K_{1}K_{2})}. \end{equation} (25)

The amounts $n_{\text{F}^{ - }}$, $n_{\text{Li}_{2}\text{F}^{ + }}$, and $n_{{\text{LiF}_{2}}^{ - }}$ can be obtained from Eqs. 15–17, respectively. n_LiF is given from Eq. 23 as follows:   

\begin{equation} n_{\text{LiF}} = n_{\text{Li,t}} - (n_{\text{Li${^{+}}$}} + 2n_{\text{Li${_{2}}$F${^{+}}$}} + n_{\text{LiF${_{2}}{^{-}}$}}). \end{equation} (26)

Consequently, we can evaluate c_i (i = Li⁺, F⁻, Li₂F⁺, LiF₂⁻, FSA⁻, and Cs⁺) at any arbitrary titration points;   

\begin{align} & c_{\text{Li${^{+}}$}} = \frac{n_{\text{Li${^{+}}$}}}{V_{\text{t,Li/F}}},\quad c_{\text{F${^{-}}$}} = \frac{n_{\text{F${^{-}}$}}}{V_{\text{t,Li/F}}},\quad c_{\text{Li${_{2}}$F${^{+}}$}} = \frac{n_{\text{Li${_{2}}$F${^{+}}$}}}{V_{\text{t,Li/F}}},\quad \\ &c_{\text{LiF${_{2}}{^{-}}$}} = \frac{n_{\text{LiF${_{2}}{^{-}}$}}}{V_{\text{t,Li/F}}},\quad c_{\text{FSA${^{-}}$}} = \frac{n_{\text{Li,t}}}{V_{\text{t,Li/F}}},\quad \text{and}\quad c_{\text{Cs${^{+}}$}} = \frac{n_{\text{F,t}}}{V_{\text{t,Li/F}}}. \end{align} (27)

Limited case of no precipitation

Even in the model with both of the precipitation and triple ion formation equilibria, the precipitation would not occur when n_F,t/n_Li,t is very small or very large, as discussed in detail in the later section. Therefore, the case in which only the triple ions are formed without the precipitation is also discussed here. In this case, only K₂ is considered by ignoring K₁. The mass balance of the titration solution is given by:   

\begin{equation} n_{\text{Li,t}} = n_{\text{Li${^{+}}$}} + 2n_{\text{Li${_{2}}$F${^{+}}$}} + n_{\text{LiF${_{2}}{^{-}}$}}, \end{equation} (28)

for Li, and   

\begin{equation} n_{\text{F,t}} = n_{\text{F${^{-}}$}} + n_{\text{Li${_{2}}$F${^{+}}$}} + 2n_{\text{LiF${_{2}}{^{-}}$}}, \end{equation} (29)

for F. The equilibrium constant of K₂ (Eqs. 16 and 17) and the mass balance concept give (details in SI, Limited case of no precipitation in Section 2.1.2):   

\begin{equation} n_{\text{Li,t}} = n_{\text{Li${^{+}}$}} + K_{2}n_{\text{Li${^{+}}$}}n_{\text{F${^{-}}$}}(2n_{\text{Li${^{+}}$}} + n_{\text{F${^{-}}$}}), \end{equation} (30)

  

\begin{equation} n_{\text{F,t}} = n_{\text{F${^{-}}$}} + K_{2}n_{\text{Li${^{+}}$}}n_{\text{F${^{-}}$}}(n_{\text{Li${^{+}}$}} + 2n_{\text{F${^{-}}$}}). \end{equation} (31)

Unfortunately, it is not easy to solve Eqs. 30 and 31, but we may reasonably ignore the triple ion formation when n_F,t/n_Li,t was very small or large.

Limited case A) Assumption of $n_{{\text{LiF}_{2}}^{ - }} \approx 0$ at $n_{\text{F,t}} \ll n_{\text{Li,t}}$

In this case, the mass balance of the solution is rewritten by:   

\begin{equation} n_{\text{Li,t}} = n_{\text{Li${^{+}}$}} + 2n_{\text{Li${_{2}}$F${^{+}}$}}, \end{equation} (32)

for Li, and   

\begin{equation} n_{\text{F,t}} = n_{\text{F${^{-}}$}} + n_{\text{Li${_{2}}$F${^{+}}$}}, \end{equation} (33)

for F. Considering the equilibrium constant of K₂ (Eq. 16), the related equations are rewritten as follows (details in SI, Limited case A in Section 2.1.2):   

\begin{equation} K_{2}(n_{\text{Li${^{+}}$}})^{3} + K_{2}(2n_{\text{F,t}} - n_{\text{Li,t}})(n_{\text{Li${^{+}}$}})^{2} + n_{\text{Li${^{+}}$}} - n_{\text{Li,t}} = 0. \end{equation} (34)

The analytical solution of this cubic equation for $n_{\text{Li}^{ + }}$ can be obtained by Cardano’s method. Then, $n_{\text{Li}_{2}\text{F}^{ + }}$ and $n_{\text{F}^{ - }}$ can be calculated from Eqs. 32 and 16, respectively. Consequently, we can evaluate c_i (i = Li⁺, F⁻, Li₂F⁺, LiF₂⁻, FSA⁻, and Cs⁺) at any arbitrary titration points.   

\begin{align} & c_{\text{Li${^{+}}$}} = \frac{n_{\text{Li${^{+}}$}}}{V_{\text{t,Li/F}}},\quad c_{\text{F${^{-}}$}} = \frac{n_{\text{F${^{-}}$}}}{V_{\text{t,Li/F}}},\quad c_{\text{Li${_{2}}$F${^{+}}$}} = \frac{n_{\text{Li${_{2}}$F${^{+}}$}}}{V_{\text{t,Li/F}}},\quad \\ &c_{\text{LiF${_{2}}{^{-}}$}} = 0, \quad c_{\text{FSA${^{-}}$}} = \frac{n_{\text{Li,t}}}{V_{\text{t,Li/F}}},\quad \text{and}\quad c_{\text{Cs${^{+}}$}} = \frac{n_{\text{F,t}}}{V_{\text{t,Li/F}}}. \end{align} (35)

Limited case B) Assumption of $n_{\text{Li}_{2}\text{F}^{ + }} \approx 0$ at $n_{\text{Li,t}} \ll n_{\text{F,t}}$

In this case, the mass balance of analyte solution is rewritten by:   

\begin{equation} n_{\text{Li,t}} = n_{\text{Li${^{+}}$}} + n_{\text{LiF${_{2}}{^{-}}$}}, \end{equation} (36)

for Li, and   

\begin{equation} n_{\text{F,t}} = n_{\text{F${^{-}}$}} + 2n_{\text{LiF${_{2}}{^{-}}$}}, \end{equation} (37)

for F. Further rewritten by considering the equilibrium constant of K₂ (Eq. 17) gives (details in SI, Limited case B in Section 2.1.2):   

\begin{equation} K_{2}(n_{\text{F${^{-}}$}})^{3} + K_{2}(2n_{\text{Li,t}} - n_{\text{F,t}})(n_{\text{F${^{-}}$}})^{2} + n_{\text{F${^{-}}$}} - n_{\text{F,t}} = 0. \end{equation} (38)

The analytical solution of Eq. 38 for $n_{\text{F}^{ - }}$ can be obtained by Cardano’s method. $n_{{\text{LiF}_{2}}^{ - }}$ and $n_{\text{Li}^{ + }}$ can be given from Eqs. 37 and 17, respectively. Consequently, we can evaluate c_i (i = Li⁺, F⁻, Li₂F⁺, LiF₂⁻, FSA⁻, and Cs⁺) at any arbitrary titration points.   

\begin{align} & c_{\text{Li${^{+}}$}} = \frac{n_{\text{Li${^{+}}$}}}{V_{\text{t,Li/F}}},\quad c_{\text{F${^{-}}$}} = \frac{n_{\text{F${^{-}}$}}}{V_{\text{t,Li/F}}},\quad c_{\text{Li${_{2}}$F${^{+}}$}} = 0,\quad c_{\text{LiF${_{2}}{^{-}}$}} = \frac{n_{\text{LiF${_{2}}{^{-}}$}}}{V_{\text{t,Li/F}}}, \quad \\ &c_{\text{FSA${^{-}}$}} = \frac{n_{\text{Li,t}}}{V_{\text{t,Li/F}}},\quad \text{and}\quad c_{\text{Cs${^{+}}$}} = \frac{n_{\text{F,t}}}{V_{\text{t,Li/F}}}. \end{align} (39)

3.2.2 Mg²⁺/F⁻ hybrid electrolyte solution

The following equilibria are considered in mixed solutions of Mg(TFSA)₂ and CsF: the precipitation equilibrium and the ion association equilibrium;   

\begin{equation} \text{Mg$^{2+}$} + \text{2F$^{-}$} \rightleftharpoons \text{MgF$_{2}{}\downarrow$},\quad K_{\text{SP,MgF${_{2}}$}} = \frac{c_{\text{Mg${^{2+}}$}}(c_{\text{F${^{-}}$}})^{2}}{(c^{\circ})^{3}}, \end{equation} (40)

  

\begin{equation} \text{Mg$^{2+}$} + \text{F$^{-}$} \rightleftharpoons \text{MgF$^{+}$},\quad K_{\text{A}} = \frac{c_{\text{MgF${^{+}}$}}c^{\circ}}{c_{\text{Mg${^{2+}}$}}c_{\text{F${^{-}}$}}}, \end{equation} (41)

where $K_{\text{SP,MgF}_{2}}$ and K_A are the equilibrium constants for the MgF₂ precipitation and the formation of MgF⁺, respectively. The Cs⁺ and TFSA⁻ were assumed to be inert, as in the case of Li⁺/F⁻ hybrid electrolyte solution. The total amount of Mg can be expressed with the initial volume, $V_{\text{Mg(TFSA)}_{2}}$, and the initial concentration, $c_{\text{0,Mg(TFSA)}_{2}}$, of the Mg(TFSA)₂ solution as an analyte.   

\begin{equation} n_{\text{Mg,t}} = c_{\text{0,Mg(TFSA)${_{2}}$}}V_{\text{Mg(TFSA)${_{2}}$}}. \end{equation} (42)

The total amount of F at an arbitrary titration point can be given in the same way by the volume, V_CsF, and the concentration, c_0,CsF, of the CsF solution as a titrant, as follows:   

\begin{equation} n_{\text{F,t}} = c_{\text{0,CsF}}V_{\text{CsF}}. \end{equation} (43)

The total volume of the titration solution at an arbitrary titration point is:   

\begin{equation} V_{\text{t,Mg/F}} = V_{\text{CsF}} + V_{\text{Mg(TFSA)${_{2}}$}}. \end{equation} (44)

Equations 40 and 41 can be rewritten with n_i (i = Mg²⁺, F⁻, and MgF⁺):   

\begin{align} & K_{\text{SP,MgF${_{2}}$}} = \frac{n_{\text{Mg${^{2+}}$}}(n_{\text{F${^{-}}$}})^{2}}{(c^{\circ} V_{\text{t,Mg/F}})^{3}},\\ & K_{3} = n_{\text{Mg${^{2+}}$}}(n_{\text{F${^{-}}$}})^{2}\quad (K_{3} \equiv K_{\text{SP,MgF${_{2}}$}}(c^{\circ} V_{\text{t,Mg/F}})^{3}). \end{align} (45)

  

\begin{align} & K_{\text{A}} = \frac{n_{\text{MgF${^{+}}$}}c^{\circ} V_{\text{t,Mg/F}}}{n_{\text{Mg${^{2+}}$}}n_{\text{F${^{-}}$}}},\\ & K_{4} = \frac{n_{\text{MgF${^{+}}$}}}{n_{\text{Mg${^{2+}}$}}n_{\text{F${^{-}}$}}}\quad \left(K_{4} \equiv \frac{K_{\text{A}}}{c^{\circ} V_{\text{t,Mg/F}}}\right). \end{align} (46)

3.2.2.1 Precipitation equilibrium model without the ion association

As in the case of the Li⁺/F⁻ hybrid electrolyte solution, we first consider the equilibrium values of c_i during the titration by assuming that only precipitation equilibrium occurs, and that no association ion is formed. In this case, only K₃ is considered (but K₄ is ignored). The mass balance of the titration solution is given by:   

\begin{equation} n_{\text{Mg,t}} = n_{\text{Mg${^{2+}}$}} + n_{\text{MgF${_{2}}$}}, \end{equation} (47)

for Mg, and   

\begin{equation} n_{\text{F,t}} = n_{\text{F${^{-}}$}} + 2n_{\text{MgF${_{2}}$}}, \end{equation} (48)

for F, where $n_{\text{MgF}_{2}}$ is the amount of MgF₂ precipitate. The mass balance concept and the equilibrium constant of K₃ (Eq. 45) yield (details in SI, Section 2.2.1):   

\begin{equation} (n_{\text{F${^{-}}$}})^{3} - (2n_{\text{Mg,t}} - n_{\text{F,t}})(n_{\text{F${^{-}}$}})^{2} - 2K_{3} = 0. \end{equation} (49)

Thus, we can obtain $n_{\text{F}^{ - }}$. $n_{\text{Mg}^{2 + }}$ is given by Eq. 45, while $n_{\text{MgF}_{2}}$ is given by:   

\begin{equation} n_{\text{MgF${_{2}}$}} = n_{\text{Mg,t}} - n_{\text{Mg${^{2+}}$}}. \end{equation} (50)

Consequently,   

\begin{align} & c_{\text{Mg${^{2+}}$}} = \frac{n_{\text{Mg${^{2+}}$}}}{V_{\text{t,Mg/F}}},\quad c_{\text{F${^{-}}$}} = \frac{n_{\text{F${^{-}}$}}}{V_{\text{t,Mg/F}}},\quad c_{\text{TFSA${^{-}}$}} = \frac{2n_{\text{Mg,t}}}{V_{\text{t,Mg/F}}}, \quad \\ &\text{and}\quad c_{\text{Cs${^{+}}$}} = \frac{n_{\text{F,t}}}{V_{\text{t,Mg/F}}}. \end{align} (51)

3.2.2.2 Model with both of the precipitation and ion association equilibria

We consider here the equilibrium values of c_i during the titration by assuming both of the precipitation and the ion association equilibria with K₃ and K₄. The mass balance is given by:   

\begin{equation} n_{\text{Mg,t}} = n_{\text{Mg${^{2+}}$}} + n_{\text{MgF${_{2}}$}} + n_{\text{MgF${^{+}}$}}, \end{equation} (52)

for Mg, and   

\begin{equation} n_{\text{F,t}} = n_{\text{F${^{-}}$}} + 2n_{\text{MgF${_{2}}$}} + n_{\text{MgF${^{+}}$}}, \end{equation} (53)

for F, where $n_{\text{MgF}^{ + }}$ is the amount of MgF⁺. Considering the equilibrium constants of K₃ and K₄ (Eqs. 45 and 46), the related equations are rewritten as follows (details in SI, Section 2.2.2):   

\begin{equation} (n_{\text{F${^{-}}$}})^{3} + (2n_{\text{Mg,t}} - n_{\text{F,t}})(n_{\text{F${^{-}}$}})^{2} - K_{3}K_{4}n_{\text{F${^{-}}$}} - 2K_{3} = 0, \end{equation} (54)

Thus, we can obtain $n_{\text{F}^{ - }}$, and then $n_{\text{Mg}^{2 + }}$ and $n_{\text{MgF}^{ + }}$ are given by Eqs. 45 and 46, respectively, while $n_{\text{MgF}_{2}}$ can be obtained from Eq. 52 as follows:   

\begin{equation} n_{\text{MgF${_{2}}$}} = n_{\text{Mg,t}} - (n_{\text{Mg${^{2+}}$}} + n_{\text{MgF${^{+}}$}}). \end{equation} (55)

Consequently,   

\begin{align} & c_{\text{Mg${^{2+}}$}} = \frac{n_{\text{Mg${^{2+}}$}}}{V_{\text{t,Mg/F}}},\quad c_{\text{F${^{-}}$}} = \frac{n_{\text{F${^{-}}$}}}{V_{\text{t,Mg/F}}},\quad c_{\text{MgF${^{+}}$}} = \frac{n_{\text{MgF${^{+}}$}}}{V_{\text{t,Mg/F}}},\quad \\ &c_{\text{TFSA${^{-}}$}} = \frac{2n_{\text{Mg,t}}}{V_{\text{t,Mg/F}}},\quad \text{and}\quad c_{\text{Cs${^{+}}$}} = \frac{n_{\text{F,t}}}{V_{\text{t,Mg/F}}}. \end{align} (56)

Limited case of no precipitation

We can say that the precipitation will not occur at n_F,t/n_Mg,t ≪ 1, as discussed in detail in the later section. Therefore, the case in which only the associated ion is formed without the precipitation is discussed here using only K₄ (without K₃). The mass balance is written as follows:   

\begin{equation} n_{\text{Mg,t}} = n_{\text{Mg${^{2+}}$}} + n_{\text{MgF${^{+}}$}}, \end{equation} (57)

for Mg, and   

\begin{equation} n_{\text{F,t}} = n_{\text{F${^{-}}$}} + n_{\text{MgF${^{+}}$}}, \end{equation} (58)

for F. These mass balance concept and the equilibrium constant of K₄ (Eq. 46) yield:   

\begin{equation} n_{\text{F${^{-}}$}} = \frac{-[1 + K_{4}(n_{\text{Mg,t}} - n_{\text{F,t}})] + \sqrt{[1 + K_{4}(n_{\text{Mg,t}} - n_{\text{F,t}})]^{2} + 4K_{4}n_{\text{F,t}}}}{2K_{4}}. \end{equation} (59)

$n_{\text{Mg}^{2 + }}$ and $n_{\text{MgF}^{ + }}$ can be calculated from Eqs. 57 and 46, respectively. Consequently,   

\begin{align} & c_{\text{Mg${^{2+}}$}} = \frac{n_{\text{Mg${^{2+}}$}}}{V_{\text{t,Mg/F}}},\quad c_{\text{F${^{-}}$}} = \frac{n_{\text{F${^{-}}$}}}{V_{\text{t,Mg/F}}},\quad c_{\text{MgF${^{+}}$}} = \frac{n_{\text{MgF${^{+}}$}}}{V_{\text{t,Mg/F}}},\quad \\ &c_{\text{TFSA${^{-}}$}} = \frac{2n_{\text{Mg,t}}}{V_{\text{t,Mg/F}}},\quad \text{and}\quad c_{\text{Cs${^{+}}$}} = \frac{n_{\text{F,t}}}{V_{\text{t,Mg/F}}}. \end{align} (60)

The value of κ of the titration solution can be evaluated from c_i and λ_i (Eq. 3) calculated on a given model with the corresponding equilibrium constant(s). The equation on the corresponding model was fitted to the experimental values of κ to evaluate the equilibrium constant(s).

4. Results and Discussion

4.1 Li⁺/F⁻ hybrid electrolyte solution

In the analysis, the equilibrium concentrations of all ions, c_i, in solution at each titration point are first determined using various assumptions described in section 3. The λ_i was then calculated considering the concentration dependence. Finally, λ_i and c_i were used to determine the conductivity, κ, in Eq. 3. Fitting to the measured conductivity data was performed using a nonlinear least square method with a free software Gnuplot^® Ver. 5.4.

First, Eq. 3 was fitted to the experimental results of the conductometric titration of the diluted Li⁺/F⁻ hybrid electrolyte solution by the CsF solution, by assuming only the precipitation equilibrium as described in Section 3.2.1.1. However, the fitting did not go well. Figure 1 shows the results of the conductometric measurements (square dots). The dotted, dashed, and single-dashed lines are given by the calculation with several values of K_SP,LiF. Solid purple line shows the calculated conductivity by assuming that all ions are inert precluding the LiF precipitation. As a result, the calculated conductivity surpassed the experimental values in the whole range of the titration curve. None of the other calculated curves, with the LiF precipitation taken into account, reproduce the experimental results either. This means that the ion equilibrium is not able to be explained on the model including only the precipitation equilibrium.

Figure 1.

Experimental results of the conductometric titration of the diluted Li⁺/F⁻ hybrid electrolyte solution with the CsF solution (square dots). The lines were calculated in the model described in Section 3.2.1.1 at various values of K_SP,LiF (K_SP,LiF = 3.0 × 10⁻⁷ for dotted red line, 3.0 × 10⁻⁶ for dashed green line, and 3.0 × 10⁻⁵ for single-dashed blue line). The purple solid line was calculated by assuming that all ions are inert and do not interact with each other. The raw data and the experimental conditions used in the calculations are listed in Tables S2 and S3.

Next, the fitting was performed on the model described in Section 3.2.1.2 by considering both of the precipitation and the triple ion formation equilibria. Here, we assume that the molar ionic conductivities of the triple ions, $\lambda_{\text{Li}_{2}\text{F}^{ + }}$ and $\lambda_{{\text{LiF}_{2}}^{ - }}$, are given as follows:   

\begin{align} \lambda_{\text{Li${_{2}}$F${^{+}}$}} & = \lambda_{\text{T}}^{\infty} - S_{\text{T}}\sqrt{c_{\text{Li${_{2}}$F${^{+}}$}}},\\ \lambda_{\text{LiF${_{2}}{^{-}}$}} & = \lambda_{\text{T}}^{\infty} - S_{\text{T}}\sqrt{c_{\text{LiF${_{2}}{^{-}}$}}}, \end{align} (61)

where   

\begin{align} \lambda_{\text{T}}^{\infty} & \equiv \lambda_{\text{Li${_{2}}$F${^{+}}$}}^{\infty} = \lambda_{\text{LiF${_{2}}{^{-}}$}}^{\infty},\\ S_{\text{T}} & \equiv S_{\text{Li${_{2}}$F${^{+}}$}} = S_{\text{LiF${_{2}}{^{-}}$}}. \end{align} (62)

The fitting was performed with a four-parameter set, K_SP,LiF, K_T, $\lambda_{\text{T}}^{\infty }$, and S_T. The refined parameters with asymptotic standard errors are shown in the caption of the Fig. S2. The fitting appears, at first glance, to be in good agreement with the experimental results (Fig. S2). However, the calculation with the refined parameters, K_SP,LiF and K_T, indicates that no precipitate occurs at n_F,t/n_Li,t ≤ 0.25 and n_F,t/n_Li,t = 4 under the experimental conditions. Hence, the fitting was reperformed on the same in the limited range of 0.33 < n_F,t/n_Li,t < 3.5. The refined results are shown as the solid red line in Fig. 2. The experimental data are well reproduced on the model, indicating the generation of LiF as well as the triple ions (Li₂F⁺, ${\text{LiF}_{2}}^{ - }$) in this range of n_F,t/n_Li,t. The refined values of K_SP,LiF and K_T were close to those obtained by the fitting for the entire range of n_F,t/n_Li,t (see the caption of Fig. S2).

Figure 2.

The refined results on the model with both of the precipitation and the triple ion formation equilibria (solid red line). The refined parameters are: K_SP,LiF = (7 ± 1) × 10⁻⁷, K_T = (9 ± 4) × 10⁵, $\lambda_{\text{T}}^{\infty } = 22 \pm 3$ S cm² mol⁻¹, and S_T = (1.0 ± 0.4) × 10² S cm² mol⁻¹ M^−1/2. The lines were calculated with the center values of the refined parameters. The blue dashed lines are the refined values on the model by considering the triple ion formation without the precipitation. Note that the experimental data (square dots) are the same as those shown in Fig. 1. The raw data and the experimental conditions used in the calculations are listed in Tables S2 and S3.

We also attempted to reproduce the conductometric data in the ranges of n_F,t/n_Li,t ≤ 0.33 and 3.5 ≤ n_F,t/n_Li,t, where no precipitation is expected based on the refined values of K_SP,LiF and K_T, though the triple ions may be generated. The dashed blue lines in Fig. 2 are the refined ones by assuming that $n_{{\text{LiF}_{2}}^{ - }} \approx 0$ at small n_F,t/n_Li,t and $n_{\text{Li}_{2}\text{F}^{ + }} \approx 0$ for large n_F,t/n_Li,t on the models described in Limited cases A and B in Section 3.2.1.2, respectively. In this case, we used the values of K_T, $\lambda_{\text{T}}^{\infty }$, and S_T refined by the fitting for the data in the limited range of 0.33 < n_F,t/n_Li,t < 3.5. The blue lines well reproduce the experimental results. Small discrepancy between the red and blue calculated lines is due to the approximations used in Limited cases A and B in Section 3.2.1.2. The concentration profile of each ion is shown in Fig. 3.

Figure 3.

The calculated concentration profile of the ions during the titration of the diluted Li⁺/F⁻ hybrid electrolyte solution with the CsF solution.

In conclusion, it has been confirmed that the precipitation does not occur in the range of n_F,t/n_Li,t < 0.3 and 4 < n_F,t/n_Li,t due to the Li₂F⁺ and LiF₂⁻ triple ion formation, respectively. The addition of excess amounts of Li⁺ or F⁻ ions to LiF precipitate-containing suspensions solubilizes LiF by the formation of the triple ions Li₂F⁺ or LiF₂⁻.

In this study, we have discussed the ion equilibrium on the diluted Li⁺/F⁻ hybrid electrolyte solutions for the conductometric titration. The proposed model and the refined values of the equilibrium constants may be used to roughly evaluate the ion equilibrium situation of the original non-diluted Li⁺/F⁻ hybrid electrolyte solutions (mixtures of 1 M LiFSA and 50 mM CsF at various volume ratios). The photo images on the mixing of CsF and LiFSA solutions have been reported by Kawasaki et al. at different Li⁺/F⁻ molar ratios (Fig. S3).¹⁴ The data indicate that no precipitation occurs at least at n_F,t/n_Li,t < 0.15 even at such high concentrations, which is almost in agreement with the rough evaluation.

4.2 Mg²⁺/F⁻ hybrid electrolyte solution

Figure 4 (square dots) shows the experimental results of the conductometric titration of an Mg(TFSA)₂ solution with a CsF solution. The κ values for the titration were first analyzed on the model described in Section 3.2.2.1 by assuming only the precipitation equilibrium (dashed lines in Fig. 4) with various values of $K_{\text{SP,MgF}_{2}}$. The calculated curves did not reproduce the experimental data in the whole.

Figure 4.

Experimental results for the concentration dependence of the conductivity (square dots). The lines show the theoretical values calculated on the model by assuming only precipitation equilibrium, where $K_{\text{SP,MgF}_{2}} = 10^{ - 10}$ for dotted red, 10⁻⁹ for dashed green, and 10⁻⁸ for single-dashed blue. For comparison, the conductivity calculated by assuming that all ions are inert and do not interact with each other is shown by the solid purple line. The raw data and the experimental conditions used in the calculations are listed in Tables S4 and S5.

Therefore, in next, the fitting was performed on the model described in Section 3.2.2.2 that considers the associated ion formation as well as the precipitation. The molar ionic conductivity of the associated ion, $\lambda_{\text{MgF}^{ + }}$, was supposed as follows:   

\begin{equation} \lambda_{\text{MgF${^{+}}$}} = \lambda_{\text{MgF${^{+}}$}}^{\infty} - S_{\text{MgF${^{+}}$}}\sqrt{c_{\text{MgF${^{+}}$}}}. \end{equation} (63)

The concentration calculation on the model by considering both of the precipitation and the association equilibria occur requires the solution of the cubic equation (Eq. 54), which makes it slightly difficult to fit the data. Therefore, we performed the fitting for the data at a limited range of n_F,t/n_Mg,t with a three-parameter set, K_A, $\lambda_{\text{MgF}^{ + }}^{\infty }$, and $S_{\text{MgF}^{ + }}$, on the model described in Limited case of no precipitation in Section 3.2.2.2, in which the association equilibrium is only considered by ignoring the precipitation. The refined parameters of the associated ion were then used to perform the fitting for the experimental data not used in the previous fitting on the model with both of the precipitation and the association ion formation occur (Section 3.2.2.2). The $K_{\text{SP,MgF}_{2}}$ was used as the fitting parameter. Then, it was confirmed from the refined equilibrium constant that the assumptions regarding the precipitation did not contradict.

Figure 5 shows the refined results. The dashed blue line is the curve refined on the model of limited case of no precipitation in Section 3.2.2.2 with the assumption of no precipitation. The fitting was performed for the experimental data of n_F,t/n_Mg,t ≤ 0.75 (square blue dots). The solid red line is the curve refined on the model in Section 3.2.2.2 by considering both the precipitation and association ion formation. The fitting was performed for the experimental data of n_F,t/n_Mg,t ≥ 1.0 (square black dots). The concentration calculation with the refined $K_{\text{SP,MgF}_{2}}$ and K_A shows that no precipitation occurs at n_F,t/n_Mg,t ≤ 0.8. Kawasaki et al. evaluated the value of $K_{\text{SP2,MgF}_{2}} = c_{\text{MgF}^{ + }}c_{\text{F}^{ - }}/(c^{\circ} )^{2} = 5 \times 10^{ - 6}$ by NMR measurement.¹⁴ The value is very close to that refined here ($K_{\text{SP2,MgF}_{2}}( = K_{\text{SP,MgF}_{2}}K_{\text{A}}) = 3 \times 10^{ - 6}$). The concentration profile of each ion is shown in Fig. 6.

Figure 5.

The refined results on the models by assuming the association equilibrium with (solid red line) and without (dashed blue line) precipitation. The fitting parameters are: $K_{\text{SP,MgF}_{2}} = 3 \times 10^{ - 9}$, K_A = 1 × 10³, $\lambda_{\text{MgF}^{ + }}^{\infty } = 20$ S cm² mol⁻¹, and $S_{\text{MgF}^{ + }} = 1 \times 10^{2}$ S cm² mol⁻¹ M^−1/2. Note that the experimental data (square black and blue dots) are the same as those shown in Fig. 4. The raw data and the experimental conditions used in the calculations are listed in Tables S4 and S5.

Figure 6.

The concentration profile of the consisting ions during the titration of Mg(TFSA)₂ solution by CsF solution.

Consequently, it has been confirmed that the precipitation does not appear at n_F,t/n_Mg,t < 0.8 due to the MgF⁺ associated ion formation. That is, the addition of excess amounts of Mg²⁺ to MgF₂-containing suspension solubilizes MgF₂ by the formation of the associated ion MgF⁺.

5. Conclusions

Conductometric titrations were carried out to prove the triple and associate ion formation in Li⁺/F⁻ and Mg²⁺/F⁻ hybrid electrolyte solutions, respectively. The experimental data of κ were well reproduced by the values calculated on the model involving both triple/associated ion formation and precipitation equilibria. The refined equilibrium constants indicate that the low solubility products (LiF and MgF₂) are solubilized by the addition of excess amounts of Li⁺ or F⁻ for LiF to form the triple ions (Li₂F⁺ and LiF₂⁻) and Mg²⁺ for MgF₂ to form the associated ion (MgF⁺).

In the conductivity calculations in this work, the c_I dependence of Λ_I was considered on Kohlrausch’s square root law, which is established by considering the ionic atmosphere for single salt. In our calculations, we used the S parameters that were experimentally determined by the c_I dependence of Λ_I for the given single salt (Fig. S1). However, in the titration solutions used here, the ions interact with not only the counter ion of the original salt but the other ones. Unfortunately, it would be very difficult to evaluate the real value of S parameters in the titration solutions at any arbitrarily titration points. This is one of problems in this approach with conductometric titration to analyze the ion equilibrium. The c_I dependence of Λ_I should become negligible by decreasing c_I. However, conductometry becomes difficult and triple/associated ions are hardly formed in such diluted solutions.

Despite the above problem and some approximations used in the limited cases in the models for the analysis, we can evaluate the order of the equilibrium constants of the triple/associated ion formation as well as the precipitation. In this work, we have proved that the addition of excess amounts of cation/anion can dissolve the F⁻-containing precipitates. This concept and method are very useful for the preparation of electrolyte solution containing high total concentrations of F⁻ and can greatly contribute to the development of fluoride shuttle batteries.

Acknowledgment

This work is based on result obtained from projects “Research and Development Initiative for Scientific Innovation of New Generation Batteries (RISING2)”, JPNP16001, and “RISING3”, JPNP21006, commissioned by the New Energy and Industrial Technology Development Organization (NEDO).

Data Availability Statement

The data that support the findings of this study are openly available under the terms of the designated Creative Commons License in J-STAGE Data at https://doi.org/10.50892/data.electrochemistry.22186777.

• 1) Fluoride shuttle batteries are expected to be innovative with high energy density superior to lithium-ion ones as on-board power-source for electric vehicles. However, the low solubility of fluoride ion in organic solvents makes it difficult to choose electrolytes. Kawasaki et al. have reported that no precipitation appears in γ-butyrolactone solution of CsF at high concentrations in the presence of excess amounts of Li⁺ or Mg²⁺ cation [M. Kawasaki, et al., J. Electrochem. Soc., 169, 110508 (2022).]. Such solutions are called hybrid electrolyte ones. In this study, we performed conductometric titration of Li⁺- or Mg²⁺-containing solutions with CsF solutions to elucidate the ion equilibrium. The titration curves were analyzed on models including precipitation of LiF or MgF₂ and formation of Li₂F⁺ and LiF₂⁻ triple ion or MgF⁺ associated one to evaluate the corresponding equilibrium constants. The results indicate that no precipitation appears in certain ranges of the F⁻/Li⁺ or F⁻/Mg²⁺ molar ratio by the triple/associated ion formation. Details of the conductometric analysis are discussed together with some problems involved in the proposed method.

CRediT Authorship Contribution Statement

Yuko Yokoyama: Data curation (Lead), Methodology (Equal), Writing – original draft (Lead), Writing – review & editing (Lead)

Mitsuo Kawasaki: Conceptualization (Supporting), Supervision (Equal), Writing – review & editing (Supporting)

Takeshi Abe: Funding acquisition (Lead), Supervision (Equal)

Zempachi Ogumi: Supervision (Equal)

Kenji Kano: Conceptualization (Lead), Methodology (Equal), Writing – original draft (Supporting), Writing – review & editing (Lead)

Conflict of Interest

The authors declare no conflict of interest in the manuscript.

Funding

New Energy and Industrial Technology Development Organization: JPNP16001

New Energy and Industrial Technology Development Organization: JPNP21006

Footnotes

Y. Yokoyama and M. Kawasaki: ECSJ Active Members

T. Abe and K. Kano: ECSJ Fellows

Z. Ogumi: ECSJ Honorary Member

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