Nondimensional model

Scaling

The model has been rescaled to obtain nondimensional quantities and homogeneize truncation errors, this scaling is inspired in Ayerbe et al. [9]. The rescaled variables are obtained with the following relations:

• Spatial and temporal dimensions:

  \[\begin{split}\begin{gathered} x = (L_\mathrm{a}+L_\mathrm{s}+L_\mathrm{c}) \hat{x} = L_0 \hat{x}; \qquad \nabla = \frac{1}{L_0} \hat{\nabla} \\ r = R_\mathrm{s} \hat{r};\qquad r_\mathrm{\scriptscriptstyle SEI} = \delta_\mathrm{\scriptscriptstyle SEI} \hat{r}_\mathrm{\scriptscriptstyle SEI} + R_\mathrm{s} ;\qquad t=t_\mathrm{c}\hat{t} \end{gathered}\end{split}\]

• Potentials:

  \[\begin{gathered} \Phi_T = \frac{R T_0}{\alpha F} ; \qquad \Phi_\mathrm{s} = \frac{I_0 L_0}{\sigma_\mathrm{ref}} ; \qquad \Phi_\mathrm{l} = \Phi_T \end{gathered}\]

  \[\begin{gathered} \varphi_\mathrm{e}=\varphi_\mathrm{e}^\mathrm{ref}+\Phi_\mathrm{l}\hat{\varphi}_\mathrm{e} ;\qquad \varphi_\mathrm{s}=\varphi_\mathrm{s}^\mathrm{ref}+\Phi_\mathrm{s}\hat{\varphi}_\mathrm{s} \end{gathered}\]

  \[\begin{gathered} U_\mathrm{eq} = \varphi_\mathrm{s}^\mathrm{ref} - \varphi_\mathrm{e}^\mathrm{ref} + \Phi_T\hat{U}_\mathrm{eq} ; \qquad \eta=\Phi_T \hat{\eta} \end{gathered}\]

• Lithium concentrations:

  \[\begin{gathered} c_\mathrm{e}=c_\mathrm{e}^\mathrm{ref}+\Delta c_\mathrm{e}^\mathrm{ref} \hat{c}_\mathrm{e} ;\qquad \Delta c_\mathrm{e}^\mathrm{ref}=\frac{I_0 L_0 (1-t_+^0)}{D_\mathrm{e}^\mathrm{eff,ref}F} ;\qquad c_\mathrm{s}= c_\mathrm{s}^\mathrm{max} \hat{c_\mathrm{s}} \end{gathered}\]

• Current densities:

  \[\begin{gathered} a_\mathrm{s} j= \frac{I_0}{L_0} \hat{i_n} ;\qquad I_\mathrm{app} = I_0 \hat{I}_\mathrm{app} ; \qquad I_0 = \frac{Q}{A t_\mathrm{c}} \end{gathered}\]

• Temperature:

  \[\begin{gathered} T = T_0+\Delta T_\mathrm{ref}\hat{T} ; \qquad \Delta T_\mathrm{ref} = \frac{I_0 t_\mathrm{c}}{L_0 \rho^\mathrm{ref} c_p^\mathrm{ref} } \Phi_T \end{gathered}\]

• SEI thickness and solvent concentration:

  \[\begin{gathered} \delta_\mathrm{\scriptscriptstyle SEI} = \delta_\mathrm{ref} \hat{\delta} \qquad c_\mathrm{\scriptscriptstyle EC}=c_\mathrm{\scriptscriptstyle EC}^\mathrm{ref} \hat{c}_\mathrm{\scriptscriptstyle EC} \end{gathered}\]

• Stresses:

  \[\begin{gathered} \sigma_{\mathrm{h}} = E_{\mathrm{ref}} \hat{\sigma_{\mathrm{h}}} \end{gathered}\]

Dimensionless numbers

With the mentioned scaling and proper arrangement in the equations, we have defined the following dimensionless quantities:

\(\tau_\mathrm{e}\)	\(\frac{D_\mathrm{e}^\mathrm{eff,ref} t_\mathrm{c}}{L_0}\)	\(\delta_\kappa\)	\(\frac{L_0 I_0}{\kappa_\mathrm{eff}^\mathrm{ref} \Phi_\mathrm{l}}\)	\(\delta_{\kappa_D}\)	\(\frac{\delta_\kappa}{2\alpha (1-t_+^0)(1+\frac{\partial \ln f_{\pm}}{\partial \ln c_\mathrm{e}})} \frac{\Phi_\mathrm{l}}{\Phi_T} \frac{c_\mathrm{e}^\mathrm{ref}}{\Delta c_\mathrm{e}^\mathrm{ref}}\)
\(\delta_{\sigma}\)	\(\frac{I_0 L_0}{\sigma_\mathrm{ref} \Phi_\mathrm{s}}\)	\(\tau_\mathrm{s}\)	\(\frac{D_\mathrm{s}^\mathrm{ref} t_\mathrm{c}}{R_\mathrm{s}^2}\)	\(S\)	\(\frac{R_\mathrm{s} I_0}{a_\mathrm{s} D_\mathrm{s}^\mathrm{ref} c_\mathrm{s}^\mathrm{max} F L_0}\)
\(\hat{k}_0\)	\(\frac{F k_0 L_0 }{I_0} c_\mathrm{e}^\mathrm{ref} (c_\mathrm{s}^\mathrm{max})^\alpha\)	\(\tau_q\)	\(\frac{t_\mathrm{c} k_T^\mathrm{ref} }{\rho^\mathrm{ref} c_p^\mathrm{ref} L_0^2}\)	\(\delta_{\lambda}\)	\(\frac{L_0^2 \rho^\mathrm{ref} c_p^\mathrm{ref} }{t_\mathrm{c} \lambda^\mathrm{ref}}\)
\(\delta_\mathrm{ref}\)	\(\frac{t_\mathrm{c} I_0 M_\mathrm{\scriptscriptstyle SEI}}{2 F \rho a_\mathrm{s} L_0}\)	\(\tau_{\scriptscriptstyle \mathrm{LAM}}\)	\(\beta t_c \left ( \frac{E_{\mathrm{ref}}}{\sigma_{\mathrm{cr}}}\right )^m\)	\(\delta_{\sigma_{\mathrm{h}}}\)	\(\frac{2\Omega}{9\left(1-\nu\right)}c_{\mathrm{s}}^{\mathrm{max}}\)

With the specified scaling and dimensionless numbers, the models equation have been reformulated.

Electrochemical Model

• Mass transport in the electrolyte:

  \[\begin{gathered} \frac{\varepsilon_\mathrm{e}}{\tau_\mathrm{e}}\frac{\partial\hat{c}_\mathrm{e}}{\partial \hat{t}} = \hat{\nabla}\cdot \left(\frac{D_\mathrm{e}^\mathrm{eff}}{D_\mathrm{e}^\mathrm{eff,ref}} \hat{\nabla} \hat{c}_\mathrm{e} \right) + \sum_{i=0}^{n_\mathrm{mat}} \hat{j}_{i} \end{gathered}\]

• Charge transport in the electrolyte:

  \[\begin{gathered} - \hat{\nabla}\cdot \left( \frac{1}{\delta_K} \frac{\kappa_\mathrm{eff}}{\kappa_\mathrm{eff}^\mathrm{ref}} \hat{\nabla}\hat{\varphi}_\mathrm{e} - \frac{1}{\delta_{K_D}} \frac{\kappa_\mathrm{eff}}{\kappa_\mathrm{eff}^\mathrm{ref}} \frac{1+\frac{\Delta T}{T_\mathrm{ref}} \hat{T}}{1+\frac{\Delta c_\mathrm{e}}{c_\mathrm{e,ref}} \hat{c}_\mathrm{e}} \hat{\nabla} \hat{c}_\mathrm{e} \right) = \sum_{i=0}^{n_\mathrm{mat}} \hat{j}_i \end{gathered}\]

• Charge transport in the electrodes and current collectors:

  \[\begin{gathered} -\hat{\nabla}\cdot \left( \frac{1}{\delta_{\sigma}} \frac{\sigma_\mathrm{eff}}{\sigma_\mathrm{eff}^\mathrm{ref}} \hat{\nabla} \hat{\varphi}_\mathrm{s} \right) = -\sum_{i=0}^{n_\mathrm{mat}} \hat{j}_i ;\quad \frac{1}{\delta_{\sigma}} \frac{\sigma_\mathrm{eff}}{\sigma_\mathrm{eff}^\mathrm{ref}} \frac{\partial \hat{\varphi}_\mathrm{s}}{\partial \mathbf{n}} \Bigg|_\mathrm{tab} = \hat{I}_\mathrm{app} \end{gathered}\]

• Mass transport in the active material (pseudodimension):

  The mass transport in the active material is calculated using Legendre polynomials.

  \[\begin{gathered} \frac{1}{\tau_\mathrm{s}} \frac{\partial \hat{c}_\mathrm{s}}{\partial \hat{t}} = \frac{1}{\hat{r}^2}\frac{\partial}{\partial \hat{r}} \left( \hat{r}^2 \frac{D_\mathrm{s}}{D_\mathrm{s}^\mathrm{ref}} \frac{\partial \hat{c}_\mathrm{s}}{\partial \hat{r}} \right) ; \quad \frac{D_\mathrm{s}}{D_\mathrm{s}^\mathrm{ref}} \frac{\partial \hat{c}_\mathrm{s}}{\partial \hat{r}} \Bigg|_{\hat{r}=1} = S \hat{j}_i \end{gathered}\]

• Exchange between the electrolyte and the electrode by lithium intercalation:

  The intercalation exchange current between the electrolyte and the active materials is calculated as follows

  \[\begin{gathered} \hat{j}_i = \hat{k}_0 \left( \left( 1+\frac{\Delta c_\mathrm{e}}{c_\mathrm{e,ref}} \hat{c}_\mathrm{e} \right) \hat{c}_\mathrm{s}|_{\hat{r}=1} (1-\hat{c}_\mathrm{s}|_{\hat{r}=1}) \right)^{0.5} 2 \sinh{\hat{\eta}} \end{gathered}\]

• Overpotential

  \[\begin{gathered} \hat{\eta} = \frac{\Phi_\mathrm{s}}{\Phi_T} \hat{\varphi}_\mathrm{s} - \frac{\Phi_\mathrm{l}}{\Phi_T} \hat{\varphi}_\mathrm{e} - \hat{U}_\mathrm{eq} \end{gathered}\]

Thermal Model

• Energy conservation

  \[\begin{split}\begin{gathered} \frac{\rho c_p}{\rho^\mathrm{ref} c_p^\mathrm{ref}} \frac{\partial \hat{T}}{\partial \hat{t}} = \frac{1}{\delta_{\lambda}}\hat{\nabla}\cdot \left( \frac{\lambda}{\lambda^\mathrm{ref}} \hat{\nabla} \hat{T} \right) + \hat{q} \\ \frac{\lambda}{\lambda^\mathrm{ref}} \frac{\partial \hat{T}}{\partial \mathbf{n}} \Bigg|_{\Gamma} = \frac{L_0 h}{\lambda^\mathrm{ref} \Delta T_\mathrm{ref}} \left(T_0-T_\mathrm{ext} + \Delta T_\mathrm{ref} \hat{T} \right) \end{gathered}\end{split}\]

• Heat generation:

  Several heat sources have been considered.

  \[\begin{gathered} \hat{q} = \hat{q}_\mathrm{ohm} + \hat{q}_\mathrm{rev} + \hat{q}_\mathrm{irr} \end{gathered}\]

  • Ohmic heat source

    This corresponds to the heat generated by the transport of charge within the cell.

    \[\begin{split}\begin{align*} \hat{q}_\mathrm{ohm} &= \hat{q}_\mathrm{solid} + \hat{q}_\mathrm{liquid} \\ \hat{q}_\mathrm{solid} &= \frac{1}{\delta_{\sigma}} \frac{\sigma_\mathrm{eff}}{\sigma_\mathrm{eff}^\mathrm{ref}} \frac{\Phi_\mathrm{s}}{\Phi_T} \hat{\nabla} \hat{\varphi}_\mathrm{s} \hat{\nabla} \hat{\varphi}_\mathrm{s} \\ \hat{q}_\mathrm{liquid} &= \frac{\Phi_\mathrm{l}}{\Phi_T} \frac{\kappa_\mathrm{eff}}{\kappa_\mathrm{eff}^\mathrm{ref}} \left(\frac{1}{\delta_{\kappa}} \hat{\nabla} \hat{\varphi}_\mathrm{e} \hat{\nabla} \hat{\varphi}_\mathrm{e} - \frac{1}{\delta_{\kappa_D}} \frac{1+\frac{\Delta T}{T_\mathrm{ref}} \hat{T}}{1+\frac{\Delta c_\mathrm{e}}{c_\mathrm{e,ref}} \hat{c}_\mathrm{e}} \hat{\nabla} \hat{c}_\mathrm{e} \hat{\nabla} \hat{\varphi}_\mathrm{e} \right) \end{align*}\end{split}\]

  • Reversible reaction heat source

    The reversible heat caused by the reaction is proportional to the entropy change, that is approximated with the variation of Open Circuit Potential.

    \[\begin{gathered} \hat{q}_\mathrm{rev} = \sum_{i=0}^{n_\mathrm{mat}} \hat{j}_{i} \frac{T}{\Phi_T} \frac{\partial U_i(c^\mathrm{surf}_\mathrm{s})}{\partial T} \end{gathered}\]

  • Irreversible polarization heat source

    This represents the irreversible heating due to the polarization heat generated by the exchange current at the electrolyte-electrode interface.

    \[\begin{gathered} \hat{q}_\mathrm{irr} = \sum_{i=0}^{n_\mathrm{mat}} \hat{j}_{i} \hat{\eta} \end{gathered}\]

Degradation Models

• SEI Models: solvent-diffusion

  The model considers that the SEI is originated by the electrochemical reaction between a EC solvent molecule, two lithium ions and two electrons at the electrode surface:

  \[\begin{gathered} \rm EC + 2 Li^+ + 2 e^- \rightarrow V_\mathrm{\scriptstyle SEI} \end{gathered}\]

  Therefore, the reaction equation reads:

  \[\begin{gathered} \hat{j}_\mathrm{\scriptscriptstyle SEI} = \frac{F L_0 k_\mathrm{\scriptscriptstyle SEI}}{I_0} c_\mathrm{\scriptscriptstyle EC}^\mathrm{ref} \hat{c}_\mathrm{\scriptscriptstyle EC} e^{-\frac{\beta}{\alpha}(\hat{\eta} - (\hat{U}_\mathrm{\scriptscriptstyle SEI} - \hat{U}_\mathrm{eq}))} \end{gathered}\]

  where the concentration of EC solvent in the SEI is modelled according to the transport equation:

  \[\begin{gathered} \frac{\partial \hat{c}_\mathrm{\scriptscriptstyle EC}}{\partial \hat{t}} - \frac{\hat{x}}{\hat{\delta}_\mathrm{\scriptscriptstyle SEI}} \frac{\partial \hat{\delta}_\mathrm{\scriptscriptstyle SEI}}{\partial \hat{t}} \hat{\nabla} \hat{c}_\mathrm{\scriptscriptstyle EC} = \hat{\nabla}\cdot \left( \frac{t_\mathrm{c} D_\mathrm{\scriptscriptstyle EC} }{\delta_\mathrm{ref}^2} \frac{\hat{\nabla} \hat{c}_\mathrm{\scriptscriptstyle EC}}{\hat{\delta}_\mathrm{\scriptscriptstyle SEI}^2} - \frac{ \partial \hat{\delta}_\mathrm{\scriptscriptstyle SEI}}{\partial \hat{t}} \hat{c}_\mathrm{\scriptscriptstyle EC} \right) \end{gathered}\]

  with the following boundary conditions:

  \[\begin{gathered} \left( \frac{t_\mathrm{c} D_\mathrm{\scriptscriptstyle EC} }{\delta_\mathrm{ref}^2} \frac{\hat{\nabla} \hat{c}_\mathrm{\scriptscriptstyle EC}}{\hat{\delta}_\mathrm{\scriptscriptstyle SEI}^2} - \frac{ \partial \hat{\delta}_\mathrm{\scriptscriptstyle SEI}}{\partial \hat{t}} \hat{c}_\mathrm{\scriptscriptstyle EC} \right) \Bigg|_{\hat{x}=0} = \frac{2 \rho_\mathrm{\scriptscriptstyle SEI}}{M_\mathrm{\scriptscriptstyle SEI} c_\mathrm{\scriptscriptstyle EC}^\mathrm{ref}} \hat{j}_\mathrm{\scriptscriptstyle SEI} \quad ; \quad \hat{c}_\mathrm{\scriptscriptstyle EC} \big|_{\hat{x}=1} = 1 \end{gathered}\]

  The SEI growth can be calculated from the reaction rate and the physical properties of the SEI layer:

  \[\begin{gathered} \frac{\partial \hat{\delta}_\mathrm{\scriptscriptstyle SEI}}{\partial \hat{t}} = - \hat{j}_\mathrm{\scriptscriptstyle SEI} \end{gathered}\]

  Thus, the total exchange current has two components:

  \[\begin{gathered} \hat{j}_\mathrm{tot} = \hat{j}_\mathrm{int} + \hat{j}_\mathrm{\scriptscriptstyle SEI} \end{gathered}\]

  And the overpotential has now an additional component corresponding to the voltage drop caused by SEI resistance:

  \[\begin{gathered} \hat{\eta} = \frac{\Phi_\mathrm{s}}{\Phi_T} \hat{\varphi}_\mathrm{s} - \frac{\Phi_\mathrm{l}}{\Phi_T} \hat{\varphi}_\mathrm{e} - \hat{U}_\mathrm{eq} - \frac{\delta_\mathrm{ref} I_0}{\kappa_\mathrm{\scriptscriptstyle SEI} L_0 a_\mathrm{s} \Phi_T} \hat{\delta}_\mathrm{\scriptscriptstyle SEI} \hat{j}_\mathrm{tot} \end{gathered}\]

• LAM model

  The model computes the loss of active material due to particle cracking driven by stresses. Therefore, the decrease of the volume fraction of active material is computed as

  \[\begin{gathered} \hat\sigma_{\mathrm{h}}=\delta_{\sigma_\mathrm{h}}\left ( 3\int_{0}^{1}\hat{c}\hat{r}^2d\hat{r}-\hat{c} \right ) \end{gathered}\]

  And the hydrostatic stress is computed from the equilibrium of stresses of a spherical electrode particle

  \[\begin{gathered} \frac{\partial \varepsilon_\mathrm{s}}{\partial \hat{t}}= -\tau_{\mathrm{\scriptscriptstyle LAM}}\left(\hat\sigma_{\mathrm{h}}\right)^m \qquad \hat\sigma_{\mathrm{h}}>0 \end{gathered}\]