Buckling analysis of cylindrical silicon electrodes considering the two-phase lithiation process

Abstract

Nanowire silicon electrodes have an essential potential application in next-generation lithium-ion battery. It is of significance for the development and application of nanowire silicon electrodes to explore the state of charging (SOC) and evaluate the influencing factors. In recent years, many studies show that the lithiation process in silicon electrodes is a two-phase lithiation process, which depends on the rate of interface reaction. In this paper, the effect of interface reaction rate on SOC is studied based on the buckling analysis of a nanowire silicon electrode using a reaction–diffusion model. Temporal evolution of the resultant axial force and the critical force for the buckling of electrode during lithiation, was illustrated and discussed. The analysis of stress was carried out in terms of the theories of finite deformation as well as small deformation. Results show the interface reaction rate has an obvious influence on SOC. Generally, SOC decreases with the increase of the interface reaction rate, but the SOC obtained by a finite reaction rate is always higher than that obtained by the conventional diffusion model.

Appendix

The expressions of $X_{n} \left( {\overline{R}} \right)$ and $T_{n} \left( {\overline{t}} \right)$ are

$$ X_{n} \left( {\overline{R}} \right) = Y_{0} \left( {\mu_{n} } \right)J_{0} \left( {\mu_{n} \overline{R}} \right) - J_{0} \left( {\mu_{n} } \right)Y_{0} \left( {\mu_{n} \overline{R}} \right) $$

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$$ \begin{aligned} T_{n} \left( {\overline{t}} \right) & = \left[ {\frac{{ - \overline{k}N_{n}^{2} \left( {\mu_{n} ,\overline{S}} \right){ + }\left( {\overline{S} \cdot \overline{k} - 1} \right)N_{n}^{1} \left( {\mu_{n} ,\overline{S}} \right)}}{{\left( {1 - \overline{k}\overline{S} + \overline{k}} \right)N_{n} \left( {\mu_{n} ,\overline{S}} \right)}} - \frac{{\overline{k}N_{n}^{0} }}{{\mu_{n}^{{2}} N_{n} \left( {\mu_{n} ,\overline{S}} \right)\left( {1 - \overline{k}\overline{S} + \overline{k}} \right)}}} \right]e^{{{ - \mu_{n}{2}} \overline{t}}} \\ & \quad + \frac{{\overline{k}N_{n}^{0} }}{{\mu_{n}^{{2}} N_{n} \left( {\mu_{n} ,\overline{S}} \right)\left( {1 - \overline{k}\overline{S} + \overline{k}} \right)}} \\ \end{aligned} $$

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where $J_{0}$ and $Y_{0}$ are the zero-order Bessel functions of the first type and the second type, respectively. Here, $\mu_{n}$ (n = 1, 2, 3,...) are the positive roots of the following equation:

$$ Y_{{0}} \left( \mu \right)\left[ {\mu J_{{1}} \left( {\mu {\overline{\text{S}}}} \right) + \overline{k}J_{0} \left( {\mu {\overline{\text{S}}}} \right)} \right] - J_{0} \left( \mu \right)\left[ {\mu Y_{{1}} \left( {\mu {\overline{\text{S}}}} \right) + \overline{k}Y_{0} \left( {\mu {\overline{\text{S}}}} \right)} \right] = 0 $$

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and

$$ \begin{aligned} N_{n} \left( {\mu_{n} ,\overline{S}} \right) & { = }\frac{1}{2}\left[ {Y_{{0}} \left( {\mu_{n} } \right)J_{{1}} \left( {\mu_{n} } \right) - J_{{0}} \left( {\mu_{n} } \right)Y_{{1}} \left( {\mu_{n} } \right)} \right]^{2} \\ & \quad - \frac{{\overline{S}^{2} }}{2}\left( {1 + \frac{{\overline{k}^{2} }}{{\mu_{n}^{2} }}} \right)\left[ {Y_{{0}} \left( {\mu_{n} } \right)J_{0} \left( {\mu_{n} \overline{S}} \right) - J_{{0}} \left( {\mu_{n} } \right)Y_{0} \left( {\mu_{n} \overline{S}} \right)} \right]^{2} \\ \end{aligned} $$

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$$ N_{n}^{0} \left( {\mu_{n} ,\overline{S}} \right) = \int_{{\overline{S}}}^{1} {X_{n} \left( {\overline{R}} \right)d\overline{R}} $$

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$$ N_{n}^{1} \left( {\mu_{n} ,\overline{S}} \right) = \frac{{\overline{S}}}{{\mu_{n} }}\left[ {J_{0} \left( {\mu_{n} } \right)Y_{1} \left( {\mu_{n} \overline{S}} \right) - J_{1} \left( {\mu_{n} \overline{S}} \right)Y_{0} \left( {\mu_{n} } \right)} \right] + \frac{2}{{\pi \mu_{n}^{2} }} $$

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$$ N_{n}^{2} \left( {\mu_{n} ,\overline{S}} \right) = \int_{{\overline{S}}}^{1} {\overline{R}^{2} X_{n} \left( {\overline{R}} \right)d\overline{R}} $$

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where $J_{1}$ and $Y_{1}$ are the first-order Bessel functions of the first type and the second type, respectively. The expressions of the three coefficients $A^{\alpha } ,B^{\alpha }$ and $A^{\beta }$ are

$$ \begin{aligned} A^{\alpha } & = \frac{{\left[ {\frac{{E^{\beta } }}{{\left( {1 + \nu^{\beta } } \right)\left( {1 - 2\nu^{\beta } } \right)}} + \frac{{E^{\alpha } }}{{\left( {1 + \nu^{\alpha } } \right)}}} \right] \cdot \frac{{E^{\alpha } \Omega }}{{3\left( {1 - \nu^{\alpha } } \right)R_{0}^{2} }}\int_{S}^{{R_{0} }} {CRdR} }}{D} \\ B^{\alpha } & = \frac{{\left[ {\frac{{E^{\alpha } }}{{\left( {1 + \nu^{\alpha } } \right)\left( {1 - 2\nu^{\alpha } } \right)}} - \frac{{E^{\beta } }}{{\left( {1 + \nu^{\beta } } \right)\left( {1 - 2\nu^{\beta } } \right)}}} \right] \cdot \frac{{E^{\alpha } \Omega S^{2} }}{{3\left( {1 - \nu^{\alpha } } \right)R_{0}^{2} }}\int_{S}^{{R_{0} }} {CRdR} }}{D} \\ A^{\beta } & = \frac{{\left[ {\frac{{E^{\alpha } }}{{\left( {1 + \nu^{\alpha } } \right)\left( {1 - 2\nu^{\alpha } } \right)}} + \frac{{E^{\alpha } }}{{\left( {1 + \nu^{\alpha } } \right)}}} \right] \cdot \frac{{E^{\alpha } \Omega }}{{3\left( {1 - \nu^{\alpha } } \right)R_{0}^{2} }}\int_{S}^{{R_{0} }} {CRdR} }}{D} \\ \end{aligned} $$

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where

$$ \begin{aligned} D & = \frac{{E^{\alpha } E^{\beta } }}{{\left( {1 + \nu^{\beta } } \right)\left( {1 - 2\nu^{\beta } } \right)\left( {1 + \nu^{\alpha } } \right)\left( {1 - 2\nu^{\alpha } } \right)}} \\ & \quad + \frac{{\left( {E^{\alpha } } \right)^{2} }}{{\left( {1 + \nu^{\alpha } } \right)^{2} \left( {1 - 2\nu^{\alpha } } \right)}}\left( {1 - \frac{{S^{2} }}{{R_{0}^{2} }}} \right) + \frac{{E^{\alpha } E^{\beta } S^{2} }}{{\left( {1 + \nu^{\beta } } \right)\left( {1 - 2\nu^{\beta } } \right)\left( {1 + \nu^{\alpha } } \right)R_{0}^{2} }} \\ \end{aligned} $$

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