## 4 Results and Discussions  
### Analysis of Decay Behavior of Foam Microstructure  
Foam is a gas-liquid two-phase conditioner. The picture of foam microstructure before treatment under 40-times magnification is clearly shown in Figure 7, and the pictures of microstructure under different decay times are shown in Figure 8. The inside nucleus was filled with air and the surface is covered with liquid film. After 0.5 min of decay, the generated bubbles were separated by the thick liquid film full of free water and microbubbles. The shape of bubble particle was approximately circle or oval regardless of bubble size, as shown in Figure 8a. After 5 min of decay, the relatively large size bubbles became bigger, while the relatively small bubbles become smaller gradually until vanishment. The shape of bubbles started to be polygon, especially for large-size bubbles, as shown in Figure 8b. After 10 min of decay, the bubble size further became larger, and the bubbles were firmly attached to each other because of thinner liquid film induced by the loss of free water. The number of bubble particles decreased further on account of the vanishment of microbubbles and small bubbles because  
Figure 5: Warning-Blender method: (**a**) Waring-Blender tester, (**b**) generated foam. Figure 6: Addition of foam agent: (**a**) Injection of foam before mixing, (**b**) homogenous mixtures after mixing. of the merger. The shape of the remaining bubbles became polygon, while the pentagon and hexagon were the most common, as shown in Figure 8c. After 15 min, 20 min, 25 min, and 30 min of decay, it can be seen that the bubble size further became larger and the liquid film further became thinner. The liquidity of the bubbles dried up gradually and the bubbles no longer had obvious deformations, as shown in Figure 8d-g. After 60 min of decay, the film was adsorbed and bent by the micro solid body and the bubble structure had all but disappeared (see Figure 8h). The above phenomena were caused by the drainage of water in liquid film and air fusion among bubbles [16]. The drainage of water in the liquid film was caused by gravity and pressure difference. The free water flowed under the influence of gravity when the liquid film was full of free water. Meanwhile, a plateau boundary was formed at the intersection of adjacent bubbles (see Figure 9a). The movement of free water caused by pressure difference can be expressed by the Laplace equation as follows:  
\[P_{A}-P_{B}\ =\ \frac{\sigma}{R} \tag{5}\]  
where \(P_{A}\), \(P_{B}\) are the pressure at A point and B point, respectively; \(\sigma\) is surface tension of liquid film; and \(R\) is radius of constant volume circle. Free water flows from A point to B point when \(P_{A}>P_{B}\)  
Figure 8: Variation of foam microstructure under 40-times magnification after treatment (**a**) 0.5 min; (**b**) 5 min; (**c**) 10 min; (**d**) 15 min; (**e**) 20 min; (**f**) 25 min; (**g**) 30 min; (**h**) 60 min. Figure 7: Microstructure of liquid film before treatment. as shown in Figure 9a. The existence of microbubbles in the liquid film hinders the movement of the free water and contributes to the stability of the bubble microstructure. When the angle of plateau boundary is 120\({}^{\circ}\), the pressure at A point and B point is the minimum, and the bubble is stable. Therefore, the shape of bubble in two-dimension plane is hexagon, which expresses the phenomena of existence of hexagon in Figure 8c-e. Figure 9b shows the sketch of curved surface in three dimensional. The pressure difference can be expressed as follows:  
\[\Delta P\ =\ \gamma\bigg{(}\frac{1}{R_{1}}-\frac{1}{R_{2}}\bigg{)} \tag{6}\]  
where \(\Delta P\) is pressure difference, \(\gamma\) is coefficient of surface tension, and \(R_{1}\) and \(R_{2}\) are curvature radius at orthogonal plane.