PXD model

In the following, all the symbols that are not introduced directly in the text are collected in the table at the end of the page

Electrolyte

The mass and charge conservation in the electrolyte are given by

$$\begin{array}{rl}\frac{\partial }{\partial t}({\epsilon }_{\text{elyte}}{c}_{\text{elyte}})+\mathrm{\nabla }\cdot {\mathbf{\text{N}}}_{\text{elyte}}& =R_{\text{elyte}},\\ \mathrm{\nabla }\cdot {\mathbf{\text{j}}}_{\text{elyte}}& =F{R}_{\text{elyte}},\end{array}$$

where the fluxes are given by

$$\begin{array}{rl}{\mathbf{\text{j}}}_{\text{elyte}}& =-{\kappa }_{\text{elyte},\text{eff}}\mathrm{\nabla }{\varphi }_{\text{elyte}}-{\kappa }_{\text{elyte},\text{eff}}\frac{1-t_{+}}{z_{+}F}\left(\frac{\partial \mu }{\partial c}\right)\mathrm{\nabla }{c}_{\text{elyte}},\\ {\mathbf{\text{N}}}_{\text{elyte}}& =-D_{\text{elyte},\text{eff}}\mathrm{\nabla }{c}_{\text{elyte}}+\frac{t_{+}}{z_{+}F}{\mathbf{\text{j}}}_{\text{elyte}}.\end{array}$$

The volumetric reaction rate is given as $R_{\text{elyte}}=-\sum _{\text{elde}}{\gamma }_{\text{elde}}{R}_{\text{elde}}$ where ${\gamma }_{\text{elde}}$ is the volumetric surface area and the expression for $R_{\text{elde}}$ is given below. Note that the reaction rates depends on the spatial variable $x$. For the chemical potential, we use $\mu =2RT\log (c_{\text{elyte}})$. The effective quantities are computed from the intrinsic properties and the volume fraction using a Bruggemann coefficient, denoted $b$, which yields ${\kappa }_{\text{elyte},\text{eff}}={\epsilon }_{\text{elyte}}^b{\kappa }_{\text{elyte}}$ and $D_{\text{elyte},\text{eff}}={\epsilon }_{\text{elyte}}^b{D}_{\text{elyte}}$. For the electrolyte, we have a spatially dependent Bruggeman coefficient.

Electrode

In the electrode, the charge conservation equation is given by

$$-\mathrm{\nabla }\cdot ({\kappa }_{\text{elde},\text{eff}}\mathrm{\nabla }{\varphi }_{\text{elde}})=F{\gamma }_{\text{elde}}{R}_{\text{elde}}.$$

We use a pseudo particle model for $c_{\text{elde}}(t,x,r)$

$$\frac{\partial }{\partial t}{c}_{\text{elde}}-\frac{1}{r^2}\frac{\partial }{\partial r}(r^2{D}_{\text{elde}}\frac{\partial }{\partial r}{c}_{\text{elde}})=0.$$

with boundary condition

$$-4\pi r_p^2{D}_{\text{elde}}\frac{\partial c_{\text{elde}}}{\partial r}(t,x,r_p)=\frac{{\gamma }_{\text{elde}}{R}_{\text{elde}}}{{\epsilon }_{\text{elde}}}\frac{4\pi r_p^3}{3}.$$

Reaction kinetics

The reaction rate $R_{\text{elde}}$ at each electrode is given

$$R_{\text{elde}}=j_{\text{elde}}(c_{\text{elde}},c_{\text{elyte}},T)(e^{\alpha F\frac{{\eta }_{\text{elde}}}{RT}}-e^{-(1-\alpha )F\frac{{\eta }_{\text{elde}}}{RT}}).$$

where ${\eta }_{\text{elde}}$ and $j_{\text{elde}}$ denote the overpotential and the reaction exchange current density. The overpotential ${\eta }_{\text{elde}}$ is given by

$${\eta }_{\text{elde}}={\varphi }_{\text{elde}}-{\varphi }_{\text{elyte}}-U_{\text{elde}}(c_{\text{elde}},T).$$

where $U_{\text{elde}}$ denotes the open circuit potential, given as a function of the Lithium concentration in the electrode and the temperature. The exchange current density is given by

$$j_{\text{elde}}=k_{\text{elde},0}{e}^{-\frac{E_a}{R}(1/T-1/T_{\text{ref}})}{(c_{\text{elyte}}(c_{\text{elde},max}-c_{\text{elde}})c_{\text{elde}})}^{\frac{1}{2}}.$$

Symbol list

Symbol	Definition
$c_{\text{elyte}}$, $c_{\text{elde}}$	Lithium concentration in electrolyte, electrode
${\varphi }_{\text{elyte}}$, ${\varphi }_{\text{elde}}$	Electrolyte potential in electrolyte, electrode
$T$	Temperature
$t_{+}$	Transference number
$z_{+}$	Charge number
$F$	Faraday coefficient
$r_p$	Particle radius
$R_{\text{elde}}$	Reaction rate at electrode
${\gamma }_{\text{elde}}$	Volumetric surface area
$k_{\text{elde},0}$	Reaction rate constant
${\epsilon }_{\text{elde}}$	Volume fraction electrode
${\epsilon }_{\text{elyte}}$	Volume fraction electrolyte
$E_q$	Activation energy of reaction
$R$	Ideal gas constant