Lithium Diffusion and Stress in a Polycrystalline Film Electrode

Abstract

In the present work, the two-dimensional analytical solution for Li diffusion in grains and grain boundaries of a polycrystalline film electrode is established with consideration of Li-segregation at the grain boundary. The stress field induced by the inhomogeneity of Li concentration, called chemical stress here for brevity, is analyzed via the finite element simulation. The effects of the grain boundary including its size, its diffusion coefficient as well as the segregation phenomenon on the solute concentration and the chemical stress are examined. It shows that grain boundaries can assist fast diffusion and significantly affect the stress profile in the whole film. It proves that tailoring the grain boundary size or other grain boundary-related parameters may provide an alternative strategy for improving the overall diffusivity and mechanical stability of battery electrodes.

Appendices

Appendix A

1.1 1. Derivation of the Characteristic Equation in Eq. (10b)

By substituting the solutions given by Eqs. (10a) and (10b) into the flux continuity condition in Eq. (3) and the segregation condition in Eq. (4), we can obtain the following two equations

$$\begin{aligned}&{D}^{\prime } \sum _{n=0}^\infty {\sum _{m=1}^\infty {{A}_{{\textit{nm}}}^{\prime } {\alpha }_{{\textit{nm}}}^{\prime } \sinh ({-{\alpha }_{{\textit{nm}}}^{\prime } a})} \cos ({\beta _n y}) \exp \left[ {-D^{\prime }t\left( {-{\alpha }_{{\textit{nm}}}^{\prime 2} +\beta _n^2} \right) } \right] } \nonumber \\&\quad =-D\sum _{n=0}^\infty {\sum _{m=1}^\infty {A_{{\textit{nm}}} \alpha _{{\textit{nm}}} \sin ({\alpha _{{\textit{nm}}} L} )} \cos ({\beta _n y})\exp \left[ {-Dt\left( {\alpha _{{\textit{nm}}}^2 +\beta _n^2} \right) } \right] } \end{aligned}$$

(A1)

$$\begin{aligned}&\sum _{n=0}^\infty {\sum _{m=1}^\infty {{A}_{{\textit{nm}}}^{\prime } \cosh ({-{\alpha }_{{\textit{nm}}}^{\prime } a})} \cos ({\beta _n y})\exp \left[ {-D^{\prime }t\left( {-{\alpha }_{{\textit{nm}}}^{\prime 2} +\beta _n^2} \right) } \right] } \nonumber \\&\quad =K\sum _{n=0}^\infty {\sum _{m=1}^\infty {A_{{\textit{nm}}} \cos ({\alpha _{{\textit{nm}}} L})} \cos ({\beta _n y})\exp \left[ {-Dt\left( {\alpha _{{\textit{nm}}}^2 +\beta _n^2} \right) } \right] } \end{aligned}$$

(A2)

By considering the orthogonal relation of

$$\begin{aligned} \int _0^{Y_0} {\cos ({\beta _n y})\cos ({\beta _k y})} \hbox {d}y=\left\{ \begin{array}{lr} Y_0 /2 \quad &{} (n=k\ne 0) \\ Y_0 \quad &{} (n=k=0) \\ 0 \quad &{} (n\ne k) \\ \end{array}\right. \end{aligned}$$

(A3)

Equations (A1) and (A2) become

$$\begin{aligned}&{D}^{\prime } {A}_{{\textit{nm}}}^{\prime } {\alpha }_{{\textit{nm}}}^{\prime } \sinh ({-{\alpha }_{{\textit{nm}}}^{\prime } a} ) \exp \left[ {-D^{\prime }t\left( {-{\alpha }_{{\textit{nm}}}^{\prime 2} +\beta _n^2} \right) } \right] \nonumber \\&\quad =-DA_{{\textit{nm}}} \alpha _{{\textit{nm}}} \sin ({\alpha _{{\textit{nm}}} L} )\exp \left[ {-Dt\left( {\alpha _{{\textit{nm}}}^2 +\beta _n^2} \right) } \right] \end{aligned}$$

(A4)

$$\begin{aligned}&{A}_{{\textit{nm}}}^{\prime } \cosh ({-{\alpha }_{{\textit{nm}}}^{\prime } a}) \exp \left[ {-D^{\prime }t\left( {-{\alpha }_{{\textit{nm}}}^{\prime 2} +\beta _n^2} \right) } \right] \nonumber \\&\quad =KA_{{\textit{nm}}} \cos ({\alpha _{{\textit{nm}}} L}) \exp \left[ {-Dt\left( {\alpha _{{\textit{nm}}}^2 +\beta _n^2} \right) } \right] \end{aligned}$$

(A5)

By combining Eqs. (A4) and (A5), the characteristic equation in Eq. (11b) can be obtained.

2. Derivation of Coefficients $A_{{\textit{nm}}}$ for the Case When GB Size is Very Small

By considering the orthogonal relation in Eq. (A3), Eq. (12) becomes

$$\begin{aligned} \sum _{m=1}^\infty {A_{{\textit{nm}}} X_{{\textit{nm}}} (x)} =\frac{J_0 Y_0}{D}\frac{2 ({-1})^{n+1}}{n^{2}\pi ^{2}} \end{aligned}$$

(A6)

for $n \ne 0$ and

$$\begin{aligned} \sum _{m=1}^\infty {A_{0m} X_{0m} (x)} =-\frac{J_0 Y_0}{6D} + \frac{a}{L+aK}\frac{KJ_0 -J_0^{\prime }}{2DY_0}x^{2}-\frac{KJ_0 -J_0^{\prime }}{2K{D}^{\prime } Y_0}\frac{a^{2}L}{L+aK} \end{aligned}$$

(A7)

for $n=0$.

It should be noted that $X_{{\textit{nm}}} (x)$ in Eq. (10c) has no obvious orthogonal property, and we have to deal with it before moving on.

Technically, we have the relation of

$$\begin{aligned} \frac{\partial ^{2}X_{{\textit{nm}}}}{\partial x^{2}}=-\,\alpha _{{\textit{nm}}}^2 X_{{\textit{nm}}} \end{aligned}$$

(A8)

By considering the relation in Eq. (A8), we have

$$\begin{aligned} \int _0^L {X_{{\textit{nm}}}} X_{nl} \hbox {d}x= & {} -\frac{1}{\alpha _{{\textit{nm}}}^2 -\alpha _{nl}^2}\int _0^L {\left( {\frac{\partial ^{2}X_{{\textit{nm}}}}{\partial x^{2}}X_{nl} -\frac{\partial ^{2}X_{nl}}{\partial x^{2}}X_{{\textit{nm}}}} \right) } \hbox {d}x \nonumber \\= & {} -\left( {X_{nl} \frac{\partial X_{{\textit{nm}}}}{\partial x}} \right) _0^L +\int _0^L {\frac{\partial X_{nl}}{\partial x}} \frac{\partial X_{{\textit{nm}}}}{\partial x}\hbox {d}x+\left( {X_{{\textit{nm}}} \frac{\partial X_{nl}}{\partial x}} \right) _0^L -\int _0^L {\frac{\partial X_{nl}}{\partial x}\frac{\partial X_{{\textit{nm}}}}{\partial x}} \hbox {d}x \nonumber \\\\= & {} \alpha _{{\textit{nm}}} \cos ({\alpha _{nl} L})\sin ({\alpha _{{\textit{nm}}} L})-\alpha _{nl} \cos ({\alpha _{{\textit{nm}}} L}) \sin ({\alpha _{nl} L})\nonumber \end{aligned}$$

(A9)

for $m \ne l$ and

$$\begin{aligned} \int _0^L {X_{nl}} X_{nl} \hbox {d}x=\int _0^L {\cos ({\alpha _{nl} x})} \cos ({\alpha _{nl} x})dx = \frac{L}{2}+\frac{\sin ({2\alpha _{nl} L})}{4\alpha _{nl}} \end{aligned}$$

(A10)

for $m=l$.

Coefficients $A_{{\textit{nm}}}$ for the case of $n \ne 0$ will be determined first. By substituting Eqs. (A9) and (A10) into Eq. (A6), we can obtain

$$\begin{aligned}&A_{nl} \left[ {\frac{L}{2}+\frac{\sin ({2\alpha _{nl} L})}{4\alpha _{nl}}} \right] +\sum _{m\ne l} {A_{{\textit{nm}}} \frac{\alpha _{{\textit{nm}}} \cos ({\alpha _{nl} L}) \sin ({\alpha _{{\textit{nm}}} L})-\alpha _{nl} \cos ({\alpha _{{\textit{nm}}} L})\sin ({\alpha _{nl} L})}{\alpha _{{\textit{nm}}}^2 -\alpha _{nl}^2}} \nonumber \\&\quad =\frac{J_0 Y_0}{D}\frac{2 ({-1})^{n+1}}{n^{2}\pi ^{2}}\frac{\sin ({\alpha _{nl} L})}{\alpha _{nl}} \end{aligned}$$

(A11)

By considering the characteristic equation in Eq. (13), we have

$$\begin{aligned} -Ka\cos ( {\alpha _{{\textit{nm}}} L} )\cos ( {\alpha _{nl} L}) =\frac{\alpha _{{\textit{nm}}} \sin ( {\alpha _{{\textit{nm}}} L}) \cos ( {\alpha _{nl} L} )-\alpha _{nl} \sin ( {\alpha _{nl} L} )\cos ( {\alpha _{{\textit{nm}}} L} )}{\alpha _{{\textit{nm}}}^2 -\alpha _{nl}^2} \end{aligned}$$

(A12)

By substituting the obtained Eq. (A12) into Eq. (A11), we can get

$$\begin{aligned} A_{nl} \left[ {\frac{L}{2}+\frac{\sin ({2\alpha _{nl} L})}{4\alpha _{nl}}} \right] -Ka\cos ({\alpha _{nl} L}) \sum _{m\ne l} {A_{{\textit{nm}}} \cos ({\alpha _{{\textit{nm}}} L})} =\frac{J_0 Y_0}{D}\frac{2 ({-1})^{n+1}}{n^{2}\pi ^{2}}\frac{\sin ({\alpha _{nl} L})}{\alpha _{nl}} \end{aligned}$$

(A13)

From Eq. (A6), we have

$$\begin{aligned} Ka\cos ^{2}( {\alpha _{nl} L} )A_{nl} +Ka\cos ( {\alpha _{nl} L}) \sum _{m=1}^\infty {A_{{\textit{nm}}} \cos ( {\alpha _{{\textit{nm}}} L})} =\frac{J_0 Y_0}{D}\frac{2( {-1} )^{n+1}}{n^{2}\pi ^{2}}Ka\cos ({\alpha _{nl} L}) \end{aligned}$$

(A14)

By combining Eqs. (A13) and (A14), we can obtain the expression of $A_{{nm}}$ for $n \ne 0$ as given in Eq. (14a). Similarly, $A_{{nm}}$ for $n =0$ can be determined by Eq. (A7) instead of Eq. (A6). But it should be noted that when $\alpha _{0m} =0$, the limit has to be taken for obtaining the expression in Eq. (14c).