Effect of electrode processing on the stability of electrode structure

Abstract

Compared with a large number of researches on the stress analysis of lithiation active particles and electrode plates, the study on structural stability of the electrode composites during the service of lithium-ion batteries (LIBs) is bare. In order to account for the large deformation that occurs during the axial force loading, the models of geometrically nonlinear deformation are proposed, and the effects of the drying process, chemo-mechanical coupling and concentration-dependent material properties on the stability are investigated. The imperfect interface properties of the collector/active layer are modeled by a cohesive zone model (CZM), which originates from different drying conditions. As a result, the competition mechanism of the interface failure and buckling instability of the laminated electrode structure is formulated on the basis of our experimental observations and analytical modeling. Moreover, the effects of drying rate, geometric conditions, and charge/discharge rates on the stability of the electrode structure are systematically examined. Our findings can provide valuable insights for designing next-generation mechanically stable electrode composites.

Appendix A

In order to obtain the effective modulus of electrode composites, it is essential to evaluate the volume fractions of different components under various drying conditions. The electrode slurry with a thickness h after drying time t is $h = h_{0} - \int_{0}^{t} \gamma {\text{d}}t$, where $\gamma$ is the drying rate, and $h_{0}$ is the initial thickness. According to the work of Font et al. [40], the concentration of inactive components $C_{{{\text{ICs}}}}$ can be defined by a convective-diffusive equation,

$$\frac{\partial }{\partial t}\left[ {\left( {1 - V_{{{\text{GM}}}} } \right)C_{{{\text{ICs}}}} } \right] = \frac{\partial }{\partial z}\left[ {D_{{{\text{eff}}}} \frac{{\partial C_{{{\text{ICs}}}} }}{\partial z} - F_{{\text{l}}} C_{{{\text{ICs}}}} } \right],$$

(21)

where $V_{{{\text{GM}}}}$ is the volume fraction of graphite material (GM) (i.e., active components). Note that GM is stabilized and can be regarded as uniformly distributed in the electrode slurry according to the previous work [50]. Therefore, $V_{{{\text{GM}}}}$ can be calculated by its initial value $V_{{{\text{GM}}}}^{0}$ as $V_{{{\text{GM}}}} = V_{{{\text{GM}}}}^{0} h_{0} /h$. As reported in Ref. [40], the volume fraction of GM at the beginning and end of drying was 0.28 and 0.5, respectively. Here,$D_{{{\text{eff}}}} = D_{{{\text{ICs}}}} (1 - V_{{{\text{GM}}}} )^{3/2}$ is the effective diffusivity of the tortuous electrode. $D_{{{\text{ICs}}}}$ is the diffusivity of ICs in the solvent. In this equation, $F_{{\text{l}}}$ denotes the liquid volume fraction flux and is given by $F_{{\text{l}}} = - z{\text{d}}V_{{{\text{GM}}}} /{\text{d}}t$. Meanwhile, the drying process model satisfies the boundary conditions

$$\left. {\left( {D_{{{\text{eff}}}} \frac{{\partial C_{{{\text{ICs}}}} }}{\partial z} - F_{{\text{l}}} C_{{{\text{ICs}}}} } \right)} \right|_{z = 0} = 0,\;\left. {\left( {D_{{{\text{eff}}}} \frac{{\partial C_{{{\text{ICs}}}} }}{\partial z} - F_{{\text{l}}} C_{{{\text{ICs}}}} } \right)} \right|_{z = h} = \left( {1 - V_{{{\text{GM}}}} } \right)C_{{{\text{ICs}}}} \frac{{{\text{d}}h}}{{{\text{d}}t}}.$$

(22)

Combining Eqs. (21), (22), we obtain the volume fractions of components in this electrode composites as

$$V_{{{\text{GM}}}}^{{{\text{end}}}} = V_{{{\text{GM}}}}^{{0}} h_{0} /h_{{{\text{end}}}} ,\;\;V_{{{\text{ICs}}}}^{{{\text{end}}}} = \left( {1 - V_{{{\text{GM}}}}^{{{\text{end}}}} } \right)C_{{{\text{ICs}}}}^{{{\text{end}}}} M_{{{\text{ICs}}}} /\rho_{{{\text{ICs}}}}$$

(23)

where $h_{{{\text{end}}}}$ and $C_{{{\text{ICs}}}}^{{{\text{end}}}}$ are the electrode thickness and concentration of ICs of dried composites, and $h_{{\text{a}}} = h_{{{\text{end}}}}$. $M_{{{\text{ICs}}}}$ and $\rho_{{{\text{ICs}}}}$ are the molar mass and density of ICs, and $\rho_{{{\text{ICs}}}}$ can be calculated by the density of binder and conductive additive. Moreover, the effective mechanical properties of electrode composites can be estimated by the combination of open cell theory and the S-combining rules [51]and written by

$$K_{a} = E_{ICs} \frac{{1 + V_{GM}^{end} \xi_{lk} \chi_{k} }}{{1 - V_{GM}^{end} \Psi_{k} \chi_{k} }}\left( {\frac{{V_{GM}^{end} }}{{1 - V_{AM}^{end} }}} \right)^{2} ,G_{a} = \frac{{3E_{ICs} }}{8}\frac{{1 + V_{GM}^{end} \xi_{\lg } \chi_{g} }}{{1 - V_{GM}^{end} \Psi_{g} \chi_{g} }}\left( {\frac{{V_{ICs}^{end} }}{{1 - V_{GM}^{end} }}} \right)^{2}$$

(24)

where $E_{{{\text{ICs}}}}$ is elastic modulus of ICs, which is equal to 2.5 GPa [49]. $\xi_{{{\text{lk}}}}$, $\xi_{\lg }$, $\chi_{{\text{k}}}$, $\chi_{{\text{g}}}$, $\Psi_{{\text{k}}}$, and $\Psi_{{\text{g}}}$ are related to the material properties of active and inactive components and written by

$$\begin{gathered} \xi_{lk} = \frac{{2(1 - 2v_{PICs} )}}{{(1 + v_{PICs} )}},\quad \xi_{\lg } = \frac{{(7 - 5v_{PICs} )}}{{(8 - 10v_{PICs} )}}, \hfill \\ \chi_{k} = \frac{{K_{GM} - K_{PICs} }}{{K_{GM} + \xi_{lk} K_{PICs} }}{\kern 1pt} ,\quad \chi_{g} = \frac{{G_{GM} - G_{PICs} }}{{G_{GM} + \xi_{\lg } G_{PICs} }}, \hfill \\ \Psi_{k} = 1 + \frac{{V_{GM} V_{PICs} (1 - \Gamma V_{PICs} )(K_{GM} - K_{PICs} )(\xi_{uk} - \xi_{lk} )}}{{K_{GM} + \xi_{uk} (V_{AM} K_{GM} + V_{PICs} K_{PICs} )}}, \hfill \\ \Psi_{g} = 1 + \frac{{V_{GM} V_{PICs} (1 - \Gamma V_{PICs} )(G_{GM} - G_{PICs} )(\xi_{ug} - \xi_{\lg } )}}{{G_{GM} + \xi_{ug} (V_{GM} G_{GM} + V_{PICs} G_{PICs} )}} \hfill \\ \end{gathered}$$

(25)

where $v_{{{\text{PICs}}}}$ is the Poisson’s ratio of the ICs porous matrix, and is equal to 1/3. $\Gamma = \left( {2\lambda - 1} \right)/\lambda$, $\lambda$ represents the critical volume fraction and is assumed as 2/3 [52]. $K_{{{\text{PICs}}}}$ and $G_{{{\text{PICs}}}}$ are the corresponding bulk and shear moduli and can be given by

$$K_{{{\text{PICs}}}} = E_{{{\text{ICs}}}} \left( {\frac{{V_{{{\text{ICs}}}}^{{{\text{end}}}} }}{{1 - V_{{{\text{GM}}}}^{{{\text{end}}}} }}} \right)^{2} ,\;\;\;G_{{{\text{PICs}}}} = \frac{3}{8}E_{{{\text{ICs}}}} \left( {\frac{{V_{{{\text{ICs}}}}^{{{\text{end}}}} }}{{1 - V_{{{\text{GM}}}}^{{{\text{end}}}} }}} \right)^{2}.$$

(26)

Considering the concentration-dependent property of graphite material, the bulk and shear modulus $K_{{{\text{GM}}}}$ and $G_{{{\text{GM}}}}$ in Eq. (25) can be obtained by the elastic modulus $E_{{{\text{GM}}}}$, where $E_{{{\text{GM}}}} = \left( {19.25 + 82.234 \times {c \mathord{\left/ {\vphantom {c {c_{\max } }}} \right. \kern-\nulldelimiterspace} {c_{\max } }}} \right){\text{GPa}}$ and 0$v_{{{\text{GM}}}} = 0.26$ [48]. With Eqs. (21)–(26), we can obtain the effective elastic modulus and Poisson’s ratio of electrode composites and have $E_{{\text{a}}} \left( z \right) = 9G_{{\text{a}}} K_{{\text{a}}} /(3K_{{\text{a}}} + G_{{\text{a}}} )$ and $v_{{\text{a}}} = \left( {3K_{{\text{a}}} - 2G_{{\text{a}}} } \right)/(6K_{{\text{a}}} + 2G_{{\text{a}}} )$, respectively. Moreover, the line strain of the composite electrode layer due to lithiation can be calculated as

$$\varepsilon_{{\text{a}}} = \varepsilon_{{{\text{GM}}}} V_{{{\text{GM}}}} + \varepsilon_{{{\text{PICs}}}} V_{{{\text{PICs}}}}$$

(27)

where $\varepsilon_{{{\text{GM}}}}$ is the lithiation strain of a graphite particle and estimated as ${{\Omega c} \mathord{\left/ {\vphantom {{\Omega c} 3}} \right. \kern-\nulldelimiterspace} 3}$. $\varepsilon_{{{\text{PICs}}}}$ is the strain of the porous matrix (i.e. binder, carbon black, and porosity). And $\varepsilon_{{{\text{PICs}}}}$ is equal to zero because the porous matrix does not influence the lithiation reaction.