Force-chemical coupling analysis of nanocomposite anode during charging and discharging process

Abstract

Anode materials are a key part of lithium-ion batteries, of which silicon-based anodes are considered the most promising electrode materials due to their high theoretical specific capacity. However, during the operation of the battery, the silicon material undergoes a huge volume change resulting in damage to the electrodes. Currently, the use of additive particles to make composite electrodes is a more reasonable approach. During the charge/discharge cycle of the battery, the electrodes may undergo cyclic expansion/contraction and may undergo elastic deformation and inelastic deformation, and lead to electrode ratchet deformation and capacity degradation. In this paper, a coupled nanoelectrode mechanics-electrochemistry model is developed to investigate the electrode stress evolution and strain accumulation of nanocomposite negative electrode during charge/discharge cycling under different influencing factors, and to analyze the change of electrochemical properties due to diffusion stress. Our work shows that the use of composite matrix materials, on the other hand, exhibits better mechanical stability, has smaller inelastic strains after a cycle, and produces less accumulation of irreversible strains. The established analytical model of nanocomposite electrode helps electrode design and is instructive for the preparation and structural design of nanocomposite negative electrodes.

Appendices

Appendix A

In order to better characterize the degree of agglomeration of GNS in composites, the following two agglomeration parameters ζ and λ are introduced:

$$\zeta = \phi_{a}$$

(A.1)

$$\lambda = \frac{{\phi_{r}^{a} }}{{\phi_{r} }}$$

(A.2)

where ${\phi }_{a}$ is the volume fraction of the agglomeration domain,${\phi }_{r}$ is the average volume fraction of GNS and ${\phi }_{r}^{a}$ is the volume fraction of GNS in the agglomeration domain.

The mechanical properties of the composites were calculated using the Mori–Tanaka method when the reinforced GNS is distributed in the matrix along a random direction, assuming that the effective modulus of the matrix and agglomerate domains is

$$K_{m} = K_{0} + \frac{{\phi_{r} \left( {\delta_{r} - 3K_{0} \alpha_{r} } \right)\left( {1 - \lambda } \right)}}{{3\left[ {1 - \zeta - \phi_{r} \left( {1 - \lambda } \right) + \phi_{r} \left( {1 - \lambda } \right)\alpha_{r} } \right]}}$$

(A.3)

$$= G_{m} G_{0} + \frac{{\phi_{r} \left( {\eta_{r} - 2G_{0} \beta_{r} } \right)\left( {1 - \lambda } \right)}}{{2\left[ {1 - \zeta - \phi_{r} \left( {1 - \lambda } \right) + \phi_{r} \left( {1 - \lambda } \right)\beta_{r} } \right]}}$$

(A.4)

$$K_{a} = K_{0} + \frac{{\left( {\delta_{r} - 3K_{0} \alpha_{r} } \right)\phi_{r} \lambda }}{{3\left[ {\zeta - \phi_{r} \lambda + \phi_{r} \lambda \alpha_{r} } \right]}}$$

(A.5)

$$G_{a} = G_{0} + \frac{{\left( {\eta_{r} - 2G_{0} \beta_{r} } \right)\phi_{r} \lambda }}{{2\left[ {\zeta - \phi_{r} \lambda + \phi_{r} \lambda \alpha_{r} } \right]}}$$

(A.6)

Based on previous work,${k}_{\alpha }=678{\text{GPa}}$,${l}_{r}=245{\text{GPa}}, {m}_{r}={p}_{r}=368{\text{GPa}}$ and ${n}_{r}=1099{\text{GPa}}$ were selected.

For the effective modulus of the composite, it can be calculated

$$K = K_{m} \left[ {1 + \frac{{\zeta \left( {\frac{{K_{a} }}{{K_{m} }} - 1} \right)}}{{1 + \alpha \left( {1 - \zeta } \right)\left( {\frac{{K_{a} }}{{K_{m} }} - 1} \right)}}} \right]$$

(A.7)

$$G = G_{m} \left[ {1 + \frac{{\zeta \left( {\frac{{G_{a} }}{{G_{m} }} - 1} \right)}}{{1 + \beta \left( {1 - \zeta } \right)\left( {\frac{{G_{a} }}{{G_{m} }} - 1} \right)}}} \right]$$

(A.8)

where $\alpha =\frac{3{K}_{m}}{3{K}_{m}+4{G}_{m}}$ and $\beta =\frac{6({K}_{m}+{G}_{m})}{5(3{K}_{m}+4{G}_{m})}$.

The elastic modulus of the composite is given by

$$E = \frac{9KG}{{3K + G}}$$

(A.9)

Appendix B

For sheet-reinforced GNS, the modulus of elasticity along the length and width directions are described by the Halpin–Tsai equations, respectively, as

$$\frac{{E_{l} }}{{E_{0} }} = \frac{{1 + \zeta_{l} \chi_{l} \phi }}{{1 - \chi_{l} \phi }}$$

(B.1)

$$\frac{{E_{w} }}{{E_{0} }} = \frac{{1 + \zeta_{w} \chi_{w} \phi }}{{1 - \chi_{w} \phi }}$$

(B.2)

where E is the modulus of the composite material, and the subscripts l, w, 0, and s represent the longitudinal, transverse, initial matrix, and additives, respectively, $\zeta$ is the geometric measurement parameter depending on the load condition, and the subscripts l and w represent the longitudinal and transverse direction, respectively. In the case of such a volume fraction, there is $\zeta =2\frac{a}{b}$, in which a is the length along the force direction, b is the width in the cross-sectional direction.$\upchi =\frac{{E}_{s}/{E}_{0}-1}{{E}_{s}/{E}_{0}+\zeta }$ is the composite modulus control factor which is related to the material properties of the initial matrix and the additive.$\phi$ is the volume fraction of the additives. It should be noted that $\phi$ here has the same meaning as ${\phi }_{r}$ in “Appendix A.”