Anomalous interfacial stress generation and role of elasto-plasticity in mechanical failure of Si-based thin film anodes of Li-ion batteries

Abstract

To understand the reason for mechanical failure and capacity fading of Si-based composite thin film batteries, we have developed a lithiation-induced interfacial stress model. The role of electrochemical charging reaction process, stiffness-induced elasto-plastic deformation and relative change in resistivity in detail through regional material heterogeneity sensitivity exponent has been accounted for anomalous interfacial stress generation. The insight gained from the results of state-of-health of the battery electrodes suggests that the transition of material from elastic to plastic behaviour and cracking at the interface are due to volume expansion, i.e., fully lithiated. Finally, the unified lithiation-induced stress model unravels the effect of embedded material heterogeneity parameters coupled with resistivity and stiffness and its anomalous dynamics at composite electrode–collector interface. The verification with the available experimental data in the literature has also been made and hence providing a better insight into the origin of degradation and the evaluation of advanced battery electrodes.

Appendices

Appendix

Stiffness-induced deformation

The force experienced by the electrode material and the substrate due to stiffness-induced deformation is given in equations (13) and (14), respectively. Under the assumption of equi-biaxial in-plane total strain and within the classical beam theory, the axial tensile force is in equilibrium with the axial compressive force at the electrode–substrate interface and hence it can be expressed or rearranged as follows

$$ \varepsilon_{{\text{s}}} = S^{*} \varepsilon_{{\text{e}}}. $$

(A.1)

where $S^{*}$ is the effective stiffness ratio and can be given as $S^{*} = S_{{\text{e}}} /S_{{\text{s}}} = \left( {Y_{{\text{e}}} A_{{\text{e}}} } \right)/ \left( {Y_{{\text{s}}} A_{{\text{s}}} } \right)$. The cross-sectional area of the electrode and the current collector has been written as $A_{{\text{e}}} = \left( {t_{{\text{e}}} w} \right)$ and $A_{{\text{s}}} = \left( {t_{{\text{s}}} w} \right)$, where w is the width of the electrode and hence, $S^{*} = Y_{{\text{e}}} t_{{\text{e}}} /Y_{{\text{s}}} t_{{\text{s}}}$. For the dimensional neutrality, we have used the NES ratio, i.e., $S = S^{*} /S_{{{\text{min}}}}^{*}$ in our calculation. In other words, the normalized stiffness ratio can be obtained by normalizing the stiffness ratio with its minimum value. Further, the NES ratio has been used to maintain the interfacial continuous condition the above expression has been rewritten as follows:

$$ \left. {\varepsilon_{{\text{e}}} } \right|_{{z_{{\text{e}}} = - t_{{\text{e}}} /2}} = S\left. {\varepsilon_{{\text{s}}} } \right|_{{z_{{{\text{s}} }} = t_{{\text{s}}} /2}}. $$

(A.2)

Within the limit of $S \to 1$ and the interfacial continuous condition used in equation (A.2) is not at all destroying the nature of expression used in reference [44] for the development of modified Stoney’s and can be expressed as follows:

$$ \left. {\varepsilon_{{\text{e}}} } \right|_{{z_{{\text{e}}} = - t_{{\text{e}}} /2}} = \left. {\varepsilon_{{\text{s}}} } \right|_{{z_{{{\text{s}} }} = t_{{\text{s}}} /2}}. $$

(A.3)

Further, by using equations (4–8) and equation (A.2), we get

$$ \frac{F}{{Y_{{\text{s}}} t_{{\text{s}}} w}} + \kappa \left( {\frac{{t_{{\text{s}}} }}{2}} \right) = S\left( { - \frac{F}{{Y_{{\text{e}}} t_{{\text{e}}} w}} + \kappa \left( { - \frac{{t_{{\text{e}}} }}{2}} \right) + \beta \frac{C}{{C_{{{\text{max}}}} }}} \right). $$

(A.4)

After rearranging the above equation, we get

$$ \frac{F}{w}\left( {\frac{S}{{Y_{{\text{e}}} t_{{\text{e}}} }} + \frac{1}{{Y_{{\text{s}}} t_{{\text{s}}} }}} \right) + \kappa \left( {\frac{{St_{{\text{e}}} }}{2} + \frac{{t_{{\text{s}}} }}{2}} \right) = S\beta \overline{c}. $$

(A.5)

In the above equation, the $F/w$ can be replaced from equation (12) and we get

$$ \kappa \frac{{\left( {Y_{{\text{e}}} t_{{\text{e}}}^{3} + Y_{{\text{s}}} t_{{\text{s}}}^{3} } \right)}}{{6\left( {t_{{\text{e}}} + t_{{\text{s}}} } \right)}}\left( {\frac{{Y_{{\text{e}}} t_{{\text{e}}} + SY_{{\text{s}}} t_{{\text{s}}} }}{{Y_{{\text{e}}} t_{{\text{e}}} Y_{{\text{s}}} t_{{\text{s}}} }}} \right) + \kappa \left( {\frac{{St_{{\text{e}}} + t_{{\text{s}}} }}{2}} \right) = S\beta \overline{c}. $$

(A.6)

The rearrangement of the above expression can be expressed as follows:

$$ \kappa \left\{ {\left( {Y_{{\text{e}}} t_{{\text{e}}}^{3} + Y_{{\text{s}}} t_{{\text{s}}}^{3} } \right)\left( {Y_{{\text{e}}} t_{{\text{e}}} + SY_{{\text{s}}} t_{{\text{s}}} } \right) + 3\left( {St_{{\text{e}}} + t_{{\text{s}}} } \right)\left( {t_{{\text{e}}} + t_{{\text{s}}} } \right)Y_{{\text{e}}} t_{{\text{e}}} Y_{{\text{s}}} t_{{\text{s}}} } \right\} = 6S\beta \overline{c}\left( {t_{{\text{e}}} + t_{{\text{s}}} } \right)Y_{{\text{e}}} t_{{\text{e}}} Y_{{\text{s}}} t_{{\text{s}}} $$

(A.7)

Further rearrangement will lead to

$$ \kappa \left( {Y_{{\text{e}}}^{2} t_{{\text{e}}}^{4} + SY_{{\text{e}}} t_{{\text{e}}}^{3} Y_{{\text{s}}} t_{{\text{s}}} + Y_{{\text{s}}} t_{{\text{s}}}^{3} Y_{{\text{e}}} t_{{\text{e}}} + SY_{{\text{s}}}^{2} t_{{\text{s}}}^{4} + 3SY_{{\text{e}}} t_{{\text{e}}}^{3} Y_{{\text{s}}} t_{{\text{s}}} + 3SY_{{\text{e}}} t_{{\text{e}}}^{2} Y_{{\text{s}}} t_{{\text{s}}}^{2} + 3Y_{{\text{e}}} t_{{\text{e}}} Y_{{\text{s}}} t_{{\text{s}}}^{3} + 3Y_{{\text{e}}} t_{{\text{e}}}^{2} Y_{{\text{s}}} t_{{\text{s}}}^{2} } \right) = 6\beta \overline{c}\left( {t_{{\text{e}}} /t_{{\text{s}}} + 1} \right)Y_{{\text{e}}} t_{{\text{e}}} Y_{{\text{s}}} t_{{\text{s}}}^{2}. $$

(A.8)

Dividing the above equation by $Y_{{\text{s}}}^{2} t_{{\text{s}}}^{4}$ and we get

$$ \kappa \left( {\frac{{Y_{{\text{e}}}^{2} t_{{\text{e}}}^{4} }}{{Y_{{\text{s}}}^{2} t_{{\text{s}}}^{4} }} + 4S\frac{{Y_{{\text{e}}} t_{{\text{e}}}^{3} }}{{Y_{{\text{s}}} t_{{\text{s}}}^{3} }} + 4\frac{{Y_{{\text{e}}} t_{{\text{e}}} }}{{Y_{{\text{s}}} t_{{\text{s}}} }} + S + 3\left( {S + 1} \right)\frac{{Y_{{\text{e}}} t_{{\text{e}}}^{2} }}{{Y_{{\text{s}}} t_{{\text{s}}}^{2} }}} \right) = 6\beta \overline{c}\left( {t_{{\text{e}}} /t_{{\text{s}}} + 1} \right)\frac{{Y_{{\text{e}}} t_{{\text{e}}} }}{{Y_{{\text{s}}} t_{{\text{s}}}^{2} }}. $$

(A.9)

By solving the above equation, substituting with $S^{*} = Y_{{\text{e}}} t_{{\text{e}}} /Y_{{\text{s}}} t_{{\text{s}}}$ and $t = t_{{\text{e}}} /t_{{\text{s}}}$ and rearranging the terms we get the curvature expression as follows:

$$ \kappa \left( {C,S\left( C \right)} \right) = \frac{{6SS^{*} \beta \overline{c}\left( {t + 1} \right)}}{{t_{{\text{s}}} \left[ {S + 4S^{*} + 3\left( {S + 1} \right)S^{*} t + 4SS^{*} t^{2} + S^{*2} t^{2} } \right]}} $$

(A.10)

The above expression of curvature is due to lithiation cum stiffness-driven curvature response of bilayer film electrode.