# Spontaneous Directional Transportation Surface of Water Droplet and Gas Bubble: A Review
## 2 Basic Theories  
The relevant basic theories of fluid transportation include contact angle, contact angle hysteresis, advancing contact angle, receding contact angle, sliding angle, three classical wettability models, surface tension, surface energy, and Laplace pressure. It is important to understand these theories to figure out the principle of SDT.  
Contact Angle, Contact Angle Hysteresis, Advancing Contact Angle, Receding Contact Angle, and Rolling Angle  
Contact angle \(\theta\) is often used to characterize solid wettability. It is the angle between the tangent line of the gas-liquid interface and the solid-liquid line at the junction of gas, liquid, and solid phases. According to the different contact angles of water on the solid surface, the wettability of solid surface can be divided into four types: superhydrophobic (\(0^{\circ}\leq\theta\leq 10^{\circ}\)), hydrophilic (\(10^{\circ}\leq\theta\leq 90^{\circ}\)), hydrophobic (\(90^{\circ}\leq\theta\leq 150^{\circ}\)), and superhydrophobic (\(150^{\circ}\leq\theta\leq 180^{\circ}\)) ([PERSON] and [PERSON], 2012; [PERSON], 2012; [PERSON], 2012). In the case of the wettability of underwater gas bubbles on the solid surface, the wetting interface changes from a solid-liquid interface in air to a gas-solid interface in water, but the above theory is still valid.  
For example, in the case of liquid, when delivering liquid to a droplet on a smooth and ideal solid surface, the contact line will move outward and keep the contact angle of the droplet the same. However, because the actual solid surface is rough and inhomogeneous, the contact line may not move with the increase in liquid volume. This phenomenon is called contact angle hysteresis. When delivering liquid to a droplet on a rough surface, the contact angle will increase because of contact angle hysteresis, and the contact angle at this time is named advancing contact angle \(\theta_{\text{ad}}\). The contact angle when the solid-liquid contact line is about to move forward is the maximum advancing contact angle, as shown in Figure 1a. On the contrary, removing liquid from a droplet on a rough surface causes the contact angle, called receding contact angle \(\theta_{\text{re}}\), to decrease. The contact angle when the solid-liquid contact line is about to move backward is the minimum receding contact angle, as shown in Figure 1b. The contact angle hysteresis usually causes the drag force in SDWT, and the force \(F\) can be calculated by the following equation [32]:  
\[F=\pi w\gamma(\cos\theta_{\text{re}}-\cos\theta_{\text{ad}}) \tag{1}\]  
where \(w\) is the diameter of the solid-liquid contact surface and \(\gamma\) is the surface tension. On a tilted plane, the advancing angle and the receding angle of the liquid can also be observed. Due to the existence of the contact angle hysteresis, when the plane begins to tilt, the droplet does not immediately move. When the plane rotates to a certain angle, the droplet begins to roll. This angle is called rolling angle \(\alpha\). The degree of contact angle hysteresis determines whether the droplet is easy to roll.