# Improved Electrochemical-Mechanical Parameter Estimation Technique for Lithium-Ion Battery Models
## 2 Materials and Methods
### Mechanical Deformation  
The electrodes of lithium-ion batteries are composed of porous composite structures that include active material particles, conductive additives, and a polymeric binder. The active material serves as a host for lithium ions, which are reversibly inserted and extracted during charge and discharge. These processes are called intercalation and deintercalation, respectively. As lithium ions intercalate into or deintercalate from the host structure, they cause lattice configuration changes, leading to microscopic volume expansion or contraction. The lattice structure strain can be characterized through XRD techniques by measuring the variation of the lattice parameters as a function of the corresponding state of lithiation of the electrode. The state of lithiation is the ratio between the actual and the maximum amount of lithium concentration that can be hosted from the active material (\(y\) is the cathode state of lithiation, while \(x\) is the anode one, as reported in Equation (2)). \[y=\frac{c_{s,p}}{c_{s,p}^{max}}\quad;\quad x=\frac{c_{s,n}}{c_{s,n}^{max}} \tag{2}\]  
The maximum lithium concentration depends on the lithiated active material crystal's density \(\rho^{crystal}\) and molar mass \(M_{m}\), as shown in Equation (3) [24]. \[c_{s,k}^{max}=\frac{\rho_{k}^{crystal}}{M_{m,k}} \tag{3}\]  
During lithiation, active materials exhibit phase transitions from their delithiated configuration to the fully lithiated one. LFP undergoes a single phase transition from FePO\({}_{4}\) to LiFePO\({}_{4}\). In contrast, graphite (C\({}_{6}\)) manifests multiple phase transitions before reaching its fully lithiated stage LiC\({}_{6}\)[16; 25]. In addition, graphite's stage transition is not symmetric during its lithiation-delithiation cycle. This is mainly due to the presence of stage IIL during lithium extraction. In fact, this stage is only present at low discharge current rates and gradually fades, increasing the discharge current. Therefore, there is a lithiation window where stages II, IIL, and III coexist; lower discharge rates increase the stage IIL volume fraction and smooth the volume change from stage II to III due to the lower density of stage IIL. Using the XRD measurement of LFP [26] and graphite [25; 27], the crystal volume changes as function of the state of lithiation are shown in Figure 1.  
Figure 1: Crystal lattice strain of **(a)** LFP and **(b)** graphite as a function of lithium content expressed with the state of lithiation of the electrode. The lattice parameters exhibit directional dependence in their deformation due to the intrinsic anisotropy of the lattice structure. However, this anisotropic nature is mitigated by the random orientation of the crystals within the particle. Thus, considering the volumetric strain, the active material particle deforms consistently with its crystal lattice. Then, at the upper scale, the electrode volume change \(\Delta V^{e}\) depends on the \(n^{p}\) particles' volumetric deformation \(\Delta V^{p}\) and on a possible reduction in porosity accounted for through a volumetric expansion parameter \(v\)[28], as shown in Equation (4). In addition, the volume of the \(n^{p}\) particles constitutes the overall active material volume, leading to Equation (5). \[\Delta V^{e}=vn^{p}\Delta V^{p} \tag{4}\]  
\[n^{p}V_{0}^{p}=\zeta V_{0}^{e} \tag{5}\]  
Therefore, the volumetric strain of the electrode is obtained by dividing the Equation (4) by the initial volume of the single particle \(V_{0}^{p}\) and accounting for Equation (5), as reported in Equation (6). Since the battery is assumed to be free to expand, the reduction in porosity is considered negligible, adopting the assumption of \(v=1\)[16]. \[\varepsilon_{v}^{e}(t)=\frac{\Delta V^{e}}{V_{0}^{e}}=\xi\frac{\Delta V^{p}}{V _{0}^{p}}=\xi\varepsilon_{v}^{p}(t) \tag{6}\]  
The previous equations hold under the assumption that all \(n^{p}\) particles deform uniformly. However, this is not entirely accurate due to lithium diffusion across the electrode thickness.