# Improved Electrochemical-Mechanical Parameter Estimation Technique for Lithium-Ion Battery Models
## 2 Materials and Methods
### Doyle-Fuller-Newman Model  
The DFN model provides a physics-based framework that captures the coupled electrical and chemical species transport phenomena that occur within the battery. Based on the porous electrode and concentrated solution theories, the model accounts for key processes such as lithium-ion diffusion in the solid and electrolyte phases and charge transfer kinetics at the electrode-electrolyte interface. It also captures the multiscale behavior arising from lithium diffusion within the active material particles. The DFN model is also referred to as the Pseudo-Two-Dimensional (P2D) model. This name arises from its formulation: it resolves a one-dimensional spatial domain along the electrode thickness, and at each position within this domain, it incorporates a second dimension representing lithium diffusion within spherical particles. The P2D governing equations and the corresponding boundary conditions are listed in Table 1.  
\begin{table}
\begin{tabular}{l l} \hline \hline
**Description** & **Governing Equations** & **Boundary Conditions** \\ \hline \multicolumn{3}{c}{Mass conservation} \\ \hline Active material particles, \(k\in\{p,n\}\) & \(\frac{\alpha_{i,j}}{\alpha\mu}=\
abla\cdot\left[D_{s,k}^{\text{eff}}(\frac{ \alpha_{i,j}}{\alpha\mu}-\frac{\Omega_{i,j}}{R_{s}T}\frac{\alpha\alpha_{j,j}}{ \alpha\mu})\right]\) & \(D_{s,k}^{\text{eff}}(\frac{\alpha_{i,j}}{\alpha\mu}-\frac{\Omega_{i,j}}{R_{s}T} \frac{\alpha\mu_{j,j}}{\alpha\mu})\big{|}_{r=0}=0\), \(D_{s,k}^{\text{eff}}(\frac{\alpha_{i,j}}{\alpha\mu}-\frac{\Omega_{i,j}}{R_{s} T}\frac{\alpha\mu_{j,j}}{\alpha\mu})\big{|}_{r=0}=0\), \(D_{s,k}^{\text{eff}}(\frac{\alpha_{i,j}}{\alpha\mu}-\frac{\Omega_{i,j}}{R_{s} T}\frac{\alpha\mu_{j,j}}{\alpha\mu})\big{|}_{r=R_{s,k}}=-I_{l\mu,k}\) \\ Electrolyte, \(k\in\{p,s,n\}\) & \(\epsilon_{s,k}\frac{\lambda_{i,j}}{\alpha\mu}=\frac{2}{\alpha\mu}\big{(}D_{c,k }^{\text{eff}}(\frac{\alpha_{i,j}}{\alpha\mu})+a_{s,k}(1-t_{+})I_{l\mu,k} \big{)}\) & \(\frac{\lambda_{i,j}}{\alpha\mu}\big{|}_{s=0,t_{+}t_{+}t_{+}}=0\) \\ \hline \multicolumn{3}{c}{Charge conservation} \\ \hline Active material particles, \(k\in\{p,n\}\) & \(\frac{2}{\alpha\mu}\big{(}\epsilon_{s,k}^{\text{eff}}\frac{\alpha\mu_{k,j}}{ \alpha\mu}\big{)}=a_{s,k}F_{l\mu,k}\) & \(\epsilon_{s,k}^{\text{eff}}\frac{\alpha\mu_{k,j}}{\alpha\mu}\big{)}\big{|}_{s= 0,t_{+}t_{+}t_{+}t_{+}}=F_{l\mu,k}\), \(\epsilon_{s,k}^{\text{eff}}\frac{\alpha\mu_{k,j}}{\alpha\mu}\big{|}_{s=-t_{+} t_{+}t_{+}}=0\) \\ Electrolyte, \(k\in\{p,s,n\}\) & \(\frac{2}{\alpha\mu}\big{[}\epsilon_{s,k}^{\text{eff}}\big{(}\frac{\alpha\mu_{k,j}}{ \alpha\mu}-\frac{2(1-t_{+})I_{l}}{\Gamma}\frac{\alpha\mu(\epsilon_{s,k})}{ \alpha\mu}\big{)}\big{]}=-a_{s,k}F_{l\mu,k}\) & \(\frac{\alpha\mu_{k,j}}{\alpha\mu}\big{|}_{s=0,t_{+}t_{+}t_{+}t_{+}}=0\), \(\Phi_{s,k}\big{|}_{s=0,t_{+}t_{+}t_{+}t_{+}}=0\) \\ \hline \multicolumn{3}{c}{Reaction kinetics} \\ \hline Butler–Volmer & \(I_{l\mu,k}=\begin{cases}\frac{2}{\pi}\frac{\text{i}\mu}{\pi}\sinh\frac{\text{i} \theta_{k}T}{\eta_{k}}&k\in\{p,n\}\\ 0&k=s\end{cases}\) & - \\ Exchange current & \(i_{0,k}=F_{k}^{\text{eff}}\big{[}\epsilon_{s,k}^{\text{surf}}\big{(}\frac{ \alpha\max}{\epsilon_{s,k}}-\sum_{s,k}^{\text{surf}}\big{)}\big{]}^{0.5}\) & - \\ Overpotential \(k\in\{p,n\}\) & \(\eta_{k}(z,t)=\Phi_{s,k}(z,t)-\Phi_{s,k}(z,t)-U_{k}-F_{l\mu,k}R_{c,k}\) & - \\ \hline \hline \end{tabular}
\end{table}
Table 1: Governing equations of the P2D model for LIBs. The subscripts \(n\), \(p\), and \(s\) indicate the negative electrode, positive electrode, and separator, respectively.  
The superscript _eff_ refers to the correction applied to the diffusion coefficients, conductivities, and reaction rates, accounting for porosity and tortuosity in each domain [22], as well as thermal dependence when the temperature differs from the reference value [23].