Elastic properties of diverse sodium-ion conductive materials: a first-principles study

Abstract

All-solid-state sodium batteries are expected to be low-cost energy-storage devices because they use resource-rich sodium. However, mechanical degradation remains a critical issue, and solid electrolytes must exhibit sufficient ductility to accommodate the expansion and contraction of the electrode active materials at their interface. In this study, the elastic properties of various sodium-ion conductors were evaluated using first-principles calculations and compared based on their anion elements, compositions, crystal phases, and other factors. The elastic moduli of approximately 40 sodium-ion conductors were calculated. The results revealed that sulfide-, chloride-, and hydride-based materials generally exhibit low elastic moduli, whereas oxide materials exhibit higher values. Among the evaluated sulfide compounds, β-Na₃PS₄ crystals exhibit a relatively high Pugh's ratio, suggesting sufficient ductility to resist mechanical degradation. Furthermore, Na_2.875Sb_0.875W_0.125S₄ exhibited lower elastic moduli and higher Poisson's and Pugh's ratios than Na₃SbS₄. Unsupervised clustering of the bulk and shear moduli identified two distinct categories of oxide electrolytes: those with medium elastic moduli and those with large elastic moduli. Notably, NaAl₁₁O₁₇ exhibited the highest elastic modulus and Pugh's ratio among all the ox electrolytes, which was attributed to its high oxide ion concentration. A nonlinear correlation was observed between the elastic moduli and the mean atomic volume across most materials; however, hydrides deviated from this trend, exhibiting low elastic moduli despite their small mean atomic volumes.

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Introduction

Driven by the global transition to renewable energy, the demand for sustainable energy storage devices has increased significantly in recent years. Among the various types of energy storage devices, all-solid-state sodium batteries (ASSBs) are expected to be promising alternatives to conventional lithium-ion batteries owing to their safety, high energy densities, and the abundance of sodium resources.^1–3 The use of solid electrolytes instead of liquid organic electrolytes offers ASSBs various advantages, including improved safety and high thermal stability. In addition, they enable the use of sodium metal as a negative electrode, which leads to high energy densities. However, several issues remain regarding the development of high-performance ASSBs. Mechanical degradation at the solid–solid interface between the electrode active materials and solid electrolytes is caused by the expansion and contraction of the electrode active material during charging and discharging. These volumetric changes induce cracks and voids inside the electrode active materials and solid electrolytes.⁴ To prevent mechanical degradation, solid electrolytes must have sufficient deformability and ductility to withstand the expansion and contraction of the electrode active materials.

The elastic modulus is a fundamental property that quantifies the resistance of a material to deformation under microscopic elastic strain. During elastic deformation, the stress and strain are proportional, with the elastic modulus serving as a proportionality constant. Solid electrolytes with lower elastic moduli deform more readily under stress, facilitating better interfacial contact with active electrode materials and accommodating volumetric changes during cycling.

The elastic properties are described by various proportionality constants that depend on the plane and direction of the stress and strain. These constants are organized into a four-dimensional elastic tensor, which is often simplified to a two-dimensional form for easier interpretation and computation. The Voigt–Reuss–Hill approximation is widely used to estimate elastic moduli from the tensor components.^5–7 The principal elastic moduli used to evaluate crystalline materials include the polycrystalline Young's modulus (E), bulk modulus (B), and shear modulus (G). Uniaxial Young's modulus represents the proportionality between stress and strain in the uniaxial direction, whereas the polycrystalline Young's modulus (E) provides an average value for polycrystalline materials. B quantifies the relationship between isotropic stress and volumetric strain, whereas G describes the angular displacement under an equilibrium shear force. Another parameter essential for mechanical characterization is Poisson's ratio (ν), which represents the ratio of transverse strain to axial strain under uniaxial stress.

Pugh's ratio, defined as the ratio of the bulk modulus to the shear modulus (B/G),⁸ is a critical metric for evaluating the mechanical ductility of solid electrolytes.⁹ Initially proposed by Pugh for polycrystalline metals, this criterion assesses the tendency of a material for plastic deformation, which occurs when the material is strained beyond its elastic limit. Pugh suggested that materials with B/G > 1.75 are more likely to exhibit plastic deformation.⁸ However, while this standard is well-established for metals, its applicability to ceramic materials, such as those used in solid electrolytes, remains unverified.

Deng et al.¹⁰ performed comprehensive first-principles calculations of the elastic moduli for representative solid electrolytes, including principal sodium-ion conductors. Their work provided a solid foundation for understanding the mechanical properties of these materials. Building on this important groundwork, the present study aims to extend the scope of the investigation to include recently developed sodium-ion conductive solid electrolytes, thereby deepening the insight into their elastic behavior and potential applications.

Sodium-ion-conductive solid electrolytes have been extensively studied, mainly in oxide and sulfide materials. β″-alumina¹¹ and NASICON,^12,13 discovered in the 1970s, are well-known oxide electrolytes with high ionic conductivity. Sulfide electrolytes are promising due to their higher ionic conductivity and mechanical ductility than those of oxide electrolytes.¹⁴ Furthermore, sulfide electrolytes exhibit greater formability, and they can achieve densification simply by pressing at room temperature.¹⁵ Na₃PS₄ glass-ceramic electrolytes with high sodium-ion conductivity have enabled the development of ASSBs operable at room temperature.^16,17 Various Na₂S–M_xS_y (M: the central group 13, 14, or 15 cations) sulfide electrolytes have been reported, such as Na₅AlS₄ and Na₄SnS₄.^18–28 Na₃SbS₄ is considered especially promising because it combines high ionic conductivity (>10⁻³ S cm⁻¹) with improved air stability, exhibiting minimal hydrogen sulfide (H₂S) evolution.^26–28 Furthermore, Na_2.88Sb_0.88W_0.12S₄ electrolytes, which are tungsten-substituted structures of Na₃SbS₄, exhibit ionic conductivity exceeding 10⁻² S cm⁻¹.^29–31 In addition to oxides and sulfides, various electrolytes incorporating other anionic elements—such as selenides, halides, and hydrides—have been reported. Selenides with compositions similar to those of sulfides, such as Na₃PSe₄ and Na₃SbSe₄.^32,33 Promising chloride sodium-ion conductors, such as Na₃MCl₆ (M = Y or Er), Na₂ZrCl₆, and NaTaCl₆, have recently been explored.^34–38 Borohydride electrolytes are also promising materials,^39–43 for example, the ionic conductivity of Na₂(CB₉H₁₀)(CB₁₁H₁₂) is higher than 7 × 10⁻² S cm⁻¹.⁴⁴

In this study, the elastic properties of various sodium-ion conducting solid electrolytes are evaluated using first-principles calculations. Materials were also compared based on factors such as anion elements, compositions, and crystal phases. Furthermore, the elastic moduli of the tungsten-substituted structures of Na₃SbS₄ (Na_3−xSb_1−xW_xS₄, x = 0.0625 and 0.125) were calculated to investigate the effect of the substitution on their mechanical properties. In addition, the distribution of the elastic moduli of sodium-ion conductive solid electrolytes is analyzed using unsupervised clustering. Correlations between the mean atomic volume (MAV) and elastic moduli are also examined, providing insights into the structural factors that directly determine the elastic properties.

Computational details

The elastic properties of crystal structures were obtained using first-principles calculations based on density functional theory (DFT).^45,46 All first-principles calculations were implemented using the Vienna ab initio simulation package (VASP)^47,48 with the generalized gradient approximation exchange correlation functional in the scheme of Perdew–Burke–Ernzerhof (PBE).^49,50 The projected augmented-wave (PAW) method^51,52 was adopted to describe the pseudopotential of inner-shell electrons. The cut-off energies for the plane wave basis set and electric energy convergence were set at 500 and 10⁻⁶ eV, respectively. The Monkhorst–Pack scheme⁵³ was adopted to determine the k-point distributions and irreducible Brillouin zones. Both the lattice constants and ionic positions of the ordinary crystal structures were fully relaxed, and the final forces on all the relaxed atoms were less than 0.01 eV Å⁻¹. The on-site Coulomb term (U) values of 2.50, 3.50, and 4.00 eV for Ti-3d, Zr-4d, and Nb-4d, respectively, were set based on previous studies.^54–57 Spin polarization was not considered in these calculations. The DFT-D3 method with Becke–Johnson damping was adopted to incorporate the van der Waals dispersion energy in the first-principles calculations.⁵⁸ MAV was defined as the ratio of the cell volume and the total number of atoms.¹⁰ Note that this cell volume was calculated from lattice constants after structural optimization.

The elastic properties were calculated through the following procedure. The original crystal structures were obtained from the Materials Project⁵⁹ and Inorganic Crystal Structure Database (ICSD).⁶⁰ Exceptionally, the Na₂ZrCl₆ crystal structure was generated using the supercell program⁶¹ based on the CIF obtained from ICSD, since the Na sites are randomly occupied and a specific configuration needed to be selected from possible candidates. The atomic geometries of crystal structures were then fully optimized using first-principles calculations. Subsequently, the elastic tensors were evaluated for the optimized structures, and the elastic properties of the polycrystalline phase were derived using the Voigt–Reuss–Hill approximation.

The following is an outline of the Voigt–Reuss–Hill approximation method.^5–7 The bulk modulus (B) and shear modulus (G) were obtained as the average values (the Hill's prediction values) of those calculated using Voigt's (B_V and G_V) and Reuss's (B_R and G_R) prediction methods (eqn (1)–(6)). In these equations, each component of the inverse elasticity tensor is represented by S_ij. The polycrystalline Young's modulus (E) and Poisson's ratio (ν) were estimated using eqn (7) and (8), respectively.

9B_V = (C₁₁ + C₂₂ + C₃₃) + 2(C₁₂ + C₂₃ + C₃₁) (1)

1/B_R = (S₁₁ + S₂₂ + S₃₃) + 2(S₁₂ + S₂₃ + S₃₁) (2)

B = (B_V + B_R)/2 (3)

15G_V = (C₁₁ + C₂₂ + C₃₃) − (C₁₂ + C₂₃ + C₃₁) + 3(C₄₄ + C₅₅ + C₆₆) (4)

15/G_R = 4(S₁₁ + S₂₂ + S₃₃) − 4(S₁₂ + S₂₃ + S₃₁) + 3(S₄₄ + S₅₅ + S₆₆) (5)

G = (G_V + G_R)/2 (6)

E = 9BG/(3B + G) (7)

ν = (3B − 2G)/2(3B + G) (8)

The universal anisotropy (A^U) was calculated using eqn (9).^62,63 If the crystal structure is completely isotropic, A^U = 0: the farther the A^U deviates from 0, the higher the degree of anisotropy.

These elastic properties were extracted from the elastic tensors using VASPKIT.⁶⁴ Unsupervised clustering was performed using the k-means algorithm implemented in the scikit-learn library⁶⁵ in Python 3.10. To examine the effect of elemental substitution, the atomic charges of the Na_3−xSb_1−xW_xS₄ crystal structures were calculated using the Bader analysis.⁶⁶

Results and discussion

The calculated elastic moduli for various crystal structures of the sodium-ion conductors and reported experimental ambient ionic conductivities are summarized in Table 1. This table includes the first-principles calculation data of previous studies.^10,67 Before the discussion, the experimentally obtained elastic moduli of previously reported Na-ion conductors were summarized. Representative crystal structures are shown in Fig. 1. Previous data on the elastic moduli of Na₂S–P₂S₅ glass sulfide electrolytes experimentally obtained by the ultrasonic method are summarized in Table S1.⁷⁵ It is important to note that the calculations in this study represent theoretical predictions based on crystal morphology. Consequently, the mechanical properties of the corresponding powder compacts should be investigated independently to validate these findings. There is a valuable previous study for the experimental elastic moduli of NASICON electrolytes based on Na₃Zr₂Si₂PO₁₂,⁷⁶ although they contain several impurities and the relative densities of the samples are not close to 100%. The Young's modulus of β-alumina has been experimentally determined using the hardness indentation method, yielding two different values, 215 and 174 GPa, depending on the plane of indentation.⁷⁷

Table 1 First-principles calculation results for the elastic properties on the crystal structures of diverse sodium-ion conductors, which were obtained from both this study and previous studies. These data include the reported data of their ambient ionic conductivities

Type	Formula	Crystal system	B /GPa	G /GPa	E /GPa	ν	B/G	MAV cm^−3 mol^−1	A ^U	Ref.	Experimental ionic conductivity/S cm^−1 (25–30 °C)
Sulfide	α-Na3PS4	Tetragonal	29.85	13.17	34.44	0.308	2.27	12.20	0.72	This study	>10^−6 (ref. 16)
Sulfide	α-Na3PS4	Tetragonal	25.3	13.1	33.6	0.28	2.29	—	0.22	10
Sulfide	β-Na3PS4	Cubic	30.38	9.86	26.68	0.354	3.08	12.13	2.27	This study	4.6 × 10^−4 (glass-ceramic)^16,17
Sulfide	β-Na3PS4	Cubic	21.5	13.1	32.6	0.25	1.64	—	0.25	10
Sulfide	NaPS3	Monoclinic	20.15	7.72	20.55	0.33	2.61	13.3	2.55	This study	—
Sulfide	Na3SbS4	Cubic	26.06	11.85	30.87	0.303	2.20	13.28	0.5	This study	1.0–3.0 × 10^−3 (ref. 26–28)
Sulfide	Na3BS3	Monoclinic	26.43	17.43	42.87	0.23	1.52	11.74	0.28	This study	1.1 × 10^−5 (glass)^18
Sulfide	Na5AlS4	Orthorhombic	30.06	14.07	36.52	0.298	2.14	13.23	0.1	This study	3.2 × 10^−7 (ref. 19)
Sulfide	Na3InS3	Monoclinic	27.23	11.35	29.89	0.317	2.40	14.64	0.92	This study	6.8 × 10^−6 (glass)^21
Sulfide	Na4SnS4	Tetragonal	30.59	12.41	32.79	0.321	2.47	13.74	0.96	This study	1.4 × 10^−8 (ref. 24)
Sulfide	Na3AsS4	Tetragonal	28.91	12.49	32.76	0.311	2.31	12.61	0.55	This study	3.1 × 10^−6 (ref. 25)
Selenide	Na3PSe4	Cubic	27.44	10.34	27.56	0.333	2.65	13.95	0.32	This study	1.1 × 10^−4 (ref. 32)
Selenide	Na3PSe4	Cubic	20	9	23	0.31	2.22	—	0.34	67
Selenide	Na3SbSe4	Cubic	24.76	9.18	24.51	0.335	2.70	15.04	0.13	This study	8.5 × 10^−4 (ref. 33)
Anti-perovskite	Na3OBr	Cubic	35.09	25.8	62.16	0.205	1.36	11.11	0.11	This study	—
Anti-perovskite	Na3OBr	Cubic	34	23.6	57.4	0.22	1.44	—	0.14	10
Anti-perovskite	Na3OCl	Cubic	35.48	25.98	62.65	0.206	1.37	10.66	0.32	This study	—
Anti-perovskite	Na3OCl	Cubic	36.4	24.6	60.2	0.22	1.48	—	0.20	10
Anti-perovskite	Na3OBr0.5Cl0.5	Cubic	35.18	25.64	61.88	0.207	1.37	10.88	0.2	This study	—
NASICON	NaZr2(PO4)3	Trigonal	88.42	46.82	119.39	0.275	1.89	8.76	0.32	This study	8.9 × 10^−6 (ref. 68)
NASICON	NaZr2(PO4)3	Trigonal	86.3	47.7	120.9	0.27	1.81	—	0.41	10
NASICON	NaZr2(PO4)3	Trigonal	93	46	118	0.29	2.02	—	0.33	67
NASICON	NaTi2(PO4)3	Trigonal	115.22	57.55	148.01	0.286	2.00	7.79	0.25	This study	10^−4–1.1 × 10^−3 (ref. 69)
NASICON	NaTi2(PO4)3	Trigonal	109	56	144	0.28	1.95	—	0.24	67
NASICON	Na3Sc2(PO4)3	Trigonal	64.79	43.11	105.85	0.228	1.50	7.89	3.01	This study	3 × 10^−4 (ref. 70)
NASICON	Na3Sc2(PO4)3	Monoclinic	89.15	46.58	119.01	0.277	1.91	7.92	0.51	This study	—
NASICON	Na4Zr2(SiO4)3	Trigonal	98.16	56.45	142.11	0.259	1.74	7.89	0.36	This study	—
NASICON	Na4Zr2(SiO4)3	Trigonal	93	57	141	0.25	1.63	—	0.37	67
NASICON	Na3Zr2Si2PO12	Triclinic	98.33	48.8	125.63	0.287	2.01	8.34	0.63	This study	6.7 × 10^−4 (ref. 71)
NASICON	Na3Zr2Si2PO12	Monoclinic	78.73	50.5	124.81	0.236	1.56	8.16	0.63	This study	—
NASICON	Na3Hf2Si2PO12	Monoclinic	90.59	55.42	138.10	0.246	1.63	7.96	0.58	This study	1.48 × 10^−3 (ref. 72)
Na-β-alumina	NaAl11O17	Hexagonal	153.94	62.76	165.75	0.321	2.45	6.42	2.11	This study	—
Other oxide	NaAlO2	Orthorhombic	86.27	45.36	115.78	0.276	1.90	7.44	0.68	This study	—
Other oxide	Na5AlO4	Orthorhombic	59.85	33.79	85.31	0.262	1.77	7.91	0.05	This study	—
Other oxide	Na5AlO4	Orthorhombic	51	30	74	0.26	1.70	—	0.06	67
Other oxide	Na7Al3O8	Triclinic	68.28	39.53	99.40	0.257	1.73	7.68	0.31	This study	—
Other oxide	Na7Al3O8	Triclinic	68	46	111	0.25	1.48	—	0.52	67
Other oxide	Na14Al4O13	Monoclinic	66.60	37.38	94.48	0.264	1.78	8.04	0.06	This study	—
Other oxide	Na17Al5O16	Monoclinic	67.30	37.74	95.39	0.264	1.78	7.99	0.05	This study	—
Other oxide	Na17Al5O16	Monoclinic	58	34	86	0.25	1.71	—	0.12	67
Other oxide	Na3BO3	Monoclinic	58.58	27.47	71.28	0.297	2.13	7.19	0.66	This study	2.0 × 10^−8 (glass)^73
Other oxide	Na3BO3	Monoclinic	52	25	64	0.29	2.08	—	0.67	67
Other oxide	Na4SiO4	Triclinic	67.41	36.47	92.70	0.271	1.85	7.61	0.3	This study	—
Other oxide	Na4SiO4	Triclinic	58	32	81	0.27	1.81	—	0.43	67
Other oxide	Na3SbO4	Monoclinic	80.34	52.08	128.47	0.233	1.54	7.34	0.37	This study	—
Other oxide	Na3SbO4	Monoclinic	67	44	108	0.23	1.52	—	0.87	67
Other oxide	Na3NbO4	Monoclinic	79.04	51.59	127.12	0.232	1.53	7.32	0.28	This study	—
Other oxide	Na3NbO4	Monoclinic	69	42	104	0.25	1.64	—	0.62	67
Chloride	NaAlCl4	Orthorhombic	13.72	6.88	17.68	0.285	2.00	15.09	0.17	This study	3.9 × 10^−6 (ref. 74)
Chloride	NaAlCl4	Orthorhombic	5	3	7	0.26	1.67	—	3.92	67
Chloride	NaTaCl6	Monoclinic	11.67	5.97	15.31	0.281	1.95	15.44	0.72	This study	6.2 × 10^−5 (ref. 38)
Chloride	Na2ZrCl6	Trigonal	18.67	8.28	21.65	0.307	2.25	15.49	0.29	This study	1.8 × 10^−5 (ref. 35)
Chloride	Na3ErCl6	Monoclinic	24.71	11.37	29.57	0.301	2.17	14.58	0.42	This study	—

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Fig. 1 Various typical crystal structures which have been used as sodium-ion conductors.

Generally, sulfides, selenides, and chlorides tend to have lower elastic moduli, whereas oxides tend to have higher elastic moduli. The overall order of the elastic moduli follows the trend: oxides > sulfides > selenides > chlorides. The general trend observed for Pugh's ratio (B/G) is selenides > sulfides > chlorides > oxides. Many sodium-ion conductors, except for some sulfides and oxides, exceed the ductility/brittleness criterion for polycrystalline metals (B/G = 1.75), suggesting that they are ductile electrolytes at the crystal structure scale. The A^U values were below 1 for most systems, except for a few cases such as β-Na₃PS₄ and NaPS₃, suggesting that the elastic properties of most sodium-ion conductors are isotropic.

Sulfides are characterized by relatively low elastic moduli, and many materials exhibit relatively high B/G values. Among the polymorphs of Na₃PS₄, the high-temperature phase, β-Na₃PS₄, has significantly lower G and E than the low-temperature phase, α-Na₃PS₄. Notably, β-Na₃PS₄ has an exceptionally high B/G ratio of 3.08, whereas all other materials listed in Table 1 exhibit B/G values below 3. This suggests that β-Na₃PS₄ possesses particularly higher mechanical ductility. Notably, NaPS₃ exhibits relatively low elastic moduli. In addition, Na₃BS₃ displays a remarkably low B/G ratio of 1.4, indicating low ductility. This behavior contrasts with the general trend observed for sulfides. Selenides generally exhibit slightly lower elastic moduli and higher B/G ratios than sulfides. The elastic moduli of the glassy NaPS₃ electrolyte tended to be lower than those of the crystalline Na₃PS₄. It was predicted that this tendency is due to the larger MAV of the glassy phase, resulting in weaker chemical bonding compared to the crystalline phase. In contrast, the Poisson's and Pugh's ratios of NaPS₃ were almost the same for both glassy and crystalline phases. The Poisson's and Pugh's ratios of glassy NaPS₃ were intermediate between those of β- and α-Na₃PS₄, relatively closer to those of the β-phase.

The elastic moduli of the oxide electrolytes can generally be classified into two groups, high and moderate, except for anti-perovskite electrolytes that exhibit lower elastic moduli and NaAl₁₁O₁₇ with extremely high elastic moduli. The elastic moduli of the oxides largely depend on their oxygen content. Materials with an oxygen content exceeding 50% tend to be classified as “high,” and those with a lower oxygen content as “moderate.” These two classifications are discussed again in the section that presents the unsupervised clustering results. The Pugh's ratios (B/G) of most oxides ranged from 1.5 to 2.0. The distinctive mechanical properties of NaAl₁₁O₁₇ include not only significant elastic moduli but also a relatively high B/G ratio of 2.45. Materials with high elastic moduli generally exhibit low Pugh's ratios, making this property unique. This suggests that NaAl₁₁O₁₇ is not only an exceptionally rigid material but also has the potential to possess ductility.

The chloride electrolytes generally exhibit lower elastic moduli than the sulfide and selenide electrolytes, indicating their high deformability. Their Pugh's ratios (B/G) were approximately 2.0, which is relatively low compared to sulfides but exceeds the threshold value of 1.75 exhibit the lowest elastic moduli and a B/G ratio of 1.95, indicating that it is a useful electrolyte with sufficient mechanical deformability and ductility.

To predict the elastic properties of Na_2.88Sb_0.88W_0.12S₄, a sodium-ion conductor with exceptionally high ionic conductivity, we calculated the elastic moduli of analogous structures derived by partially substituting tungsten into Na₃SbS₄.

The crystal structure model of Na₃SbS₄ (six Na, two Sb, and eight S atoms) was obtained from the Materials Project database. The structure was expanded into a 2 × 2 × 2 supercell (48 Na, 16 Sb, and 64 S atoms) using the VASPKIT software. In these supercells, tungsten substitution was implemented as shown in Fig. 2. For Na_2.9375Sb_0.0625W_0.0625S₄, a single Sb atom at the center of the cell was replaced with W, and for Na_2.875Sb_0.875W_0.125S₄, the Sb atoms at both the center and corners of the cell were replaced with W atoms. In addition, it was assumed that the Na atoms in the vicinity of the W atoms would be removed because of their proximity. For Na_2.875Sb_0.875W_0.125S₄, both the Na atoms adjacent to the W atoms were removed. After structural optimization, the elastic moduli of all the compositions were calculated, as summarized in Table 2.

Fig. 2 Flowchart on creating Na_3−xSb_1−xW_xS₄ crystal model structures by substituting Sb atom with W atom and doping Na vacancies. In the substituted structures, the anion atoms are omitted from visualization.

Table 2 Calculated elastic moduli of various model crystal structures of Na_3−xSb_1−xW_xS₄

Formula	B/GPa	G/GPa	E/GPa	ν	B/G	MAV/cm^3 mol^−1	A ^U
Na3SbS4	26.06	11.85	30.87	0.303	2.20	13.28	0.50
Na2.9375Sb0.9375W0.0625S4	30.54	10.97	29.39	0.34	2.78	13.35	0.49
Na2.875Sb0.875W0.125S4	38.26	10.72	29.41	0.372	3.57	13.41	0.43

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Despite the negligible change in MAV owing to tungsten substitution, the bulk modulus increased significantly, whereas the shear modulus and polycrystalline Young's modulus decreased slightly. Consequently, the B/G ratio increased, indicating a transition to a more ductile material. Before the tungsten substitution, the B/G ratio was 2.20, which increased to 2.78 and 3.57 for Na_2.9375Sb_0.0625W_0.0625S₄ and Na_2.875Sb_0.875W_0.125S₄, respectively. For the composition Na_2.875Sb_0.875W_0.125S₄, the B/G ratio was higher than that of β-Na₃PS₄ (3.08), which was the highest among the materials listed in Table 1. These results suggest that tungsten substitution significantly enhances the ductility of Na₃SbS₄, making it one of the most ductile sodium-ion conductors. The Poisson's ratio of Na_2.875Sb_0.875W_0.125S₄ was also higher than that of Na₃SbS₄, further indicating that the crystal structure was more prone to deformation under the applied stress.

The results of Bader charge analysis are shown in Table S2. The positive charge on W (+1.63) is greater than that on Sb (approximately +1.45). This indicates a stronger Coulombic interaction between the central cation and surrounding sulfur atoms, which is considered to be the cause of the higher bulk modulus. Furthermore, this enhancement of the Coulombic interaction is suggested to have no significant effect on the shear modulus and polycrystalline Young's modulus.

Fig. 3 shows the structures of the various borohydride-based sodium solid electrolytes employed in this study. Because of the presence of multiple crystal phases of NaBH₄, three different structures were prepared. The structure of NaBH₄ consists of Na⁺ ions coordinated to BH₄⁻ tetrahedra. NaB₃H₈ features a structure in which Na⁺ ions are coordinated to B₃H₈⁻ units, which can be described as clusters. Similarly, NaB₆H₆, NaB₉H₁₀C, and NaB₁₁H₁₂C exhibit structures in which the Na⁺ ions are coordinated around clusters containing multiple B atoms.

Fig. 3 Unit cell structures of various sodium-ion conductors with boron hydrides; gold: sodium, green: boron, pink: hydrogen.

The calculated elastic moduli of the borohydride-based sodium-ion conductors are summarized in Table 3. Despite having exceptionally low MAVs, these materials exhibit elastic moduli comparable to those of sulfide-based solid electrolytes. This behavior is unique compared to that of other sodium solid electrolytes. Notably, the shear modulus of NaB₁₁H₁₂C is as low as 6.92 GPa. The B/G ratio, which varies depending on the material and crystal system, had relatively high values, exceeding 2.0 for NaB₆H₆ and NaB₁₁H₁₂C. In contrast, NaB₉H₁₀C exhibited a relatively low B/G ratio, comparable to those of the oxide electrolytes. The A^U exceeded 1 in certain systems, but remained low overall. These findings suggest that the borohydride-based solid electrolytes exhibit elastic isotropy.

Table 3 Calculated elastic moduli of crystal structures of various borohydride-based sodium-ion conductors

Type	Formula	Crystal system	B/GPa	G/GPa	E/GPa	ν	B/G	MAV/cm^3 mol^−1	A ^U	Ref.	Experimental ionic conductivity/S cm^−1 (25–30 °C)
Hydride	NaBH4	Cubic	22.53	17.21	41.15	0.195	1.31	5.31	0.02	This study	≅10^−10 ^78
Hydride	NaBH4	Tetragonal	25.88	14.54	36.74	0.263	1.78	5.13	0.43	This study	—
Hydride	NaBH4	Orthorhombic	22.57	14.97	36.77	0.228	1.51	5.15	0.06	This study	—
Hydride	NaB3H8	Orthorhombic	19.21	11.00	27.71	0.26	1.75	4.76	0.31	This study	—
Hydride	NaB6H6	Monoclinic	20.21	8.68	22.78	0.312	2.33	5.58	1.49	This study	≅10^−8 ^79
Hydride	NaB6H6	Monoclinic	10	6	15	0.27	1.67	—	1.59	67
Hydride	NaB9H10C	Orthorhombic	18.21	9.98	25.32	0.268	1.82	5.52	0.5	This study	0.03 ^80
Hydride	NaB11H12C	Orthorhombic	16.32	6.92	18.2	0.314	2.36	5.21	1.43	This study	≅10^−5 ^81

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The calculated elastic moduli of the Na_xY-type and Na₃PO₄ crystals are summarized in Table S3. Compared with lithium-based materials, the elastic moduli of these sodium-based compounds are generally lower, with Na₂Se and NaI exhibiting particularly low values. However, many of these materials have relatively low B/G ratios, falling below the threshold value of 1.75. Na₃PO₄ exhibited moderate elastic moduli compared to other sodium-ion conductors with a B/G ratio of 1.98, which is not particularly low.

Fig. 4 shows the distribution of the bulk and shear moduli of the sodium-ion conductors investigated in this study. Fig. 4a illustrates the distribution of materials with relatively low elastic moduli (primarily non-oxide materials), while Fig. 4b displays the distribution of oxides, excluding those with an anti-perovskite structure, which generally exhibit higher elastic moduli. As shown in Fig. 4a, many materials exhibit B/G ratios that exceed the threshold value of 1.75. In contrast, the NaBH₄, Na₃BS₃, and anti-perovskite structures have B/G ratios well below 1.75. Notably, only two materials, β-Na₃PS₄ and Na_2.875Sb_0.875W_0.125S₄, exhibit B/G ratios exceeding 3. As shown in Fig. 4b, most oxide electrolytes were distributed at approximately B/G = 1.75. However, NaAl₁₁O₁₇ is an exceptional case with a significantly high B/G ratio.

Fig. 4 Correlation between bulk Modulus (B) and shear Modulus (G) of various sodium-ion conductors: (a) electrolytes with lower elastic moduli: sulfides, selenides, anti-perovskite electrolytes, halides, and hydrides, (b) electrolytes with higher elastic moduli: oxides except for anti-perovskite electrolytes.

The results of the unsupervised clustering using the k-means method (k = 4) based on the bulk and shear moduli are shown in Fig. 5. Materials can be primarily divided into four classes: those with low, medium, and high elastic moduli and materials with exceptionally high elastic moduli. The low-elastic moduli region (cluster 0) consists of sulfides, selenides, halides (excluding fluorides), hydrides, and anti-perovskite-type electrolytes. Oxides (and fluorides) are mainly divided into those with medium elastic moduli (cluster 2) and those with high elastic moduli (cluster 1). NaTi₂(PO₄)₃ and NaAl₁₁O₁₇, exhibiting exceptionally high elastic moduli, were classified into a group of materials with extremely high elastic moduli (cluster 3).

Fig. 5 Clustering of sodium-ion conductors based on bulk modulus (B) and shear modulus (G) using k-means method with four clusters.

Fig. 6 shows the correlations between MAV and elastic properties (a: bulk modulus, b: polycrystalline Young's modulus, and c: shear modulus). A negative nonlinear correlation exists between MAV and each elastic modulus when the hydrides are excluded. However, focusing on the oxide electrolytes reveals that even with similar MAV values, the elastic moduli can differ significantly. This result confirms that factors other than MAV, such as the space group and crystal system, influence the elastic properties. Among the oxides, anti-perovskite structures, which contain halogens, tend to exhibit higher MAV and lower elastic moduli. Sulfides display a negative correlation between the MAV and elastic moduli, with the correlation being particularly pronounced for the bulk and shear moduli. By contrast, chlorides exhibit exceptionally high MAV values and notably low elastic moduli. Hydrides are exceptional because they exhibit low MAVs and low elastic moduli.

Fig. 6 Correlation between mean atomic volume and (a) bulk moduli B, (b) polycrystalline elastic moduli E, and (c) shear moduli G of various crystal structures of various sodium-ion conductors.

A comparative analysis was also conducted with Li-ion-conducting solid electrolytes. The computational results obtained for the Na-ion-conducting solid electrolytes in this study were generally similar to those for Li-ion-conducting solid electrolytes obtained from first-principles calculations.⁸² A slight difference was observed in the bulk modulus of sulfide electrolytes: while many Li-ion conductive solid electrolytes exhibited values around 30–40 GPa, most Na-based ones showed lower values of approximately 20–30 GPa. Consequently, materials with B/G ratios clearly exceeding 3.0, which are often found in Li-ion conductive solid electrolytes, were much less common in the Na-ion conductive solid electrolytes. Furthermore, experimental measurements of the elastic moduli of sulfide glass electrolytes revealed a correlation between the average atomic volume and Young's modulus.⁹ This correlation is consistent with the one observed for the Na-ion-conducting solid electrolytes in this study, and a similar trend has also been reported in the computational results for Li-ion conductive solid electrolytes.⁸²

As a reference data for the comparison, elastic moduli extracted from Materials Project were listed in Table S4. Although there are slight differences in the values due to variations in computational conditions, such as lattice constants, the elastic properties are generally consistent with those obtained in this study and in previous calculations. This study also includes the elastic moduli of many electrolytes that are not available in the Materials Project or previous reports, making it a valuable contribution in this respect.

Conclusions

In this study, the elastic properties and Pugh's ratios (B/G) of various sodium-ion conductive solid electrolytes were estimated using first-principles calculations. Sulfides, chlorides, and hydrides generally exhibited low elastic moduli, with β-Na₃PS₄ showing a notably high Pugh's ratio, suggesting high ductility. Oxide electrolytes were categorized as materials with medium-to-high elastic moduli, with a clear trend of higher elastic moduli correlated with an increased oxygen content. Among oxides, NaAl₁₁O₁₇ was identified as having the highest elastic modulus and Pugh's ratio, indicating unique mechanical properties, as materials with high elastic moduli typically show lower Pugh's ratios.

The substitution of tungsten into Na₃SbS₄ resulted in significant changes not only in ionic conductivity but also in elastic properties. Specifically, the bulk modulus and Poisson's ratio increased, whereas the polycrystalline Young's and shear moduli decreased. This resulted in an increase in Pugh's ratio, suggesting enhanced mechanical ductility. Across the studied materials, those with high ionic conductivities, such as β-Na₃PS₄ and Na_2.875Sb_0.875W_0.125S₄, were found to have relatively high Poisson's ratios. A negative nonlinear correlation was observed between the MAV and elastic moduli. Borohydride-based electrolytes deviated from this trend by exhibiting low elastic moduli despite their low MAV.

These findings provide valuable insights into the mechanical properties of sodium-ion conductive solid electrolytes and offer guidance for the design of mechanically stable materials for all-solid-state batteries.

Author contributions

Masato Torii: performed the simulations, processed the results, wrote the manuscript. Atsushi Sakuda: supervised the work, revised the manuscript. Kota Motohashi: provided critical feedback, revised the manuscript. Akitoshi Hayashi: provided critical feedback, revised the manuscript.

Conflicts of interest

There are no conflicts to declare.

Data availability

The database IDs of the solid electrolytes used in this study, the number of k-points, and their lattice constants obtained after structural optimization for the elastic tensor calculations are listed in Table S5. We have compiled the elastic modulus data in CSV format as well as the elastic tensor data on the following GitHub page. https://github.com/MasatoTorii/Na_ElasticDB.

Supplementary information (SI) is available. See DOI: https://doi.org/10.1039/d5ma00912j.

Acknowledgements

This study was supported by the MEXT/JSPS KAKENHI (grant number JP23H04633), MEXT Program: Data Creation and Utilization-Type Material Research and Development Project (JPMXP1122712807), JST Adopting Sustainable Partnerships for Innovative Research Ecosystem (JPMJAP2313), JST SPRING (grant number JPMJSP2139) and the 2024 Osaka Metropolitan University (OMU) Strategic Research Promotion Project (Priority Research). First-principles calculations were partially performed using the computing resources offered under the category of General Projects by the Research Institute for Information Technology, Kyushu University.

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