A first principles study of the mechanical properties of Li–Sn alloys

Abstract

To obtain a better understanding of the failure mechanism of lithium ion batteries during charging–discharging, we have systematically studied the mechanical properties of Li–Sn alloys based on density functional theory, including elastic constants such as the orientation-averaged bulk, shear and Young's moduli and Poisson's ratios, as well as the anisotropy and brittleness-ductility. It is shown that bulk, shear and Young's moduli of isotropic Li_xSn alloys decrease almost linearly with the increase of Li concentration. Further, based on the analysis of shear to bulk modulus and Poisson's ratios, it is inferred that Li–Sn alloys are brittle. The poor cycle performance and crushing failure of Sn anode materials during charging and discharging are mainly due to a transition of electrode material properties from ductile to brittle.

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1. Introduction

Over the last decade, the demand for high-energy storage has been rapidly increasing. There are more and more studies focusing on new active materials of batteries with high voltage, large capacity and high rate capability.^1,2 As an alternative energy storage device, lithium ion batteries (LIBs) have become the most promising portable power sources in consumer electronics and vehicle electrification. Here, one of the most critical parameters for the next generation of LIBs is the energy density or specific energy.³ As is well known, LIBs can store more energy per weight than that of their competitors such as nickel–metal hydride, nickel–cadmium and lead-acid batteries.

Up to now, graphite with a layer structure is the most widely used negative electrode material in LIBs, and during charging, the compound LiC₆ is formed from the lithium–graphite interaction. Unfortunately, its theoretical capacity is relatively low (372 mA h g⁻¹).^4,5 In contrast, Sn exhibits an extraordinarily large theoretical capacity of 994 mA h g⁻¹, which is about two and a half times of the conventional graphite anode. As an anode material, however, Sn experiences a massive volume change (∼300%) during a charge–discharge process, and binary alloys are formed during lithiation, e.g., Li₂Sn₅, LiSn, Li₇Sn₃, Li₅Sn₂, Li₁₃Sn₅, Li₇Sn₂ and Li₁₇Sn₄.⁶ Thus, the large volume expansion, phase transition and Li diffusion-induced stress within electrode materials may result in their capacity loss and poor cycling performance due to electrode damage and fracture.⁷

To improve the durability of LIBs, significant progresses have been made in understanding the Li ion diffusion-induced stress. For example, under galvanostatic or potentiostatic operation, the stress and strain evolution within a spherical particle was modeled.⁸ The effects of surface energy and elasticity on the stress evolution in spherical electrodes were also investigated.⁹ Wolfenstine demonstrated that a critical particle size for lowering the capacity fade can be estimated by using Young's modulus, fracture toughness and volume deformation.¹⁰ It is also shown that, upon lithiation, the change of the averaged Young's modulus of Li_xFePO₄ is less than 1%.¹¹ However, Young's moduli of crystalline and amorphous Li_xSi electrodes decrease almost linearly with Li concentration, and when forming the phase Li₁₅Si₄, Young's modulus drops to one-third of its original value.¹² Thus, it is necessary to have a better understanding on the Li concentration-dependent mechanical properties of LIBs.

In this paper, we calculated the elastic constants of Li–Sn alloys based on the density functional theory. These constants such as bulk, shear and Young's moduli and Poisson's ratios can be obtained with the help of the Voight–Reuss–Hill approximation.¹³ Other properties of Li–Sn alloys, including the anisotropy and brittleness-ductility, were also calculated. Finally, it is shown that there is a quantitative relationship between the mechanical properties of Li–Sn alloys and Li concentration, which is important for evaluating deformation and failure of electrode materials.

2. Calculation methodology

2.1 Structure optimization

By using the Vienna Ab initio Simulation Package with the local density approximation for exchange and correction, we calculated the structural and elastic properties of Li–Sn alloys based on the density functional theory. The projected augmented wave method was used to describe the ion–electron interaction.^14,15 Here, valence configurations for Li and Sn are 1s²2s¹ and 5s²5p² (4d¹⁰), respectively. The kinetic energy cutoff for the plane wave basis set was converged for every structure and its value in the plane wave expansion is 500 eV. In calculations, all atoms were fully relaxed by using the conjugate gradient method until residual forces on constituent atoms were less than 5 × 10⁻² eV Å⁻¹, and in determining mechanical properties, the value of convergent force increases to 1 × 10⁻³ eV Å⁻¹. In addition, the Brillouin zone was sampled with the Monkhorst-Pack k-point grid.¹⁶

2.2 Elastic constants

By applying a small strain to the equilibrium lattice, the elastic constants, C_ij, in a crystalline system can be deduced from the change of its total energy in a unit cell. The strain tensor ε_ij has six independent components, and with the Voigt notation, they can be expressed as

Under a small strain ε_ij, the relationship between the strain energy and stress σ_ij is given by

According to the generalized Hooke law, σ_ij = C_ijklε_kl, the elastic constants can be identified as the second derivative of the total energy to strain, that is

Here, the total energy of a crystal is expanded in the following Taylor form:

where V₀ is the volume of the unstrained system and E(V₀, 0) is the corresponding total energy.

Then, through selecting a specific strain ε_ij = (e₁, e₂, e₃, e₄, e₅, e₆), we can determine the total energy change before and after a set of different strains (≤2%) and quadratic coefficients. According to the Voight–Reuss–Hill approximation,¹³ bulk and shear moduli, K and G, can be calculated by

K = (K_V + K_R)/2 (5)

and

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

where subscripts ‘V’ and ‘R’ indicate the Voight and Reuss approximations, respectively. K_V, K_R, G_V and G_R are defined asandwhere S_ij are the elastic compliance constants, i.e., the inverse matrix of elastic constants. Then, Young's modulus Y and Poisson's ratio ν can be obtained by

Y = 9KG/(3K + G) (11)

and

ν = (3K − 2G)/2(3K + G) (12)

3. Results and discussion

3.1 Structure properties

Structure optimization is firstly performed to determine the lattice parameters and the positions of atoms in Li–Sn alloys. To obtain an equilibrium volume, all structures are fully relaxed. As is well known, the local density approximation usually underestimates lattice constants and overestimates elastic constants, and while the generalized gradient approximation is just opposite. Thus, we use the local density approximation for exchange-correction, and due to an over-binding problem of the local density approximation, it is uniform in estimating all structures. The structure data of crystalline Li–Sn phases obtained from calculations are summarized in Tables 1 and 2, and for comparison, the available experimental data are also listed (see Table 2). It is obvious that our results are well agreement with experimental data.

Table 1 Crystal structure information for Li, Sn and Li_xSn, where y is the ratio of Li atoms to the total number of atoms in alloys

Phase	Structure	x	y = x/(1 + x)	Space group
β-Sn		0.00	0.000	I41/amd
Li2Sn5		0.40	0.286	P4/mbm
LiSn		1.00	0.500	P2/m
Li7Sn3		2.33	0.700	P21/m
Li5Sn2		2.50	0.714	R3̄m
Li13Sn5		2.60	0.722	P3̄m1
Li7Sn2		3.50	0.778	Cmmm
Li17Sn4		4.25	0.810	F4̄3m
Li		∞	1.000	Im3m

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Table 2 Lattice constants in units of Å and k-points used in the calculation, where the experimental values are listed in parentheses

Phase	a	b	c	Volume	k-points
β-Sn	5.792(5.831)	5.792(5.831)	3.122(3.184)^32	104.78	9 × 9 × 16
Li2Sn5	10.331(10.274)	10.331(10.274)	3.169(3.125)^33	338.20	3 × 3 × 11
LiSn	5.162(5.172)	7.764(7.742)	3.233(3.182)^34	125.00	7 × 10 × 4
Li7Sn3	9.495(9.451)	8.536(8.561)	4.738(4.721)^35	369.00	4 × 7 × 4
Li5Sn2	4.725(4.740)	4.725(4.740)	119.844(19.833)^36	383.75	8 × 8 × 2
Li13Sn5	4.703(4.701)	4.703(4.701)	17.130(17.124)^37	328.14	8 × 8 × 2
Li7Sn2	9.847(9.802)	13.838(13.803)	4.712(4.752)^38	642.00	3 × 2 × 7
Li17Sn4	13.907(19.690)	13.907(19.690)	13.907(19.690)^39	7608.27	2 × 2 × 2
Li	3.436(3.513)	3.436(3.513)	3.436(3.513)^40	40.58	22 × 22 × 22

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3.2 Elastic properties

Different deformation modes are applied to obtain the elastic constants of various crystal systems. For example, for tetragonal Li₂Sn₅ and β-Sn phases, six deformation modes are considered to determine C₁₁, C₁₂, C₁₃, C₃₃, C₄₄ and C₆₆.¹⁷ Similarly, five deformation modes for hexagonal Li–Sn phases are used to calculate C₁₁, C₁₂, C₁₃, C₃₃ and C₄₄.¹⁸ In order to obtain nine independent elastic constants of orthorhombic Li₇Sn₂, we apply three monoclinic strain tensor, three orthorhombic strain tensor and expansion along three high-symmetry directions.¹⁹ For monoclinic LiSn and Li₇Sn₃ that have thirteen independent elastic constants, we apply the same nine strain tensors performed on the orthorhombic phase and an additional four.²⁰ All the calculated elastic constants are given in Table 3. In the cases of tetragonal structure β-Sn and cubic structure Li, the calculated elastic constants are C₁₁ = 77.93, C₃₃ = 94.94, C₄₄ = 20.38, C₆₆ = 25.82, C₁₂ = 55.70, C₁₃ = 46.52 and C₁₁ = 16.03, C₁₂ = 13.07, C₄₄ = 11.84, respectively. They are consistent with their corresponding experimental data such as C₁₁ = 73.4, C₃₃ = 90.7, C₄₄ = 21.9, C₆₆ = 25.82, C₁₂ = 59.9, and C₁₃ = 39.1 for β-Sn and C₁₁ = 14.39, C₁₂ = 12.04, and C₄₄ = 11.58 for Li.^21,22

Table 3 Elastic constants C_ij for Li, Sn and Li_xSn alloys, where all quantities are in units of GPa

Phase	C11	C22	C33	C44	C55	C66	C12	C13	C23
β-Sn	77.92	77.92	94.94	20.38	20.38	25.58	55.70	46.52	46.52
Li2Sn5	77.29	77.29	102.47	30.55	30.55	37.89	41.77	16.62	16.62
LiSn	57.70	87.36	58.42	16.20	15.20	17.35	14.96	26.56	21.32
Li7Sn3	56.26	76.17	80.53	10.26	37.20	35.48	18.00	18.03	−6.50
Li5Sn2	74.81	74.81	109.89	17.66	17.66	30.36	14.08	−4.46	−4.46
Li13Sn5	75.12	75.12	101.15	20.11	20.11	31.34	12.45	−3.20	−3.20
Li7Sn2	53.53	63.83	74.23	10.62	33.38	35.23	19.74	11.20	−4.13
Li17Sn4	48.30	48.30	48.30	23.78	23.78	23.78	15.41	15.41	15.41
Li	16.03	16.03	16.03	11.84	11.84	11.84	13.07	13.07	13.07

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Based on these elastic constants, other mechanical parameters for Li–Sn alloys, such as K, G, Y, and ν, can be derived by the Voight–Reuss–Hill approximation.¹³ The orientation-averaged elastic constants and ν for crystalline Li–Sn phases are plotted in Fig. 1. Here, the Li fraction y is defined as the ratio of Li atoms (x) in Li_xSn alloys, i.e., y = x/(1 + x). Obviously, such a dimensionless variable y for all the alloy phases ranges from 0 (Sn) to 1 (Li), which allows us to assess the validity of the law of mixtures.

Fig. 1 (a) Bulk modulus (K), Young's modulus (Y), and shear modulus (G) and (b) Poisson's ratio (ν) versus Li fraction y.

As shown in Fig. 1, K, G, and Y as well as ν decrease almost linearly with the increase of Li concentration, demonstrating a weakening tendency of the resistant ability of Li–Sn alloys to deformation. This is because the Sn–Sn covalent bonds are replaced by the weaker Li–Sn and Li–Li bonds during lithiation, leading to a weaker average strength of constituent chemical bonds in alloys.²³ Our results agree well with previous experimental findings.²⁴ In addition, the difference of K, G, Y and ν between Li₇Sn₃, Li₅Sn₂, Li₁₃Sn₅ and Li₇Sn₂ is minimal.

3.3 Anisotropy

Due to the anisotropy of active materials, micro-cracks can be induced in LIBs during the charging–discharging process. To study the influence of anisotropy on their mechanical properties, we have calculated the elastic anisotropy of Li–Sn alloys by using the universal anisotropy index A^U,²⁵ that is

According to Huggins²⁶ and Courtney et al.,²⁷ Li–Sn alloys can be classified into the following three categories based on their Li ion concentration x, viz. (i) the poor lithium phase (x < 2.33), (ii) the medium intercalated-lithium phase (2.33 < x < 3.5) and (iii) the rich lithium phase (3.5 < x < 4.4). As shown in Fig. 2, the elastic anisotropy of the pure metal β-Sn is similar to the poor lithium phases Li₂Sn₅ and LiSn with the universal anisotropy index A^U of 0.44, 0.46 and 0.42, respectively. Li₇Sn₃ lies in the boundary between the poor lithium and medium intercalated-lithium phases and Li₇Sn₂ in the boundary between the medium intercalated-lithium and rich lithium phases. Furthermore, both Li₇Sn₃ and Li₇Sn₂ exhibit roughly the same universal anisotropy index with the medium intercalated-lithium phases Li₅Sn₂ and Li₁₃Sn₅. This is consistent with the calculated elastic properties (see in Fig. 1), where K, G, Y and ν of these four phases are nearly similar. The full lithiation phase Li₁₇Sn₄ trends to be isotropic with A^U ≈ 0 since (K_V/K_R) ≈ 1, which is in agreement with what Ranganathan and Ostoja-Starzewski have suggested.²⁵ A clear anisotropic property can be found for pure metal Li based on the A^U value of 7.34, as indicated by former theoretical and experiment results.^22,25

Fig. 2 Universal anisotropy factor A^U at different Li fractions y: (I) the poor lithium phase (x < 2.33), (II) the medium intercalated lithium phase (2.33 < x < 3.5), and (III) the rich lithium phase (3.5 < x < 4.4).

3.4 Ductility and brittleness

According to the empirical Pugh formula,²⁸ the ratio G/K can be used to distinguish the ductile and brittle behaviors of intermetallic and metalloid compounds. In the case of G/K > 0.5, a material behaves in a brittle manner, otherwise it is ductile. Here, it is worth noting that, due to the over-simplification, the ductile to brittle transition value of 0.5 is not definitive. However, the G/K ratio during lithiation can describe the change tendency of electrode material properties from ductility to brittleness. It is seen in Fig. 3 that, for pure metal β-Sn and Li, the values of G/K are less than 0.5, and thus they are ductile materials. Obviously, Li–Sn alloys exhibit brittle behaviors because their G/K values are more than 0.5. In addition, Poisson's ratios can also determine the ductile and brittle behaviors. For ductile materials, their Poisson's ratios are generally more than 1/3; but for brittle materials, they are less than 1/3. Thus, we can conclude that Li–Sn alloys are brittle materials because of their low Poisson's ratios (see Fig. 3).

Fig. 3 (a) G/K and (b) Poisson's ratio ν versus y = x/(x + 1).

In the charging process, electrode materials experience a large-volume deformation with inserting Li ions.²⁹ In addition, electrode material properties may will change from ductility to brittleness. Hence, the brittle active materials are more seriously damaged than ductile ones at the same condition when the Li diffusion-induced stress of external electrode materials transforms from compression to tension.³⁰ In the discharging process, the volume of electrode materials shrinks due to delithiation of Li ions. In the meantime, the average stress upon external electrode materials is tensile, which promotes the propagation of cracks from the external surface to the interior of electrode materials. As the charge–discharge process continues, more cracks appear on the surface and in the interior of electrode materials. This significantly affects the efficiency of active materials and leads to degradation of cycle performance. Thus, it is the ductile–brittle transition of electrode material properties that results in the poor cycle performance and crushing failure of Sn anode materials in the charge–discharge process.

3.5 Electronic structure

To further elucidate the similarity between the mechanical properties of Li₇Sn₃, Li₅Sn₂, Li₁₃Sn₅ and Li₇Sn₂, we calculate the electronic structures of Li–Sn alloys. As shown in Fig. 4, there are finite values at the Fermi levels, indicating their metallic conductivities. In addition, the fluctuant N(E_F) values near the Fermi level (E_F) demonstrate their different electronic conductivity. The partial density of states (PDOS) shows that the Sn 5p states makes the main contribution at E_F for most of Li–Sn alloys, but its role is weaken with the increase of Li ion concentration. As shown in Fig. 4(h), in the fully lithiation phase Li₁₇Sn₄, the most contribution comes from Li s states due to Li being the dominant constituent atoms.

Fig. 4 The calculated total and partial density of states of Li–Sn alloys in the units of states per eV per unit cell and states per eV per atom, respectively.

In Li–Sn alloys (except Li₁₇Sn₄), the typical feature is the presence of a pseudo-gap (a sharp valley around the Fermi energy). There are two mechanisms for the formation of pseudo-gap in the binary alloys. One is of ionic origin and the other is owing to hybridization effects.²³ The pseudo-gap in Li–Sn alloys is believed to be due to covalent hybridization between Sn and Li atoms. Such a strong hybridization gives not only an important mixing between the states of conduction bands but also leads to a separation of the bonding states.³¹ Thus, Li₇Sn₃, Li₅Sn₂, Li₁₃Sn₅ and Li₇Sn₂ show the similar extent of covalent hybridization because of their minor difference of Li contents (∼0.008–0.078), which further results in their similar mechanical properties. It is also see that pseudo-gap appears with the insertion of Li ion, and becomes larger with increasing Li ion, and at high Li ion concentration (Li₁₇Sn₄), it disappears. These phenomena correspond to the Sn–Sn covalent bonds that are gradually replaced by the weaker Li–Sn and Li–Li bonds during lithiation. When the Li ion concentration reaches a certain extent, Sn–Sn covalent bonds disappear, and Li–Sn bonds reach the maximum. Then, metallic Li–Li bonds begin to replace Li–Sn bonds with the further increase of Li content. At last, covalent hybridization between Sn and Li atoms disappears. The influence of the change of bonds on the alloys is that the average strength of constituent chemical bonds is weakened, resulting in a weakening tendency of their resistant ability to deformation with the increase of Li concentration. Thus, it is the average weakening bond strength that results in the almost linearly decreased moduli of Li–Sn alloys with increasing Li ion concentration.

4. Conclusions

Based on the density functional theory calculations, we have systematically studied the Li concentration-dependent mechanical properties of Li–Sn alloys. The results show that the bulk, shear and Young's moduli of isotropic Li_xSn alloys decrease almost linearly with increasing Li concentration, demonstrating that the resistant ability of Li–Sn alloys to deformation is weaken with the increase of Li concentrations. Together with the ductile–brittle transition of electrode materials with the insertion of Li ion, a better understanding has been obtained on the failure of Sn anode materials during the charging–discharging process. It is believed that the results provide a meaningful guide for the improvement of electrode materials of lithium ion batteries.

Acknowledgements

This work was supported by the National Natural Science Foundation of China (Grant nos 11372267 and 11402086), and the National High Technology Research and Development Program of China (863 Program) (Grant no 2013AA032502). Simulations were conducted at iVEC through using advanced computing resources located at iVEC@Murdoch.

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