## 4 Results and Discussion  
### Surface Shape/strain Distribution of the Si Electrode  
Compared to the thickness of Si film electrode, the copper foil substrate can be considered as a semi-infinite solid, a common model in solid mechanics. Based on the semi-infinite solid model, the deformation of the copper foil has not been taken into account in the current investigation. Through the lithiation and the delithiation process, the Si film electrode can freely expand and shrink except its bonding surface with the copper substrate. The surface deformation of the electrode during its lithiation/delithiation process is presented in Figure 4, including in-plane deformation distribution and out-of-plane displacement distribution. The strain distribution \(\varepsilon_{xx}\) shows the radial strain distribution (see Figure 4i), and the strain distribution \(\varepsilon_{yy}\) shows strain distribution of the tangential direction accordingly (see Figure 4 ii). The out-of-plane displacement distribution is presented in Figure 4 iii. Due to the radial motion of lithium ions along the silicon thin film electrode, the distribution of radial deformation of the electrode is different from that of circumferential deformation. In order to facilitate the analysis, we deal with the in-plane strain distribution: the strain \(\varepsilon_{xx}\) is averaged in the y direction, and these averages are fitted, shown as Figure 4i \(b\) and \(d\). What's more, the trend of strain distribution is the opposite in the process of lithium insertion and delithiation: the strain distribution near the central area should be changed greatly in the process of lithium insertion, and the delithiation process is reversed, shown as Figure 4i. The strain \(\varepsilon_{yy}\) is averaged in the x direction, and these averages are fitted, shown as Figure 4 ii \(b\) and \(d\). The strain distribution \(\varepsilon_{xx}\) exhibits gradient along x direction. The strain distribution \(\varepsilon_{yy}\) exhibits symmetry in the y direction, and the strain distribution \(\varepsilon_{yy}\) exhibits gradient along y direction in every tiny subregion. At the initial stage of the lithiation processes, the tangential strain \(\varepsilon_{yy}\) was significantly smaller than the radial strain \(\varepsilon_{xx}\), shown as Figure 4i,ii. The out-of-plane displacement \(w\) of lithium removal is significantly smaller than the out-of-plane displacement \(w\) of lithium intercalation, shown as Figure 4 iii). Besides, the out-of-plane displacement \(w\) also shows average at single time of the electrochemical reaction, while the strain distribution \(\varepsilon_{xx}\) and \(\varepsilon_{yy}\) tends to increase. This result is due to the transport path of lithium ions. The strain distributions indicate that an electrode in the form of a circular thin film deposited on a semi-infinite solid deforms in lithiation/delithiation processes similarly to a thermal expansion process.  
Figure 4: Electrode deformation distributions of sample during the first electrochemical cycle. **(i)** The strain distribution \(\varepsilon_{xx}\) shows the radial strain distribution. **(ii)** The strain distribution \(\varepsilon_{yy}\) shows strain distribution of the tangential direction accordingly. **(iii)** The out-of-plane displacement distribution \(w\) is presented.