Activity correction in a moving-boundary model for electrochemical lithium intercalation and discharge/charge voltage in LiCoO₂ electrodes

Abstract

A moving-boundary model with activity correction for the diffusion of lithium in LiCoO₂ electrodes and the electrochemical reaction on the surface of electrode particles was formulated in a reproduction of the discharge/charge voltage process in batteries that have a LiCoO₂ cathode and a mesocarbon-microbead anode. Activity correction for the electrochemical reaction as well as the diffusion of lithium where the LiCoO₂ structural phase transition occurs during discharge and charge was considered. An accurate representation of the discharge/charge voltage profiles was obtained by using the moving-boundary model with the activity correction. The activity correction incorporated into the moving-boundary model can improve the discontinuous gaps in the discharge voltage profiles at the surface of conjugate LiCoO₂ electrode particles.

Graphical Abstract

Appendices

Appendix 1

Diffusion fluxes of lithium intercalated ${{J}_{\text{p}}}$ are given by Eq. 1, and defined by the chemical potential of lithium ${{\mu }_{\text{1,p}}}$

$$ \begin{aligned} J_{{\text{p}}} & = - D_{{1,{\text{p}}}} \frac{{c_{{1,{\text{p}}}} }}{{RTR_{{\text{p}}} }}\frac{{\partial \mu _{{1,{\text{p}}}} (t,\overline{r} )}}{{\partial \overline{r} }} = - D_{{1{\text{,p}}}} \frac{{c_{{1,{\text{p}}}} }}{{R_{{\text{p}}} }}\frac{{\partial \ln a_{{1,{\text{p}}}} (t,\overline{r} )}}{{\partial \overline{r} }} = - D_{{{\text{1,p}}}} \frac{{c_{{1,{\text{p}}}} }}{{R_{{\text{p}}} }}\frac{{\partial \ln x_{{1,{\text{p}}}} (t,\overline{r} )\gamma _{{1,{\text{p}}}} (t,\overline{r} )}}{{\partial \overline{r} }} \\ & = - D_{{1,{\text{p}}}} \frac{{c_{{{\text{1,p}}}} }}{{R_{{\text{p}}} }}\left( {\frac{{\partial x_{{1,{\text{p}}}} (t,\overline{r} )}}{{\partial \overline{r} }}} \right)\frac{{{\text{d}}\ln x_{{1,{\text{p}}}} (t,\overline{r} )\gamma _{{1,{\text{p}}}} (t,\overline{r} )}}{{{\text{d}}x_{{1,{\text{p}}}} (t,\overline{r} )}} = - D_{{{\text{1,p}}}} \frac{{c_{{1,{\text{p}}}} }}{{R_{{\text{p}}} x_{{1,{\text{p}}}} (t,\overline{r} )}}\left( {1 + \frac{{x_{{{\text{1,p}}}} (t,\overline{r} )}}{{\gamma _{{1,{\text{p}}}} (t,\overline{r} )}}\frac{{{\text{d}}\gamma _{{1,{\text{p}}}} (t,\overline{r} )}}{{{\text{d}}x_{{1,{\text{p}}}} (t,\overline{r} )}}} \right)\frac{{\partial x_{{1,p}} (t,\overline{r} )}}{{\partial \overline{r} }} \\ & = - D_{{1,{\text{p}}}} \frac{{c_{{1,{\text{p}}}} }}{{R_{{\text{p}}} x_{{1,{\text{p}}}} (t,\overline{r} )}}\left( {1 + \frac{{\partial \ln \gamma _{{1,{\text{p}}}} (t,\overline{r} )}}{{\partial \ln x_{{1,{\text{p}}}} (t,\overline{r} )}}} \right)\frac{{\partial x_{{1,{\text{p}}}} (t,\overline{r} )}}{{\partial \overline{r} }}. \\ \end{aligned} $$

(19)

On the other hand, the diffusion flux of lithium intercalated is defined by the effective diffusion coefficient.

$${{J}_{\text{p}}}=-{{D}_{\text{eff,p}}}\frac{c_{_{1,\text{p}}}^{\max }}{{{R}_{\text{p}}}}\frac{\partial {{x}_{1,\text{p}}}(t,\overline{r})}{\partial \overline{r}}.$$

(20)

Using the relation between $c_{1,\,\text{p}}^{\max }$ the maximum lithium concentration, the concentration of lithium c _1,p, and the mole fraction of lithium intercalated ${{x}_{\text{1,p}}}(t,\overline{r})$,

$$c_{_{1,\text{p}}}^{\max }{{x}_{1,\text{p}}}(t,\overline{r})={{c}_{1,\text{p}}},$$

(21)

and equating of Eq. (19) with Eq. (20), we can derive the effective diffusion coefficient.

$${{D}_{\text{eff,p}}}={{D}_{1,\text{p}}}\left( 1+\frac{\partial \ln {{\gamma }_{\text{1,p}}}(t,\overline{r})}{\partial \ln {{x}_{1,\text{p}}}(t,\overline{r})} \right)$$

(22)

which is equivalent to Eq. (3).

Appendix 2

The equilibrium potential E ^eq between the lithium intercalated in the bulk electrode and lithium metal in the reference electrode is represented as follows:

$${{E}^{\text{eq}}}={{E}^{\circ }}+\frac{RT}{F}\ln \left( \frac{{{a}_{2}}}{{{a}_{1}}} \right)={{E}^{\circ }}+\frac{RT}{F}\ln \left( \frac{{{x}_{2}}}{{{x}_{1}}} \right)+\frac{RT}{F}\ln \left( \frac{{{\gamma }_{2}}}{{{\gamma }_{1}}} \right),$$

(23)

where ${{E}^{\circ }}$ is the standard equilibrium potential; a ₁ and a ₂ are the activity for the active material intercalated by lithium and the vacant active material in the bulk electrode phase, respectively; and the activity a _i is defined by a _i  = x _i γ _i with the activity coefficient γ _i . If phase separation occurs at mole fractions $x_{i}^{\alpha }$ of phase α and $x_{i}^{\beta }$ of phase β in the discharge/charge process for a LiCoO₂ electrode, the activities of lithium and vacant active material are characterized by

$${{a}_{1}}(x_{1}^{\alpha })={{a}_{1}}(x_{1}^{\beta })\ \text{and}\ {{a}_{2}}(x_{2}^{\alpha })={{a}_{2}}(x_{2}^{\beta }).$$

(24)

We obtained the thermodynamic criteria of the phase separation in terms of the molar Gibbs energy of mixing g ^M.

$${{\left. \frac{\text{d}{{g}^{\text{M}}}}{\text{d}{{x}_{1}}} \right|}_{{{x}_{1}}=x_{1}^{\alpha }}}={{\left. \frac{\text{d}{{g}^{\text{M}}}}{\text{d}{{x}_{1}}} \right|}_{{{x}_{1}}=x_{1}^{\beta }}}\ \text{and}\ \frac{{{\text{d}}^{2}}{{g}^{\text{M}}}}{\text{d}{{x}_{1}}^{2}}\ge 0.$$

(25)

The relationship between the molar Gibbs energy of mixing and equilibrium potential is given by

$$\frac{\text{d}{{g}^{\text{M}}}}{\text{d}{{x}_{1}}}=-F({{E}^{\text{eq}}}-{{E}^{\circ }}).$$

(26)

The activity coefficients of the binary system, given by the NRTL model of Renon and Prausnitz [17], are

$$\begin{matrix} \ln {{\gamma }_{1}}=x_{2}^{2}\left[ {{\tau }_{12}}{{\left( \frac{{{G}_{21}}}{{{x}_{1}}+{{x}_{2}}{{G}_{21}}} \right)}^{2}}+\frac{{{\tau }_{12}}{{G}_{12}}}{{{\left( {{x}_{2}}+{{x}_{1}}{{G}_{12}} \right)}^{2}}} \right], \\ \ln {{\gamma }_{2}}=x_{1}^{2}\left[ {{\tau }_{12}}{{\left( \frac{{{G}_{12}}}{{{x}_{2}}+{{x}_{1}}{{G}_{12}}} \right)}^{2}}+\frac{{{\tau }_{21}}{{G}_{21}}}{{{\left( {{x}_{1}}+{{x}_{2}}{{G}_{21}} \right)}^{2}}} \right], \\\end{matrix}$$

where the binary parameters are defined as

$${{\tau }_{12}}=\frac{\Delta {{g}_{12}}}{RT}\, , \,{{\tau }_{21}}=\frac{\Delta {{g}_{21}}}{RT}$$

(27)

$$ G_{{12}} = \exp ( - \alpha _{{12}} \tau _{{12}} ),\;G_{{21}} = \exp \left( { - \alpha _{{12}} \tau _{{21}} } \right) $$

(28)

where $\Delta {{g}_{12}}$ and $\Delta {{g}_{21}}$ are the binary energy parameters characterized by species 1 and 2 interactions, respectively, and the non-randomness α ₁₂ is treated as an adjustable parameter. The adjustable parameters of the NRTL model, $\Delta {{g}_{12}}$, $\Delta {{g}_{21}}$, and α ₁₂, were obtained by fitting the models to the experimental results. For the LiCoO₂ electrode, the phase transition points of each phase $x_{1}^{\alpha }$ and $x_{1}^{\beta }$ were determined by solving the thermodynamic criteria of the phase separation with the adjustable binary parameters in the NRTL model. Figure 5a shows the equilibrium potential curve for the LiCoO₂ cathode as calculated by the NRTL model with heterogeneous treatment using Eq. (24)–(26).