Diffusion induced stress and the distribution of dislocations in a nanostructured thin film electrode during lithiation

Abstract

Li-ion battery electrode materials that undergo huge volume changes require studies on fracturing during lithiation. By analyzing the process of lithiation, a new model has been established, which considers the dislocation mechanisms of nanostructured thin film electrode materials involving diffusion induced stress to improve the battery life of Li-ion batteries undergoing potentiostatic or galvanostatic charging. In the present work, the interactions between diffusion and dislocation induced stress or strain energy are demonstrated under potentiostatic and galvanostatic charging. The stress and strain energy can evolve quite differently under potentiostatic or galvanostatic charging. At the same time, we observed that the magnitude of the stress and strain energy is influenced by dislocations. What’s more, the effect of dislocations on the total strain energy or maximum stress is larger under potentiostatic charging than under galvanostatic charging at the beginning of charging. However, the total strain energy or maximum stress under galvanostatic charging is larger than that under potentiostatic charging later in the lithiation process. The influence of the dislocation effects upon the mechanical behaviour is specified, and is more significant in terms of the distribution of the stress and strain energy in the nanostructured thin film electrode. These results also show that it is possible to control the dislocation density with the methods of nanotechnology to improve Li-ion battery life.

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1 Introduction

Due to their high energy storage density,^1–3 Li-ion batteries are used in many commercial electronic appliances. To achieve a higher energy density, new anode materials and different anode shapes are being intensively studied in the Li-ion battery. For example, Si is one of the most promising anode materials for use in Li-ion battery electrodes, because of a theoretical capacity of 4200 mA h g⁻¹.^4–6 However, owing to volume change,^7–9 there is a large degree of stress that leads to irreversible capacity loss and poor cyclability in the electrode during lithiation.¹⁰ Recent experiments have shown that mechanical failure of the electrode can be mitigated by using different nanostructured anode shapes, such as nanowires,¹¹ thin films,^12,13 and hollow nanoparticles.¹⁴ These structures can reduce the stress by managing the deformation through shape optimization and geometric restrictions.

A number of theoretical models have been developed to analyze the mechanical failure of the electrode owing to Li⁺ diffusion. Prussin¹⁵ made an analogy between thermal stress and diffusion induced stress (DIS) in a thin plate. Li¹⁶ assessed a number of analytical solutions to DIS problems in spherical, cylindrical, and thin plate geometries. Huggins et al.¹⁷ utilized Griffith’s criteria to expound the fracture phenomenon in rechargeable electrochemical systems. To avoid pre-existing cracks, Bhandakkar and Gao¹⁸ developed a cohesive model to suggest a critical characteristic dimension. Qiu¹⁹ developed a novel water-soluble Li-ion binder to enhance the cyclability of c/Li-ion phosphate cathodes, Recently, Wei. et al.²⁰ established the dislocation model to analyze the diffusion induced stresses in a spherical particle. At the same time, Chen et al.²¹ also analyzed the fracture damage of a nanowire electrode in a Li-ion battery. The stress induced by Li⁺ diffusion can lead to the fracture of the electrode, which has been confirmed through many experiments. A recent design of electrodes involves the use of nanostructured thin films, which can enhance the mechanical and chemical stability of anodes simultaneously. For example, a nanostructured Si thin film exhibited superior performance during charge cycling.¹² Thus, we consider the nanostructured thin film electrodes. It is necessary to clarify the phenomena of the mechanical properties in nanostructured thin film electrodes during lithiation.

In a charging nanoparticle electrode, with one face of the nanoparticle contacting a liquid electrolyte, a reaction front propagates progressively along the r-direction of the nanoparticle. Transmission electron microscopy (TEM) observation has shown this to cause the nanoparticle to swell.²² The process of the Li⁺ intercalation has a “Medusa zone” containing a high density of dislocations. Nucleation at the moving front will proceed, along with absorption from behind. Huang²³ also found that there is a high density of dislocation in a single SnO₂ nanowire electrode. Few reports have specifically considered the dislocation effect induced by Li⁺ diffusion in a nanostructured thin film electrode. Thus, we aimed to develop an analytical model with dislocation effects to analyze the mechanical behavior in a nanostructured thin film electrode. For this purpose, the analytical model was be explored to explain the diffusion process using finite elasticity–plasticity theory^24–26 and dislocation theory²⁷ under potentiostatic and galvanostatic operation. In addition, dislocation induced strain energy was predicted to affect the magnitude distribution of the total strain energy. Finally, we provided a theoretical method for prolonging Li-ion battery life in the nanostructured thin film electrode by taking into account the dislocation effect.

2 Analytical model for the nanostructured thin film electrode

2.1 Stress and the distribution of dislocations by solute diffusion

The insertion of Li⁺ into the electrode is a diffusion process, which involves the diffusion of Li⁺ in the electrolyte, migration of Li⁺ through the solid electrolyte interphase (SEI), charge transfer at the electrode interface, and diffusion in the electrode. So the reversible reactions will occur during lithiation in the nanostructured thin film electrode. Lithium diffusion may depend on the different crystal structures (layered, spinel, olivine). In this paper, these different crystal structures are manifested through some parameters, such as the Young’s modulus of material, molar volume, maximum concentration, lattice constant of material, Burgers vector, etc. Fig. 1 shows a thin film electrode, with width 2h, submitted to the insertion of Li⁺. The electrode material is regarded as an isotropic linear elastic solid and the deformation is assumed as quasi-static. According to the solute diffusion theory, the transport of the solute is modeled as a concentration-driven diffusion process along the thickness direction of the electrode. The diffusion equation in the thin film can be expressed as²⁸where D is the diffusion coefficient, which can be considered as constant and C is the molar concentration of the solute. The migration of Li⁺ into the host causes a swelling transformation strain β, which can be defined as Ω/3. Ω is the partial molar volume of the solute in the electrode. It is worth noting that this model may work for the “half-cell” and the effect of the SEI layer will be ignored, because the SEI layer is weakly formed in the “half-cell” as compared to the “full-cell”. The stress caused by concentration gradients is similar to that caused by temperature gradients. For the concentration gradients, we have the relation

ε_x = −βC (2)

where ε_x is a component of strain. Timoshenko and Goodier²⁶ derived an expression for the stress in a thin plate when the temperature distribution is independent of the x and z coordinates. The diffusion induced stress will be developed in a thin film electrode in a manner analogous to that of thermal stresses.where E is the Young’s modulus of the electrode material and µ is Poisson’s ratio. Eqn (3) gives the DIS in the electrode during lithiation. Thus, the DIS distribution can be identical to the concentration profile from which it was derived. It is clear that there is a datum stress of σ_u = 0 induced by an arbitrary solute distribution. It satisfies the condition that the sum of the tensile stress is equivalent to the sum of the compressive stress.

Fig. 1 Schematic of Li-ion diffusion in a thin film electrode of width 2h, which is modeled as diffusion along its thickness during insertion.

Pearson et al.²⁹ showed that the stress distribution by solute lattice contraction is sufficient to induce plastic deformation and dislocations in dislocation-free silicon. What’s more, Prussin¹⁵ proposed that the diffusion of solute atoms can lead to the generation of dislocations. Therefore, we will consider the dislocations effect due to the diffusion of Li⁺ in the nanostructured thin film electrode. As has been reported, the highest stress is found at the surface at the very beginning of the diffusion process. According to Prussin,¹⁵ if the initial stress exceeds the stress σ_g that necessary to induce dislocation, the dislocations generated must lie in the surface and their Burgers vectors must have a positive edge component in the surface. It is obvious that dislocation induced stress can act to resist the process of diffusion during lithiation. Thus, the tensile stress will be reduced when the dislocation is considered in the electrode. The new datum stress, σ_u = 0, is determined by the requirement that the sum of the compressive stress equals the sum of the tensile forces will be reformed. As a result of stress reducing, the neutral planes will be moved farther from the surfaces of the diffused lattice. The more that the stress is decreased, the closer the neutral planes will become to the depth of solute penetration.

In Fig. 2, we assume the initial stress exceeds the σ_g stress and the volume of the diffused layer lies between the planes y = a and y = h. Let C be the solute concentration at the bottom surface y and the concentration at the upper surface then becomes C + (∂C/∂y)dy. Thus, the evolution of the plastic strain tensor during lithiation is dependent on the dislocation density (ρ), according to the result given by Prussin, the expression for the dislocation density in therms of the Li⁺ concentration gradient is:

ρ = 0, 0 < |y| < a (5)

where b_y is the magnitude of the edge component of the Burgers vector of the dislocation in the y direction. With increasing diffusion, the stress imposed by lattice contraction decreases at the surface. The dislocation will move into the interior. According to Estrin’s theory,³⁰ the diffusion induced dislocation stress can be expressed by

σ_x¹ = 0, b < |y| < a (7)

where M is the average Taylor factor. The variation of M can be included in the model, however, in what follows, M will be considered constant for simplicity. G is the shear modulus and φ is a numerical constant. It is very difficult, if not impossible, to obtain the exact solution of the resultant stress distribution of the electrode when the dislocation is considered. Thus, the resultant stress at the electrode can be approximated as

Fig. 2 An element of a diffused layer of nanostructured thin film electrode which is affected by the distribution of dislocation.

Hence, the distribution of stress with the dislocation effect can be obtained if the composition profile is known.

2.2 Diffusion and dislocation under potentiostatic and galvanostatic charging

In this paper, the charging conditions of potentiostatic and galvanostatic are considered. Under the potentiostatic conditions, Li-ion battery is charged with a constant voltage. The initial and boundary conditions can be expressed as

C(h, 0) = C_h, C(0, t) = finite (10)

The initial Li⁺ concentration in the electrode is denoted as zero. The analytical solution with regard to the diffusion problem is well known and the expression during charging could be expressed by²⁸

So the stress, without considering the dislocation effect, can be obtained by substituting it into eqn (3)

The variation in the solute concentration profiles can be sketched out by using eqn (8) in Fig. 3(a) during the first charging cycle. Due to the symmetry of the problem, the results are plotted only over half of the thin film width. The concentration continuously rises with charging time at the same location during insertion. Corresponding to the conditions shown, the DIS distribution shown in Fig. 3(b) will exist without considering the dislocation effect. As indicated in Fig. 3(b), the stress is tensile near the center and compressive near the free surface of the electrode. In addition, the tensile stress at the center appears before the solute reaches there. At the center, the stress always has the same magnitude, so that the stress at the center is purely hydrostatic in tension. In particular, the maximum stress, which is compressive, occurs at the surface.

Fig. 3 The concentration solute distribution and variation in stress at dimensionless time under potentiostatic charging. (a) Concentration during lithiation, (b) DIS versus the locations, (c) DIS versus dimensionless time. The concentration is normalized by C_h, while the stresses are normalized by EβC_h/1 − µ.

At any location, the stress first increases, reaches a maximum, then decreases with charging time and tends to a steady-state, as shown in Fig. 3(c). There are some dislocations in the electrode during the diffusion process. According to the theory of the dislocation in this paper, the dislocation induced stress can be written as

σ_x¹ = 0, 0 < |y| < a (15)

For the purpose of accounting for the dislocation effect, the parameters of Li-ion battery electrode materials under potentiostatic operation, which are listed in Table 1, are cited from ref. 18, 32 and 33. For simplicity, we assume the a is about half of the width (a = h/2) of the electrode. From Fig. 4, normalized dislocation induced stress always maintains the tensile stress state with different locations and charging time during insertion. At any location, the dislocation induced stress will first increase, reach a maximum, and then decrease with the charging time. What’s more, the maximum dislocation induced stress near to the surface of the electrode will reduce as the time increases. This is due to the reason that the constant surface concentration suppresses dislocation nucleation over time. Fig. 5 shows the coupling stress between diffusion and dislocation induced stress on the mechanical behavior of the electrode during potentiostatic charging. The dotted lines represent the stresses when the dislocation effect is considered and the solid lines represent the stresses when the dislocation effect is not considered. If the dislocation effect exists, there is a significant decrease in the tensile stress at the nanoscale, and it can even make the tensile stress become compressive stress at different times. Through comparative analysis, the magnitude of the stresses is significantly decreased at any given time when the dislocation is considered.

Table 1 Nanostructured Si electrode material properties and operating parameters

Parameter	Symbol	Value
a Ref. 18.b Ref. 32.c Ref. 33.
Diffusion coefficient	D	1.2 × 10^−18^a m^2 s^−1
Young’s modulus of lithiated Si	E	90.13^a GPa
Molar volume of Si	Ω	1.2052 × 10^−5^a m^3 mol^−1
Maximum concentration	Cmax	2.0152104 × 10^4^a m^3 mol^−1
Poisson’s ratio of Si electrode	µ	0.22^a
Taylor orientation factor	M	1.732–3.06^b
Empirical constant	φ	0.33^b
Burgers vector	by	2.532^c Å
Current density	I	0.11 A m^−2

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Fig. 4 The variation in dislocation induced stress at different charging times and width locations during insertion under potentiostatic charging. The stresses are normalized by .

Fig. 5 The variation in stresses owing to the solute stress at dimensionless time under potentiostatic charging. The stresses are normalized by EβC_h/1 − µ. The dotted lines represent the stresses considering the dislocation effect and the solid lines represent the stresses without considering the dislocation effect.

Under galvanostatic charging, the charging rate is assumed to be 1 C. To understand the stress due to diffusion, the boundary conditions under galvanostatic charging can be defined as

where I is a constant current density and F = 96 486.7 C mol⁻¹ is Faraday’s constant. The analytical solution with regard to the diffusion problem is well known and the expression during the insertion can be expressed by²⁸

So the DIS can be given by

The result for insertion normalized concentration under galvanostatic charging is shown in Fig. 6(a). The concentration continuously rises with charging time and different locations. Normalizing the stress makes it straightforward to compare the results in the thin film electrode, as illustrated in Fig. 6(b). The DIS is tensile at the center and compressive at the surface, as shown in Fig. 6(b), without considering the dislocation effect. At the center, the tensile stress occurs before the solute reaches the center, and the maximum tensile stress arises when the compressive stress at the surface is of the same magnitude. At any location, the stress increases in magnitude with charging time and will tend to a steady-state. The peak stress occurs at the surface after reaching the steady state

Fig. 6 The distribution of solute concentration and diffusion induced stress under galvanostatic charging. (a) Solute concentration during lithiation and (b) DIS during lithiation. The concentration is normalized by C_hFD/Ih, while the stress is normalized by EβFD/Ih(1 − µ).

During galvanostatic charging, there are also some dislocations in the electrode because of solute diffusion. According to the theory of the dislocation in this paper, the dislocation induced stress can be written as.

σ¹(y, t) = 0, 0 < |y| < a (22)

During the charging process, the dislocation effect can contribute to the stress distribution because of Li⁺ diffusion under galvanostatic charging. The dislocation at the surface moves into the interior. Normalized dislocation induced stress increases monotonically to a steady state with charging time during lithiation, as shown in Fig. 7. For insertion, the dislocation induced stress is tensile in the electrode. The maximum dislocation induced stress occurs at the surface of the electrode and can be given by σ_max¹ = MGφ(βI/b_yFD)^1/2. Here, we will consider the coupling stress between the DIS and the dislocation induced stress in the thin film electrode. The dotted lines represent the stress when the dislocation effect is considered and the solid lines represent the stress when the dislocation effect is not considered. Fig. 8 shows the effect of dislocation on stress for the Si electrode with varying times. From Fig. 8, it is obvious that the magnitude of the tensile stress may be significantly decreased and even reverted to a state of compressive stress. Significantly, there is an evident decrease in the tensile stress at the nanoscale level.

Fig. 7 Snapshot variations of the dislocation induced stress at different charging times and locations under galvanostatic charging. The stresses are normalized by .

Fig. 8 Variations in the coupling stress in a thin film electrode under galvanostatic charging. The stresses are normalized by EβFD/Ih(1 − µ). The dotted lines represent the stresses when the dislocation effect is considered and the solid lines represent the stresses when the dislocation effect is not considered.

2.3 The maximum stress in the nanostructured thin film electrode

On top of this, the dislocation effect should be considered in the nanostructured thin film electrode. Under the charging conditions of potentiostatic and galvanostatic, the different maximum principal stresses are tensile, occurring at the center or surface of the electrode during steady insertion starting from uniform lithium concentration. The maximum stress, considering the dislocation effect in the nanostructured electrode, can be expressed as

σ_max = σ_xmax − σ_x¹max (23)

During the potentiostatic and galvanostatic processes, the maximum stresses will be proposed in this paper. Fig. 9 shows the normalized center stress and maximum stress σ_max when the dislocation stress in the electrode is considered during potentiostatic and galvanostatic charging. What’s more, the normalized center stress or maximum stress under galvanostatic charging is weaker under potentiostatic charging. The main reason is that the gradient of the Li⁺ concentration under galvanostatic charging is lower than that under potentiostatic charging. Significantly, the normalized maximum stress is greater than that for galvanostatic charging in potentiostatic charging at the beginning of charging. However, the normalized maximum stress for galvanostatic charging is greater than that for potentiostatic charging later in the lithiation process, as shown in Fig. 9. Therefore, it is obvious that we will first use the galvanostatic charging operation followed by potentiostatic charging to reduce the stress in the Li-ion battery. This dislocation induced stress may be one of the factors responsible for the observed behaviour of the fracturing of the nanoparticles used in a Li-ion battery under the two charging operations. Furthermore, the stress analysis can propose solutions for avoiding fractures because of Li⁺ diffusion in the nanostructured electrode.

Fig. 9 Normalized maximum stress taking into account the dislocation effect under potentiostatic charging and galvanostatic charging.

3 Strain energy of the nanostructured thin film electrode

To estimate the fracture resistance, we can also discuss the strain energy in the electrode. For nanoscale films, the total strain energy includes the bulk energy and dislocation strain energy. Based on the theory of the linear incremental constitutive relation,³¹ we assume the incremental deformation is infinitesimal. When small deformations are supposed, the strain energy release rates will be change linearly with the size. The bulk strain energy density can be calculated to vary as a result of the deformation for the isotropically deformed film.

Bulk strain energy, which is stored in the electrode because of elastic deformation, can be obtained by integrating the strain energy density over the entire volume of the thin film

From the above, we can define strain energy with dimensionless form as G = G_T/2Eh³(βC_h/(1 − µ))². Fig. 10(a) shows the variation of the bulk strain energy for representative values of Poisson’s ratio under potentiostatic charging without considering the dislocation induced strain energy. The difference in the strain energy profiles in the four cases is due to the different Poisson’s ratio values. In the case of the plane stress condition, the total strain energy first increases, reaches a maximum, and then decreases, because of the transient nature of the stresses. Under galvanostatic charging, similar to the potentiostatic case, Poisson’s ratio affects the stored energy significantly. However, the bulk strain energy increases monotonically initially and reaches a peak value as the charging time progresses. In addition, the bulk strain energy increases monotonically and consistently reaches a steady-state with charging time, as shown in Fig. 10(b).

Fig. 10 Strain energy versus time without considering the dislocation effect for potentiostatic and galvanostatic charging (a) under potentiostatic charging and (b) under galvanostatic charging.

Consideration is now given to the dislocation energy in the electrode. In the case of a dislocation, a cut is made in the medium to the dislocation line and the faces of the cut are displaced relatively by a Burgers vector. For a single Volterra dislocation in a strained body, the elastic energy per unit length evaluates to²⁷

When diffusion proceeds under the conditions of constant surface concentration, the total number of dislocations N per unit length of the diffused layer can be determined as follows

The strain energy due to dislocation can be expressed by

When the dislocation is considered, the strain energy can be plotted across the 200 nm width of the thin film electrode, as shown in Fig. 11. Under potentiostatic charging, Fig. 11(a) shows that the dislocation strain energy initially increases, reaches a peak value, and then decreases for the nanostructured thin film electrode. During galvanostatic charging, the dislocation strain energy initially increases and reaches a peak value as the charging time progresses. The dislocation induced deformation has a great impact on the total strain energy of the electrode, as shown in Fig. 11(b). It is shown that the dislocation induced strain energy has a comparable magnitude to that of the bulk strain energy for the nanostructured electrode. In addition, Fig. 11 shows that the dislocation effect under galvanostatic charging is greater than that under potentiostatic charging . This is because galvanostatic charging needs more charging time than potentiostatic charging. As a result, the dislocation effect would induce more strain energy under galvanostatic charging than under potentiostatic charging. Finally, the normalized total strain energy of the two charging conditions is compared. Significantly, the normalized total strain energy is greater than that with galvanostatic charging under the potentiostatic operation at the beginning of charging. However, the total strain energy for galvanostatic charging is greater than that for potentiostatic charging later in the lithiation process, as shown in Fig. 12. Therefore, it is obvious that we will first use the galvanostatic charging operation followed by the potentiostatic charging operation to avoiding fractures in the Li-ion battery. These results also show that it is possible to control the dislocation density to improve Li-ion battery life.

Fig. 11 Normalized bulk strain energy, dislocation strain energy and total strain energy with time for a 200 nm electrode at µ = 0.3 (a) under potentiostatic charging and (b) under galvanostatic charging.

Fig. 12 Normalized total strain energy versus time under potentiostatic and galvanostatic charging when the dislocation strain energy is considered.

4 Conclusions

An analytical model has been developed to analyze diffusion induced stress and the distribution of dislocations in a nanostructured electrode undergoing potentiostatic and galvanostatic charging by using a combination of diffusion kinetics and dislocation theory. Most of theoretical models have been studied the only diffusion induced stress during charging. When using the nanostructured thin film electrode, dislocation induced stress plays an important role in a Li-ion battery during lithiation. A new model was established with dislocation mechanisms for the electrode. With the dislocation effects, the magnitude of tensile stresses is significantly decreased at any given time for both galvanostatic and potentiostatic charging. It is shown that the dislocation effect would induce more stress under galvanostatic charging than under potentiostatic charging. Finally, the effect of dislocations on the maximum stress and strain energy was established during galvanostatic or potentiostatic charging. In addition, the effect of dislocations on the maximum stress and strain energy was greater under the potentiostatic operation than under the galvanostatic operation at the beginning of charging. However, the maximum stress and strain energy under potentiostatic charging were lower than those under galvanostatic charging later in the lithiation process. These results show that using the galvanostatic charging operation first and then using the potentiostatic charging operation, when taking dislocation density into consideration, can mitigate the stress in the nanostructured thin film electrode. This can provide a strategy to prolong Li-ion battery life.

Acknowledgements

This work was supported by Key Project of Chinese Ministry of Education (211061), National Natural Science Foundation of China (10502025, 10872087, 11272143), Program for Chinese New Century Excellent Talents in university (NCET-12-0712). Ph.D. programs Foundation of Ministry of Education of China (20133221110008).

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