# Spontaneous Directional Transportation Surface of Water Droplet and Gas Bubble: A Review
## 2 Basic Theories
### Three Classical Wettability Models  
The wetting behavior of a droplet on a solid surface can be summarized as follows: the Young model [33], the Wenzel model [34], and the Cassie-Baxter model [35], as shown in Figure 1c-e.  
1. Young Model  
The Young model is a model of a droplet on an ideal and smooth solid surface. It was proposed by [PERSON] in 1805, showing the connection between the surface tension and the static contact angle on an ideal smooth solid surface. The model equation is as follows:  
\[\cos\theta_{\text{e}}=\frac{\gamma_{\text{SG}}-\gamma_{\text{SL}}}{\gamma_{ \text{LG}}} \tag{2}\]  
where \(\theta_{\text{e}}\) is the eigen contact angle of the droplet to the solid; \(\gamma_{\text{SG}}\), \(\gamma_{\text{SL}}\), \(\gamma_{\text{LG}}\) are the surface tensions of the solid-gas interface, solid-liquid interface, liquid-gas interface.  
2. Wenzel Model  
Since the actual surface is not an ideal smooth surface, American researcher [PERSON] proposed the [PERSON] model in 1936 based on the [PERSON] model. This model added the roughness factor, as follows:  
\[\cos\theta_{\text{a}}=r\frac{(\gamma_{\text{SG}}-\gamma_{\text{SL}})}{\gamma_ {\text{LG}}}=r\cos\theta_{\text{e}} \tag{3}\]  
Figure 1: Wetting behavior of a droplet on a solid surface. (**a**) Advancing angle. (**b**) Receding angle. (**c**) [PERSON] model. (**d**) [PERSON] model. (**e**) Cassie-Baxter model. (**f**) Laplace pressure.  
where \(\theta_{\rm a}\) is the apparent contact angle, and \(r\) is the roughness factor which is equal to the ratio of the actual and the projection area of the rough surface. From the [PERSON] model, it can be seen that the rougher surface will make the originally hydrophilic surface become more hydrophilic, and the originally hydrophobic surface will become more hydrophobic. The [PERSON] model also explains that the surface of rose petals presents a superhydrophobic state but has a high adhesive force with water droplets. [PERSON] et al. studied and analyzed the phenomenon of the \"petal effect\" in 2008, and pointed out that the microstructure scale distribution of rose petal surface is the main factor affecting the adhesion of rose petals to water droplets [36; 37]. The model successfully promotes the Young model to realistic situations but still fails to explain the low adhesion of water droplets on lotus leaves.  
3. Cassie-Baxter Model  
To solve the limitation of the Wenzel model, [PERSON] and [PERSON] improved it in 1944. The Wenzel model pointed out that a droplet on a rough surface does not fill all the rough structures of a surface and leaves air in them. Based on this theory, they proposed the Cassie-Baxter equation:  
\[\cos\theta_{a}=f_{1}\cos\theta_{\rm e}+f_{2}\cos\theta_{\rm v} \tag{4}\]  
where \(f_{1}\) and \(f_{2}\) represent the liquid-solid contact area ratio and the liquid-air contact area ratio; \(\theta_{\rm v}\) is the eigen contact angle of the droplet to the gas. When it is a water droplet, since the eigen contact angle between water and air is \(180^{\circ}\), the equation can be simplified as follows:  
\[\cos\theta_{a}=f_{1}\cos\theta_{\rm e}-f_{2} \tag{5}\]  
From the above equation, it can be seen that when the eigen contact angle of the solid surface is a constant value, the contact angle will increase by decreasing the contact area. Since the water droplet in this model does not completely fill the rough structures of the surface, this helps the rolling of the water droplet. The reason why the droplets on lotus leaves show a superhydrophobic state and can roll off very easily can be explained using the Cassie-Baxter model. [PERSON] et al. [38] prepared a biomimetic microarray surface by using shape memory polyurethane (SMP) as a substrate. With the help of SMP, the modulation of the surface morphology can be realized, and the droplets can be changed between the Wenzel and Cassie-Baxter models to achieve the difference in surface adhesion.