## 3 Results and Discussion
### Plate Shape
#### 3.1.3 Contact Angle of Plate Shape
Since we assumed that the oil on the surface of the AAO-O-SST was thoroughly infused, the interfacial tension between the aluminum and the oil was not taken into account. \[\gamma_{sw}=\sigma_{s}+\sigma_{w}-2\sqrt{\sigma_{s}\cdot\sigma_{w}} \tag{1}\]  
\[\gamma_{Aw}=\gamma_{sw} \tag{2}\]  
\[\gamma_{ASw}=f\gamma_{sw}+(1-f)\gamma_{sw} \tag{3}\]  
where \(\gamma_{sw}\) is interfacial tension between the solid (aluminum) and the liquid (water), \(\gamma_{ow}\) is that between the oil and the water, \(\gamma_{Aw}\) is that between the AAO surfaces and the water, and \(\gamma_{ASw}\) is that between the AAO-O-SST surfaces and the water. \(\sigma_{s}\) is the surface free energy of the solid, and \(\sigma_{w}\) is the surface tension of the liquid. \(r\) is the roughness factor which is the ratio of the total surface area to the projected area of the solid in contact with liquid, and \(f\) is the fraction of the projected area of the surface that is occupied by the solid. Figure 8: Schematic cross-section diagram of bare aluminum, bare aluminum-SST (salt spray test), AAO (anodic aluminum oxide), AAO-O (oil impregnation), AAO-O-SST (salt spray test) surfaces. The roughness factor (r) was calculated by Equation (4). \[r=1+\frac{2\pi\cdot\frac{a}{2}\cdot h}{\left(a+b\right)^{2}} \tag{4}\]  
where \(a\), \(b\), and \(h\) are the pore diameter, edge-to-edge spacing, and the height, respectively. \(a\) and \(b\) were 30 and 70 nm, and \(h\) was 15 \(\upmu\)m. \(f\) is predicted to be 0.65 using the FE-SEM image of the AAO surface. There are various models to predict the interfacial tension between different substances and phases: Owens-Wendt-Rabel-Kaelble (OWRK), Wu, Oss and Good (acid-base), Fowkes, and Extended Fowkes [36; 37]. This study employed the Fowkes model. Although the Extended Fowkes model furthermore consideres polar interactions and a hydrogen bonding fraction as shown in Equation (5), those two fractions are insignificant in these cases. The surface tension of water, aluminum, and oil are 72.8, 45.0, and 25.0 mN/m, respectively. The dispersion contribution of water is 21.8 mN/m. \[\gamma_{\alpha\beta}=\sigma_{\alpha}+\sigma_{\beta}-2\bigg{(}\sqrt{\sigma_{ \alpha}^{D}\cdot\sigma_{\beta}^{D}}+\sqrt{\sigma_{\alpha}^{P}\cdot\sigma_{ \beta}^{P}}+\sqrt{\sigma_{\alpha}^{H}\cdot\sigma_{\beta}^{H}}\bigg{)} \tag{5}\]  
where \(\alpha\) is aluminum or oil, and \(\beta\) is water. \(\sigma^{D}\) is the dispersion force contribution, and \(\sigma^{P}\) is the polar interaction. \(\sigma^{H}\) indicates the hydrogen bonding fraction. The diagrams of the proposed surfaces and the interfacial tensions are tabulated in Table 1. The interfacial tension of bare aluminum and AAO-O-SST showed relatively low values, whereas that for AAO noted a considerably high value. As indicated before, the surfaces of aluminum and AAO-O-SST had a relatively high contact angle. Since the high contact angle means low interfacial tension, the calculated results of interfacial tension agreed with the results of the contact angle. The interfacial tension of the AAO-O-SST was slightly less in value than that of bare aluminum, and the contact angle of the AAO-O-SST was slightly higher than that of bare aluminum. The interfacial tensions at AAO had a significantly high value due to the high roughness factor. The deep pore depth caused the high roughness factors.