An analytical multiscale modeling of a nanocomposite anode with graphene nanosheets for lithium-ion battery

Abstract

An analytical calculation of diffusion-induced stress in a layered electrode composed of a current collector and a nanocomposite active plate for lithium-ion batteries is proposed. The active plate of the bilayer nanocomposite anode is reinforced by graphene nanosheets (GNSs), whose mechanical properties are predicted by the Mori–Tanaka micromechanical method. The reinforced GNSs will make agglomeration in the matrix, which has a certain obstacle to reducing the overall stress level of the anode. The size effect of a single GNS also affects the corresponding material properties, and the Halpin–Tsai equation can be used to acquire the reliable prediction results. The analytical solutions of the diffusion-induced stress and strain of the nanocomposite anode under galvanostatic charging operation are derived, and then, the elastoplastic stress distribution in the interior of the anode is obtained. In addition, the effect of GNS’s thickness on the migration of the plastic interface is also evaluated.

Appendices

Appendix A

When yield failure occurs in plastic materials, von Mises stress is generally used as the failure criterion. The von Mises stress is calculated according to the principal stress in three directions, and the formula is as follows:

$$ \sigma_{Y} = \sqrt {\frac{1}{2}\left[ {\left( {\sigma_{1} - \sigma_{2} } \right)^{2} + \left( {\sigma_{2} - \sigma_{3} } \right)^{2} + \left( {\sigma_{3} - \sigma_{1} } \right)^{2} } \right]} $$

(A.1)

where $\sigma_{Y}$ is the von Mises stress and $\sigma_{1}$, $\sigma_{2}$, $\sigma_{3}$ are the first, second, and third principal stresses, respectively. Based on the generalized plane strain theory of thin plates, there are $\sigma_{1} = \sigma_{{{{x}}x}}$, $\sigma_{2} = \sigma_{yy}$, and $\sigma_{3} = \sigma_{{{{zz}}}}$. Obtained by Eq. (2), $\sigma_{1} = \sigma_{2}$, $\sigma_{3} = 0$, so on the yield surface, the stress can be expressed as:

$$ \sigma_{Y} = \sigma_{xx}, $$

(A.2)

Appendix B

For elastoplastic materials, there is $\Delta \sigma = C_{T} \Delta \varepsilon$, where

$$ C_{T} = \left\{ \begin{gathered} E, \, \left| \sigma \right| < \sigma_{Y} \hfill \\ E_{t} , \, \left| \sigma \right| > \sigma_{Y} \hfill \\ \end{gathered}. \right. $$

(A.3)

The physical meaning of $E$ and $\sigma_{Y}$ is consistent with the paper, and $E_{t}$ is the strain hardening modulus. As shown in Fig. 

Fig. 9

Schematic diagram of the strain hardening process

9, the strain increment in the plastic stage is divided into elastic and plastic parts, namely

$$ \Delta \varepsilon = \Delta \varepsilon_{e} + \Delta \varepsilon_{p}. $$

(A.4)

Integrate the above equation to obtain

$$ \varepsilon_{xx} = \int {\Delta \varepsilon } {\text{d}}x = \int {\left( {\Delta \varepsilon_{e} + \Delta \varepsilon_{p} } \right)} {\text{d}}x = \varepsilon_{xx}^{e} + \varepsilon_{xx}^{p} = \frac{1}{E}\sigma_{xx} + \varepsilon_{xx}^{p} $$

(A.5)

where $\varepsilon_{xx}$ is the total strain in the plastic stage, and the superscripts $e$ and $p$ represent the elastic and plastic parts of the plastic stage, respectively. Likewise, the stress increment $\Delta \sigma$ in the plastic stage can be expressed in any of the three moduli, namely

$$ \Delta \sigma = E\Delta \varepsilon_{e} = E_{p} \Delta \varepsilon_{p} = E_{t} \Delta \varepsilon $$

(A.6)

Therefore, in the plastic domain, the stress distribution can be expressed as

$$ \sigma_{xx} = \sigma_{Y} + \int {\Delta \sigma } {\text{d}}x = \sigma_{Y} + E_{p} \varepsilon_{xx}^{p}, $$

(A.7)

Substituting Eqs. (1) and (A.5) into Eq. (A.7) yields

$$ \sigma_{xx} = \frac{{E^{{\prime }} }}{{E^{{\prime }} + E_{p} }}\left[ {\sigma_{Y} + E_{p} \left( {\varepsilon_{0} + \kappa z - \frac{1}{3}\Omega C} \right)} \right]. $$

(A.8)

Appendix C

The representative volume element model for GNSs agglomeration is shown in Fig. 

Fig. 10

Schematic diagram of a representative volume element model for GNSs agglomeration

10. As has been illustrated in the paper, the composite matrix volume $V$ is consisting of two parts: the hypothetical matrix volume $V_{m}$ which is made of the silicon initial matrix and the GNSs dispersed in random direction, and the agglomeration domain volume $V_{a}$, namely

$$ V = V_{m} + V_{a}. $$

(A.9)

According to the volume relationship of each component of the composite material, the following can be obtained:

$$ \phi_{m} = \frac{{V_{m} }}{V},\phi_{a} = \frac{{V_{a} }}{V}, $$

(A.10)

where $\phi_{m}$ is the volume fraction of the hypothetical matrix, and $\phi_{a}$ is the volume fraction of the agglomeration domain. Suppose the volume of the additive in the composite is $V_{r}$, the volume of the additive in the hypothetical matrix is $V_{r}^{m}$, and the volume of the additive in the agglomeration domain is $V_{r}^{a}$, then there is $V_{r} = V_{r}^{m} + V_{r}^{a}$. According to the volume conservation relationship, it can be obtained

$$ \begin{aligned} & \phi_{r} = \frac{{V_{r} }}{V}, \\ & \phi_{r}^{m} = \frac{{V_{r}^{m} }}{{V_{r} }}, \\ & \phi_{r}^{a} = \frac{{V_{r}^{a} }}{{V_{r} }}, \\ & \phi_{r} = \phi_{r}^{m} + \phi_{r}^{a} \\ \end{aligned} $$

(A.11)

where $\phi_{r}$ is the average volume fraction of GNSs, $\phi_{r}^{m}$ is the volume fraction of GNSs in the hypothetical matrix, and $\phi_{r}^{a}$ is the volume fraction of GNSs in the agglomeration domain.

Overall, the agglomeration parameters $\xi$ and $\lambda$ introduced in the paper are defined as

$$ \begin{aligned} & \xi = \frac{{V_{a} }}{V} = \phi_{a}, \\ & \lambda = \frac{{V_{r}^{a} }}{{V_{r} }} = \frac{{\phi_{r}^{a} }}{{\phi_{r} }}, \\ \end{aligned} $$

(A.12)

It should be noted that for a given average volume fraction of GNSs, $\phi_{r}^{a}$ must be less than $\xi$, that is, $\lambda < \frac{\xi }{{\phi_{r} }}$. Similarly, for the agglomeration domain, the volume occupied by GNSs $\frac{{V_{r}^{a} }}{{V_{a} }}$ should be larger than the average volume fraction of GNSs $\phi_{r}$, i.e., $\frac{{\phi_{r} \lambda }}{\xi } > \phi_{r}$. To sum up, it can be concluded that $0 < \xi < \lambda < 1$.