## 4 Results and Discussion
### Strain Criterion of Electrochemical/Mechanical Failures  
The diffusion-induced stress in Si film electrodes relates to various factors such as the reaction path between the electrode and the electrolyte and the material in continuity of the Si electrode. Besides, the Young's modulus of the Si electrode varies through the electrochemical process. Thus, it is hardly possible to experimentally determine the dynamic process of the inner stress distribution. However, material deformation, normally expressed with a concept of strain from a mechanical standpoint, exists objectively, which can be measured quantificationally. Since the strain is not uniform across the electrodes, how to evaluate the deformation and damage of electrode is a challenging problem. Based on the current experiments, the full field strains of the electrode in every phases of the electrochemical process are determined. To effectively describe the mechanical/electrochemical failure of the working electrode, the authors employed a parameter of MSG (mean strain gradient) that depicts the randomness and the discontinuity of the strain distribution. The mathematical description of MSG can be expressed as follows. \[\delta_{total}=\sum_{i=1}^{a}\sum_{j=1}^{b}\sqrt{f_{x}\big{(}\varepsilon_{ij} \big{)}^{2}+f_{y}\big{(}\varepsilon_{ij}\big{)}^{2}}/(A\times B) \tag{1}\]  
\[\delta_{X}=\sum_{i=1}^{a}\sum_{j=1}^{b}\sqrt{f_{x}\big{(}\varepsilon_{ij} \big{)}^{2}}/A \tag{2}\]  
\[\delta_{Y}=\sum_{i=1}^{a}\sum_{j=1}^{b}\sqrt{f_{y}\big{(}\varepsilon_{ij} \big{)}^{2}}/B \tag{3}\]  
where \(\delta_{total}\) denotes the MSG considering both \(x\) and \(y\) directions, while \(\delta_{X}\) and \(\delta_{Y}\) denote the MSG along \(x\) and \(y\) direction, respectively, \(A\) and \(B\) are the pixel dimensions of the ROI along \(x\) and \(y\) direction, respectively, \(a\) and \(b\) are the pixel dimensions of the subset along \(x\) and \(y\) direction, respectively. \(\varepsilon_{ij}\) is the strain values at a certain point (\(i\), \(j\)) in the pixel coordinate, defined with the following two equations. Figure 5: Electrode deformation distributions of the electrode. \[\varepsilon_{ij}=\left\{ \begin{array}{ll}\dfrac{\left(u_{ij}+\dfrac{ \partial u_{ij}}{\partial x}dx\right)-u_{ij}}{dx}=\dfrac{\partial u_{ij}}{ \partial x}&\text{along $x$ direction}\\ \dfrac{\left(v_{ij}+\dfrac{\partial v_{ij}}{\partial y}dy\right)-v_{ij}}{dy}= \dfrac{\partial v_{ij}}{\partial y}&\text{along $y$ direction}\end{array} \right. \tag{4}\]  
where \(u_{ij}\) and \(v_{ij}\) are the displacement value along \(x\) and \(y\) directions, respectively. \(f_{x}\big{(}e_{ij}\big{)}\) and \(f_{y}\big{(}e_{ij}\big{)}\) are the gradient values of the strain along \(x\) and \(y\) directions, respectively. The physical meaning of the strain gradient explains the strain fluctuations along a certain direction which results from the inhomogeneity of the electrochemical reactions. One big advantage to analyze the electrochemical performance of the electrode by the introduced MSG parameter is that, compared to other calculation-based stress analysis methods, the characteristics of genuine deformations over a large area on the electrode can be determined experimentally, describing the macroscopical electrochemical properties such as the reaction intensity. Indeed, results from the current experiments proved that the stress distribution expressed great inconformity. It is generally believed that tensile stress is commonly defined positive, while compressive stress negative in mechanics. The electrode is under compressive stress during lithium insertion, while the electrode is under tensile stress during lithium removal. However, genuine stresses are commonly opposite in sign at different electrode locations due to the material inhomogeneity based on the current experimental results. The development trend of the MSG along two orthogonal directions with the lithiation/delithiation procedures revolving is shown as Figures 6 and 7.