Fracture damage of nanowire lithium-ion battery electrode affected by diffusion-induced stress and bending during lithiation

Abstract

Lithium-ion battery electrode materials generally experience significant volume changes during lithium diffusion in charging and discharging. These volume changes lead to diffusion-induced stress and defect nucleation. By analyzing the stress, the bending associated with lithiation, we find that size reduction of the electrode can avoid the phenomenon of volume change. In this work, we build a relationship among the stress, bending, strain energy and size of the electrode. Also, the fracture energy of the electrode materials, which can be evaluated further by the strain energy, has been derived to optimize the electrode size for maximizing the battery life. Finally, a critical electrode size determination method is proposed.

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1. Introduction

In recent years, many investigations^1–3 have been performed to study lithium-ion battery cells as secondary battery systems because of their high energy density, high efficiency of charge–discharge, and high operating voltages compared to conventional cells. It has been recognized that the development of battery electrodes materials will be a critical factor for large-capacity energy storage, so many materials, such as Si, Sn, MnO₂, Sb, Mg, and Bi, have received attention.⁴ For example, it has been found that use of tin oxide nanowires as electrode materials can increase the discharge capacity to about twice that of lithium-ion rechargeable batteries that use graphite carbon.⁵ SnO₂ is one of the most promising anode materials to replace the carbonaceous anodes used in Li-ion batteries, because of having a theoretical capacity of 781 mAh g⁻¹ and a reversible capacity exceeding 500 mAh g⁻¹.^6–9 However, because there are very large stresses owing to volume changes (of up to 260%)^10–12 during Li insertion/extraction processes in Li-ion battery electrodes, this can lead to serious irreversible capacity and poor cyclability of the electrodes.¹³ Many experiments have demonstrated^14,15 that size reduction of nanowires is one of the effective strategies to resist fracture.

So far, some previous studies have analyzed the stresses induced by the diffusion of Li⁺. Huggins and Nix¹⁶ have used Griffith's criterion to explain fracture in a one-dimensional model, and Aifantis et al.¹⁷ used Griffith's criterion to estimate the critical crack size under which the crack will not propagate for nanostructures in Li-ion batteries. Recently, dislocations about the diffusion have been researched. Wei et al.¹⁸ analyzed the effect of dislocation mechanics on the diffusion-induced stresses within a spherical particle. To avoid pre-existing cracks, Gao and Bhandakkar^19,20 developed a cohesive model of crack nucleation and suggested a critical characteristic dimension to avoid fracture. Zhao et al.²¹ used the strain energy release rate to estimate the crack propagation and developed a fracture criterion in the electrode particles. Despite these contributions, the diffusion and damage process is still not well understood.

In this paper, we will construct a model to explain the fracture behavior during lithiation and develop a method to predict the critical electrode size. Many modeling results are compared with experiments and they have reached a good agreement on the size reduction effect on the fracture in the electrode.^14–22 Following this, we continue to do further work on how to prevent fracture behavior in the electrode materials. The stress and bending deformation will be first obtained in the Li-ion battery electrodes by finite elasticity–plasticity theory.^18–21 Then, equations to describe the strain energy for nanowires (NWs) containing pre-existing cracks will be proposed. These strain energy values will be used to derive the fracture energy and estimate the critical electrode size.

2. Stress in a cylindrical electrode

According to the theory of lithium batteries, we know that lithium ions will diffuse out of the anode when the battery discharges. The electrode material is assumed to be an isotropic linear elastic solid and the mechanics behavior is changed from elastic to elastic–perfectly plastic. Simultaneously, we assume that lithium transport is purely diffusive in active material particles. In this paper, the schematic illustration of stress in a cylindrical electrode at the surface current density under galvanostatic conditions as shown in Fig. 1. There are diffusion process equations to deal with the diffusive flow of lithium by²³where D is the diffusion coefficient as a constant, and c is the molar concentration of the solute. The transformation strain ε_t owing to insertion of solute atoms connects with the partial molar volume Ω and the concentration c:We assume that the surface lithium ion concentration is C_R and the initial lithium ion concentration is C₀ in the electrode, so the initial and boundary conditions are given by:

C(r,0) = C₀, for 0 ≤ r ≤ R (3)

C(R,0) = C_R, for r = R (4)

We consider the stress caused by Li⁺ diffusion within a nanowire of radius R. The stress–stain relationships are expressed in the spherical coordinate system using the analogy between thermal- and diffusion-induced stresses.^{18–20,24–26} And the theory of surface elasticity has been developed by Gurtin et al.²⁷ So we can know the general relationships for the stresses and strain in a cylinder,where μ is the Poisson ratio of the material and E is the Young's modulus. and the diffusion process is along the r-direction of the electrode. The equation for static mechanical equilibrium and kinematic relations in the cylinder of the NW is given by:²⁶Under the plane strain conditions, the radial, tangential and axial stresses that satisfy the boundary condition in the cylinder by the dynamics can be given by:Hence, the diffusion-induced stresses at any location of the electrode and time without considering the bending effect near the ends can be obtained if the composition profile is known.

Fig. 1 Schematic illustration of stress in a cylindrical electrode at the surface current density under galvanostatic conditions.

3. Bending stress in the electrode

Understanding the diffusion-induced stress in the cylinder electrodes, there is no bending at points a large distance from the free ends. So the stresses can be calculated from eqns (9)–(11). However, there is some bending near the end,³⁴ which will generate chemical stresses in the elastic cylinder. For the non-uniform distribution of the solute atoms, we will consider the bending deformation of the cylinder because of the diffusion-induced stress in the cylinder electrode. According the Euler–Bernoulli beam assumption through incremental deformation theory, the bending moments can be expressed asSo the bending moments that introduce the stress can be found asTo prevent the cross-sections of the strip from distorting during the diffusion-induced bending, the moment applied to the strip is μM,Thus, the stress can be approximately near the ends:To comprehend the stress due to the diffusion, the boundary conditions under galvanostatic charging can be found aswhere J_n is the surface current density. During the charging, The J_n can be obtained as

J_n = J₀(1 − C_R/C_max) (19)

where J₀ is the charging rate. Eqn (19) is a linearized form of the Butler–Volmer equation.²⁸ The solute concentration during lithiation can be expressed by¹³During charging, the Li⁺ will be extracted from the anode and inserted into the cathode. It will be the opposite in the discharging. This is only discussing the situation of lithiation. So the diffusion-induced stress and bending stress can be calculated by the stress solution and the concentration solution in the plane strain during lithiation.

The concentration profiles can be depicted using eqn (18) in Fig. 2. From Fig. 2, we can see that the concentration continuously rises with charge time; then, the concentration profile will reach a steady state after some charge time during insertion. Without considering the bending moment, the diffusion-induced stresses will be shown as in Fig. 3 during insertion. The radial stress is tensile and the maximum radial stress occurs at the center in the cylinder in the Fig. 3(a). In addition, the radial stress will be decreased with increasing the concentration. From Fig. 2 and 3(a), we see that the maximum stress occurs at the center when the time after the solute reaches the center of the cylinder. The tangential stress is compressive near the surface and tensile near the center of the cylinder in Fig. 3(b). In addition, the maximum tangential stress will be at the surface of the cylinder. Fig. 3(c) shows that the axial stress is tensile near the center and compressive near the surface of the electrode. It is worth noting that the axial stress reaches a stabilized condition and the maximum tensile stress occurs at the center. To indicate the bending impact, the bending stress is shown in the Fig. 4. The variation of the bending stress with the diffusion time and different diameters will be shown near the ends. What is more, the bending stress of the cylindrical electrode is always in a compressive state under the potentiostatic operation. In the paper, radial stress is plotted in Fig. 5 at the same charging rate and the short dotted lines represent the axial stress with consideration of the bending deformation. From Fig. 4 and 5, we can see that the bending deformation effect cannot be negligible. For insertion, the axial stress is plotted with different electrode sizes at dimensionless time T = 0.5 in Fig. 6. It is obvious that the maximum stress at the center will increase with the increase in the diameter. From Fig. 2–6, it is clear that the size of the electrode is very important to the cylindrical electrode.

Fig. 2 Lithium ion concentration profile in the electrode at different radial locations and times during insertion.

Fig. 3 Profiles of diffusion-induced stress during insertion, (a) radial stress during insertion at the different locations and charging times, (b) tangential stress at the different locations and times. (c) Axial stress at different locations and times. The stress is normalized as .

Fig. 4 Profiles of bending stress at the diffusion time and different diameters near the ends. The stress is normalized as .

Fig. 5 Radial stress during insertion for various charging times. The solid lines represent the stress without the bending effect and the short dotted lines represent the stress considering the bending effect.

Fig. 6 Profiles of the axial stress with different diameters at dimensionless T = 0.5.

4. Strain energy and fracture in the cylindrical electrode

To estimate the fracture resistance, we assume that cracks may exist in the NW of the battery. Ref. 29 shows the strain energy in the particles. For nanoscale wires, the strain energy contains the bulk energy and the bending energy. According to the theory of the linear incremental constitutive relation,³⁰ we suppose that the incremental deformation is infinitesimal. When small deformations are assumed, the strain energy release rates will be linear with the size. We can calculate the strain energy density accumulated as a result of the deformation for the isotropically deformed cylinderThe strain energy, which is stored in the electrode because of the elastic deformation, and the section strain energy can be obtained by integrating the strain energy density over the entire volume of the NW:Because of the bending moment effects, the energy due to the bending stress should be considered. A schematic illustration of bending near the ends is shown in Fig. 7. The bending of the cylinder electrode can be calculated byThe solution of eqn (23) with the moment applied to the end iswhere and ∂ = Er³/12(1 − μ²) is the flexural rigidity in the axial direction. So the largest deflection, which occurs when z becomes zero, will be expressed asThis condition could be similar to the linear stress–strain relations and the energy could be calculated asSo the total strain energy could be written as:

Fig. 7 Schematic illustration about the bending effect in a cylindrical electrode.

From above we can define the strain energy in dimensionless form as Ê_T = E/πE(ΩC/3(1 − μ))². Fig. 8 shows the variation of the strain energy under the axial loading conditions without considering the bending deformation. The difference in the strain energy profiles in the four cases is due to the different the size of NW. From this, the strain energy increases initially and reaches a peak value with the charge time. In addition, the strain energy increased with the size of the electrodes and increasing charging time. To consider the bending deformation, the stain energy is plotted for the 200 nm cylindrical electrode where the short dotted lines represent the strain energy with considering the bending moment in the Fig. 9. From Fig. 9, the bending deformation has a great impact on the electrode in terms of energy. Significantly, we can see that the strain energy is related to the charge time and size of the NW. So the strain energy should take the form of the dimensional considerations

where K may be determined by the elastic boundary-value problem and is affiliated with the length of the crack and time. Here, the stain energy reaches a maximum value E_T for a crack. Letting Γ be the fracture energy of the electrode, when the maximum stain energy is located below the fracture energy of the electrode, crack nucleation will not happen. And this has been schematically shown in Fig. 10. No fracture will occur in the area where E_T < Γ.

Based on the above, we can propose a critical NW size as follows:

It is obvious that the critical size relies on the charging rate. And we plot the critical size related to J₀ in Fig. 11. If the charging rate is known, the critical electrode size can be figured out. Many experimental observations^15,31,33 have shown that the critical size of a Si wire electrode is in the range of 220–260 nm at the rate of 0.2 mV s⁻¹. We may predict the critical size using the same experimental data and analysis. The critical size is about 240 nm by eqn (29) and using the parameters in Table 1. It agrees with the experimental data^15,31 at the same charging rate.

Fig. 8 Profiles of strain energy with different diameters The strain energy varies with time and increases with the diameters of the electrodes. The strain energy is normalized as .

Fig. 9 Strain energy at the 200 nm cylindrical electrode. The solid line represents the energy without the bending effect and the short dots line represents the energy considering the bending effect. The strain energy is normalized as .

Fig. 10 Schematically sketched concept for the criteria to avert fracture for the electrode.

Fig. 11 Critical electrode size at different fracture energies with different charging rates.

Table 1 Material properties and operating parameters

Parameter	Symbol	Value
a Ref. 20.b Ref. 32.c Ref. 22.
Diffusion coefficient	D	2 × 10^−18 m^2 s^−1 ^a
Young's modulus of lithiated Si	E	90.13 GPa^b
Molar volume of Si	Ω	1.2052 × 10^−5 m^3 mol^−1 ^a
Poisson's ratio of Si electrode	μ	0.22^a
Maximum concentration	Cmax	2.0152104 × 10^4 m^3 mol^−1 ^a
Fracture energy	Γ	2 J m^−2 ^c

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5. Conclusions

We have developed a method to evaluate the stress in a cylindrical nanostructured electrode under galvanostatic conditions using a combination of diffusion kinetics and fracture mechanics. To predict the fracture, we analyze the stresses and bending deformation associated with lithiation and use the strain energy to predict the fracture behavior. The strain energy is derived under the bending deformation considered and if the maximum strain energy is less than the fracture energy of the material, there is no crack nucleation and propagation. Finally, the critical size was obtained by the fracture energy and the boundary conditions about the plane strain of the nanowire electrodes that were considered. The critical size is related to the charging rate, material properties and fracture energy. To illustrate the theory, a numerical example of the Si NW is calculated. In this case we can see that the diameter of the Si NW, which is smaller than 220–260 nm, will not fail in the lithiation process. The conclusions are in agreement with the experiments. Thus the method in this paper can be used to optimize the size of cylindrical electrodes and maximize 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).

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