Superionic Solid Electrolyte Li7La3Zr2O12 Synthesis and Thermodynamics for Application in All-Solid-State Lithium-Ion Batteries

Solid-state reaction was used for Li7La3Zr2O12 material synthesis from Li2CO3, La2O3 and ZrO2 powders. Phase investigation of Li7La3Zr2O12 was carried out by x-ray diffraction (XRD), scanning electron microscopy (SEM) and energy-dispersive x-ray spectroscopy (EDS) methods. The thermodynamic characteristics were investigated by calorimetry measurements. The molar heat capacity (Cp,m), the standard enthalpy of formation from binary compounds (ΔoxHLLZO) and from elements (ΔfHLLZO), entropy (S0298), the Gibbs free energy of the Li7La3Zr2O12 formation (∆f G0298) and the Gibbs free energy of the LLZO reaction with metallic Li (∆rGLLZO/Li) were determined. The corresponding values are Cp,m = 518.135 + 0.599 × T − 8.339 × T−2, (temperature range is 298–800 K), ΔoxHLLZO = −186.4 kJ·mol−1, ΔfHLLZO = −9327.65 ± 7.9 kJ·mol−1, S0298 = 362.3 J·mol−1·K−1, ∆f G0298 = −9435.6 kJ·mol−1, and ∆rGLLZO/Li = 8.2 kJ·mol−1, respectively. Thermodynamic performance shows the possibility of Li7La3Zr2O12 usage in lithium-ion batteries.


Introduction
The commercial history of the lithium-ion battery was started in 1991 by Sony [1]. Since then, a lot of effort has been directed to improving the electrochemical performance of lithium-ion batteries [2]. One of the perspective methods of stabilizing lithium-ion battery electrochemical characteristics and safety is to apply solid-state inorganic electrolyte instead of liquid organic electrolyte as the traditional electrolyte for commercial lithium-ion batteries [3][4][5][6][7]. Some solid-state electrolytes have high ionic conductivity in an order of magnitude of~10 −3 S·cm −1 [8] in comparison to liquid electrolyte [9].
LLZO solid-state electrolyte attracts high attention due to its relatively high electrochemical properties. Though LLZO has lower ionic conductivity in comparison with organic liquid electrolyte (~10 −4 versus~10 −2 S·cm −1 , respectively [9]), it provides high safety performance, high chemical stability against metallic lithium, a wide electrochemical potential window, low electronic conductivity, and high stability with moisture in the air; LLZO prevents lithium dendrite growth due to high mechanical strength [11][12][13][14][15].
In this work, synthesis, structure studies and thermodynamics calculations of tetragonal Li 7 La 3 Zr 2 O 12 were performed.

Materials and Methods
Tetragonal LLZO electrolyte was produced by solid-state synthesis as one of the commonly used synthesis methods for investigation and mass manufacture [39][40][41][42][43][44][45][46][47]. Initial materials Li 2 CO 3 (Xilong Sci., 99%), La 2 O 3 (ReLAB, 99.99%), and ZrO 2 (Sinopharm, 99.9%) in stoichiometric ratio were used as sources for Li, La, and Zr, respectively. Excess of 10 wt.% of lithium was initially added to precursor to avoid lithium loss during the synthesis process at high temperatures. Lanthanum oxide was preliminarily dried at 900 • C for 24 h. The mentioned materials were mechanically milled in an agate mortar and then dissolved in acetic acid with subsequent magnetic stirring at 90 • C for 12 h to provide a homogeneous solution. Excess acetic acid was evaporated at 110 • C to get dry precursor powder. Dried precursor was then mechanically milled in an agate mortar and put into an alumina crucible for heat treatment. A muffle furnace (Nabertherm, Lilienthal, Germany) was used for solidstate reaction at air atmosphere. First, the precursor was slowly heated (heat rate was 0.5 • C/min) to 130 • C for 3 h to evaporate the remaining acetic acid. Then, the precursor was heated (heat rate was 2 • C/min) to 900 • C for 8 h to provide solid-state reaction.
In this work, synthesis, structure studies and thermodynamics calculations of tetragonal Li7La3Zr2O12 were performed.

Materials and Methods
Tetragonal LLZO electrolyte was produced by solid-state synthesis as one of the commonly used synthesis methods for investigation and mass manufacture [39][40][41][42][43][44][45][46][47]. Initial materials Li2CO3 (Xilong Sci., 99%), La2O3 (ReLAB, 99.99%), and ZrO2 (Sinopharm, 99.9%) in stoichiometric ratio were used as sources for Li, La, and Zr, respectively. Excess of 10 wt.% of lithium was initially added to precursor to avoid lithium loss during the synthesis process at high temperatures. Lanthanum oxide was preliminarily dried at 900 °C for 24 h. The mentioned materials were mechanically milled in an agate mortar and then dissolved in acetic acid with subsequent magnetic stirring at 90 °C for 12 h to provide a homogeneous solution. Excess acetic acid was evaporated at 110 °C to get dry precursor powder. Dried precursor was then mechanically milled in an agate mortar and put into an alumina crucible for heat treatment. A muffle furnace (Nabertherm, Lilienthal, Germany) was used for solid-state reaction at air atmosphere. First, the precursor was slowly heated (heat rate was 0.5 °C/min) to 130 °C for 3 h to evaporate the remaining acetic acid. Then, the precursor was heated (heat rate was 2 °C/min) to 900 °C for 8 h to provide solidstate reaction.
TAM IV Microcalorimeter (TA Instruments, Shanghai, China) was used for calorimetric investigation. Measurement parameters were as follows: temperature is 298 K, volume of the cell is 20 mL. An aqueous solution of 1 mol·dm −3 HCl was filled in the ampoule at calorimetric cell. The dissolution process of the LLZO powder was started after thermal equilibrium was established. Dissolution enthalpy value was obtained from thermoelectromotive force data during the dissolution process, providing the heat dissolution curve.

Results
The XRD pattern of synthesized LLZO is shown at Figure 1. According to diffraction data, LLZO has a I41/acd space group. The vertical lines at the bottom are related to PDF ). Images of the microstructure performance of LLZO powder were taken with a scanning electron microscope (SEM) Tescan MAIA3 (Tescan, Brno, Czech Republic) with secondary electron detection. Bruker XFlash 6-10 (Bruker, Karlsruhe, Germany) was used for energydispersive X-ray spectroscopy (EDS).
TAM IV Microcalorimeter (TA Instruments, Shanghai, China) was used for calorimetric investigation. Measurement parameters were as follows: temperature is 298 K, volume of the cell is 20 mL. An aqueous solution of 1 mol·dm −3 HCl was filled in the ampoule at calorimetric cell. The dissolution process of the LLZO powder was started after thermal equilibrium was established. Dissolution enthalpy value was obtained from thermoelectromotive force data during the dissolution process, providing the heat dissolution curve.

Results
The XRD pattern of synthesized LLZO is shown at Figure 1. According to diffraction data, LLZO has a I4 1 /acd space group. The vertical lines at the bottom are related to PDF #00-064-0140. The peak indexes and interplanar distances are shown in the Supplementary Materials (Table S1). Synthesized material contains 4 wt.% of La 2 O 3 impurity after solidstate reaction. #00-064-0140. The peak indexes and interplanar distances are shown in the Supplementary Materials (Table S1). Synthesized material contains 4 wt.% of La2O3 impurity after solid-state reaction. SEM images of LLZO powder are shown in Figure 2, made at 2×, 3.5×, 10× and 11.5× magnification, respectively. All images were performed at 10 keV landing energy.  SEM images of LLZO powder are shown in Figure 2, made at 2×, 3.5×, 10× and 11.5× magnification, respectively. All images were performed at 10 keV landing energy.  (Table S1). Synthesized material contains 4 wt.% of La2O3 impurity after solid-state reaction. SEM images of LLZO powder are shown in Figure 2, made at 2×, 3.5×, 10× and 11.5× magnification, respectively. All images were performed at 10 keV landing energy.   Figure 3 are evenly distributed. The carbon in Figure 3b is electrically conductive carbon tape for sample holder. Elemental analysis of EDS spectra is shown at Table 1. EDS elemental analysis of the LLZO powder shows lanthanum excess in the solid electrolyte powder, expressed in terms of Li7La3Zr2O12 and La2O3 compounds. Elemental analysis based on Table 1 shows an excess of 3.1 wt.% of lanthanum oxide (III).

The Standard Formation Enthalpy
The formation enthalpy of Li7La3Zr2O12 (ΔoxHLLZO) from Li2CO3, La2O3, and ZrO2 is calculated according to Equation (1) from the Experimental Section. The subscript ox means "oxides", which relates to the initial compounds from Equation (1).
The following thermodynamic cycle was used for enthalpy calculation, Figure 4:   Table 1 shows an excess of 3.1 wt.% of lanthanum oxide (III).

The Standard Formation Enthalpy
The formation enthalpy of Li 7 La 3 Zr 2 O 12 (∆ ox H LLZO ) from Li 2 CO 3 , La 2 O 3, and ZrO 2 is calculated according to Equation (1) from the Experimental Section. The subscript ox means "oxides", which relates to the initial compounds from Equation (1).
The following thermodynamic cycle was used for enthalpy calculation, Figure 4: the subscript (aq) indicates "aqueous". The calorimeter was used for the standard enthalpy (∆ d H LLZO ) measurement. The received value after calorimetry measurement was equal to −1911 ± 37 J·g −1 , Table 2.
The formation enthalpy value of ∆ ox H LLZO is estimated by the next formula: The  Table 2 to compare with the measured and calculated values in this work. The value of the formation enthalpy from binary oxides ∆ ox H LLZO has the same order as corresponding values for lithium zirconate compounds and complex oxides (Table 3), thus it can be concluded that the measurements are correct. The subscripts (s) mean "solid".
Finally, the enthalpy of Li 7 La 3 Zr 2 O 12 formation from elements can be calculated by the following formula: The corresponding handbook's materials were used to define the standard enthalpies [53], Table 4. The subscripts (s) mean "solid".
The formation enthalpy value of the Li 7 La 3 Zr 2 O 12 compound, calculated by formula (8) is −9327.65 ± 7.9 kJ·mol −1 , Table 4. The enthalpy of formation value, rated by Equation (8), can be recommended for use in further thermodynamic calculations of Li 7 La 3 Zr 2 O 12 reactivity. Figure 5 shows the isobaric heat capacity of the Li 7 La 3 Zr 2 O 12 as a function of temperature (C p = f (T)). Pay attention to the certain amount of La 2 O 3 ( Figure 1 and Table 1) in LLZO synthesized powder material, the measured isobaric heat capacity for the two-phase system must be recalculated by the following additive rule: where C p is a specific heat capacity (p = const), and m is mass. In our case, the two-phase system consists of the solid electrolyte compound (LLZO) and La 2 O 3 . Thus, the heat capacity of Li 7 La 3 Zr 2 O 12 is expressed from Equation (9) as:

The Isobaric Heat Capacity
The calculated from tabular data [53] heat capacity from Equation (16) is shown on Figure 5 and Table 5.   (N-K)  The Neumann-Kopp rule and recalculated heat capacity of Li7La3Zr2O12 are in good correlation. Experimental data is for the LLZO compound with La2O3 impurity. The heat capacity temperature dependence (Equation (16)) was calculated using tabular data [53,55]. XRD and EDS quantitative analysis gives accurate enough results to define a small quantity of impurity compounds in the material.

Entropy
The Third Law of Thermodynamics states, "The entropy of a perfect crystal is zero when the temperature of the crystal is equal to absolute zero (0 K)." Thus, the entropy absolute value can be valued by the equation: where S is entropy, Tk is temperature of the k-th phase transition (0 < Tk < T), and ΔHk is enthalpy of the k-th phase transition. The Neumann-Kopp rule for entropy calculation can be expressed as follows (considering absence of phase transition at calculating temperature range): The impurity compound weight can be recalculated from the total weight of the sample, with the known mass fraction of lanthanum oxide, ω(La 2 O 3 ): and Considering Equations (11) and (12), Equation (13) can be expressed as follows: Equation (13) shows, that the Li 7 La 3 Zr 2 O 12 heat capacity C p (LLZO) can be evaluated by the measured heat capacity (C p ), tabulated heat capacity of La 2 O 3 C p (La 2 O 3 ), and La 2 O 3 mass fraction ω(La 2 O 3 ). The dependence of La 2 O 3 specific heat capacity from temperature is required for Equation (13) calculation. For this, tabular data is required to define temperature dependence for the lanthanum oxide heat capacity [53]. The heat capacity polynomial, commonly used for the low temperature range (for 300-800 K in our case) can be expressed as follows: where a, b, and c are empirical coefficients, and T is the absolute temperature. The La 2 O 3 received coefficients are a = 119.604 J·mol −1 ·K −1 , b = 14.514 × 10 −3 J·mol −1 ·K −2 , and c = 13.452 × 10 5 J·mol −1 ·K. Considering the La 2 O 3 impurity presence, the LLZO heat capacity can be recalculated via Equations (13) and (14) for the 300-800 K temperature interval. According to XRD and EDS data ( Figure 1 and Table 1, respectively) LLZO contains about 3.1 ± 0.12 wt.% La 2 O 3 . Figure 5 and Table 5 shows measured and recalculated LLZO heat capacity temperature dependence. The Neumann-Kopp (N-K) rule was used for the empirical value calculation of the heat capacity. The N-K rule approves "that the molecular heat capacity of a solid compound is the sum of the atomic heat capacities of the elements composing it; the elements having atomic heat capacities lower than those required by the Dulong-Petit law retain these lower values in their compounds." [54]. This rule commonly gives reproducible results for room temperatures, not for high temperatures. To achieve more accurate results, binary materials were used instead of single elements (accurate results are usually obtained for the same aggregate state of materials): C p is a molar heat capacity (p = const), n is a stoichiometric coefficient, and CO and BO are complex and binary oxide, respectively. Equation (15), considering Equation (1), can be expressed for LLZO as follows: The calculated from tabular data [53] heat capacity from Equation (16) is shown on Figure 5 and Table 5.   (16)) was calculated using tabular data [53,55]. XRD and EDS quantitative analysis gives accurate enough results to define a small quantity of impurity compounds in the material.

Entropy
The Third Law of Thermodynamics states, "The entropy of a perfect crystal is zero when the temperature of the crystal is equal to absolute zero (0 K)." Thus, the entropy absolute value can be valued by the equation: where S is entropy, T k is temperature of the k-th phase transition (0 < T k < T), and ∆H k is enthalpy of the k-th phase transition. The Neumann-Kopp rule for entropy calculation can be expressed as follows (considering absence of phase transition at calculating temperature range): where BO is the binary oxide compound (see Equation (15)). Equation (18) can be rewritten taking into account Equations (15) and (16): The Li 7 La 3 Zr 2 O 12 entropy is equal to 607.18 J·mol −1 ·K −1 (T = 298 K) by calculating Equation (19) using tabulated data [53,55]. The additive rule for entropy calculation can be used if the following term is met: the complex compound molar volume slightly differs of the molar volumes sum of binary compounds [55]. Thus, the molar volume for Li 2 CO 3 is 35.0 cm 3 ·mol −1 (density is ρ = 2.11 g·cm −3 [56]), for La 2 O 3 is 50.1 cm 3 ·mol −1 (density is ρ = 6.51 g·cm −3 [57]), for ZrO 2 is 21.2 cm 3 ·mol −1 (density is ρ = 5.56 g·cm −3 [57]), and for Li 7 La 3 Zr 2 O 12 is 165.0 cm 3 ·mol −1 (density is ρ = 5.09 g·cm −3 [58]). The sum of the molar volumes of Li 2 CO 3 , La 2 O 3, and ZrO 2 with their corresponding stoichiometric coefficients is 240.05 cm 3 ·mol −1 and differs about 45.5% of the LLZO molar volume, which does not allow one to apply the additive rule.
Excepting the additive calculation rule, the W. Herz rule can be used for the LLZO entropy calculation [59]: where K H is the Herz constant, M is molar mass, C p,298 is isobaric heat capacity, and m is atoms per formula. The Herz constant K H has a good correlation with average values of anion molar mass [60]: Considering C p,298 from Table 5 and Equation (21), calculated by Equation (20) the LLZO entropy is equal to 362.3 J mol −1 ·K −1 . The calculated value of LLZO entropy by the W. Herz rule is in good correlation with Ref. [60]. Hence, the N-K rule cannot be used for the entropy calculations, as follows from molar masses principle.

The Standard Gibbs Free Energy
Calculated formation enthalpy and entropy allows one to rate the standard Gibbs free energy (∆ f G 0 298 ) of LLZO formation (T = 298 K): For Equation (22), the ∆ f G 0 298 value of LLZO is equal to −9435.6 kJ·mol −1 . The stability against metallic lithium can be estimated by the Gibbs free energy calculation of the following reaction at room temperature: The Gibbs free energy of reaction (∆ r G LLZO/Li ) can be expressed as the difference between the and the Gibbs energy values of reactants and resultants of the reaction. The ∆ f G 0 298 for single elements is equal to zero, for Li 2 O is −561.2 kJ·mol −1 , and for La 2 O 3 is −1706.7 kJ·mol −1 [53]. The Li 7 La 3 Zr 2 O 12 Gibbs free energy has been calculated above. Thus, the Gibbs free energy for reaction (23) is ∆ r G LLZO/Li = 8.2 kJ·mol −1 ; this means that the reaction is thermodynamically impossible. Finally, Li 7 La 3 Zr 2 O 12 is stable against metallic lithium at room temperature.

Conclusions
The thermodynamic characteristics were determined for Li 7 La 3 Zr 2 O 12 solid-state electrolyte material for lithium-ion battery. Solid-state reaction was used as the synthesis method of Li 7 La 3 Zr 2 O 12 from Li 2 CO 3 , La 2 O 3, and ZrO 2 . The synthesized material had 3.1 wt.% of the lanthanum oxide (La 2 O 3 ) impurity according to XRD and EDS data. Probably, this amount of La 2 O 3 is unreacted oxide from the synthesis process. The enthalpy of Li 7 La 3 Zr 2 O 12 formation from binary oxides (and from Li 2 CO 3 ) ∆ ox H LLZO and from the elements ∆ f H LLZO were calculated according to the measured enthalpy of dissolution of reagents and the products of the Li 7 La 3 Zr 2 O 12 formation reaction. The obtained values are equal to −186.4 ± 7.3 kJ·mol −1 and −9327.65 ± 7.9, respectively. The formation enthalpy from binary oxides ∆ ox H LLZO is in good correlation with similar zirconate compounds, which confirms the correctness of the measurements.
The recalculated LLZO heat capacity considering La 2 O 3 presence is in good correlation with that calculated by the Neumann-Kopp rule. Finally, the temperature dependence of the LLZO heat capacity can be expressed by the formula C p (T) = 518.135 + 0.599 × T − 8.339 × T −2 (T is absolute temperature). The LLZO entropy is S 0 298 = 362.3 J·mol −1 ·K −1 , the Gibbs free energy of formation of Li 7 La 3 Zr 2 O 12 is −9435.6 kJ mol −1 . Li 7 La 3 Zr 2 O 12 material is stable against metallic lithium, according to the Gibbs free energy of the LLZO reaction with metallic Li. All thermodynamic values and functions measured and calculated for Li 7 La 3 Zr 2 O 12 can be used for modelling and further calculations of all-solid-state batteries.

Data Availability Statement:
The data presented in this study are available on request from the corresponding author.

Conflicts of Interest:
The authors declare no conflict of interest.