## 2 Materials and Methods
### Theoretical Analysis on the Anti-Soil-Adhesion Mechanism of the Unsmooth Surface
The distances from \(p_{1}\) to \(p_{0}\) and from \(p_{2}\) to \(p_{0}\) are \(d_{1}\) and \(d_{2}\), respectively. Correspondingly, \(m_{1}\) and \(m_{2}\) are the ratios of \(d_{1}\)/\(S_{0}\) and \(d_{2}\)/\(S_{0}\), respectively. Therefore, the width parallel to the S axis of each strip is (\(m_{2}\)-\(m_{1}\))-\(S_{0}\). The total area of all the strips shows the area occupied by the gas-liquid interface. Therefore, the area of the gas-liquid contact interface under the liquid drop is  
\[ \begin{array}{rl}A_{g1}&=\sum\limits_{n=1}^{N-1}\left\{\pi[(n-1)+m_{2}]^{2}S _{0}^{2}-\pi[(n-1)+m_{1}]^{2}S_{0}^{2}\right\}\\ &=\pi S_{0}^{2}\sum\limits_{n=1}^{N-1}\left\{[(n-1)+m_{2}]^{2}-[(n-1)+m_{1}]^{ 2}\right\}\end{array}  \tag{6}\]  
where \(A_{g1}\) is the interface area between the gas and the liquid at the composite interface, \(n\) is the \(n\)th wave, and \(N\) is the number of waves covered by the liquid drop. The projected area of the rough surface \(A_{p}\) is  
\[A_{\rm P}=\pi R^{2} \tag{7}\]  
Hence, according to the definition of \(\eta\), \(\eta\) can be expressed as  
\[ \begin{array}{rl}\eta&=\frac{A_{\rm gd}}{A_{\rm P}}=\frac{\pi S_{0}^{2}\sum \limits_{n=1}^{N-1}[(n-1)+m_{2}]^{2}-[(n-1)+m_{1}]^{2}}{\pi R^{2}}\\ &=\frac{S_{0}^{2}\sum\limits_{n=1}^{N-1}[(n-1)+m_{2}]^{2}-[(n-1)+m_{1}]^{2} \right\}\end{array}  \tag{8}\]  
The real contact interface area between the soil and the rough surface can be calculated as follows:  
\[ \begin{array}{rl}&A_{\rm sl}=\sum\limits_{n=1}^{N-1}\left\{\int\limits_{n 5_{0}}^{(n+m_{1})S_{0}}2\pi S\left[1+\left(\frac{dh}{dS}\right)^{2}\right]^{ 1/2}dS+\int\limits_{(n+m_{2})S_{0}}^{(n+1)S_{0}}2\pi S\left[1+\left(\frac{dh} {dS}\right)^{2}\right]^{1/2}dS\right\}\\ &+\int\limits_{NS_{0}}^{R}2\pi S\left[1+\left(\frac{dh}{dS}\right)^{2}\right]^ {1/2}dS\\ &=2\pi\sum\limits_{n=1}^{N-1}\left\{\int\limits_{n5_{0}}^{(n+m_{1})S_{0}}S \left[1+\left(\frac{dh}{dS}\right)^{2}\right]^{1/2}dS+\int\limits_{(n+m_{2})S _{0}}^{(n+1)S_{0}}S\left[1+\left(\frac{dh}{dS}\right)^{2}\right]^{1/2}dS \right\}\\ &+2\pi\int\limits_{NS_{0}}^{R}S\left[1+\left(\frac{dh}{dS}\right)^{2}\right]^ {1/2}dS\end{array}  \tag{9}\]  
Here,  
\[\int\limits_{n5_{0}}^{(n+m_{1})S_{0}}S\left[1+\left(\frac{dh}{dS}\right)^{2} \right]^{1/2}dS=\int\limits_{(n+m_{2})S_{0}}^{(n+1)S_{0}}S\left[1+\left(\frac {dh}{dS}\right)^{2}\right]^{1/2}dS \tag{10}\]\[\int\limits_{NS_{0}}^{R}S\left[1+\left(\frac{dh}{dS}\right)^{2}\right]^{1/2}dS \approx\int\limits_{(n+m_{2})S_{0}}^{(n+1)S_{0}}S\left[1+\left(\frac{dh}{dS} \right)^{2}\right]^{1/2}dS \tag{11}\]  
Let,  
\[\int\limits_{NS_{0}}^{(n+m_{1})S_{0}}S\left[1+\left(\frac{dh}{dS}\right)^{2} \right]^{1/2}dS=\int\limits_{(n+m_{2})S_{0}}^{(n+1)S_{0}}S\left[1+\left(\frac{dh }{dS}\right)^{2}\right]^{1/2}dS=a_{n} \tag{12}\]  
Thus, from Equations (11) and (12), we can obtain  
\[\int\limits_{NS_{0}}^{R}S\left[1+\left(\frac{dh}{dS}\right)^{2}\right]^{1/2}dS \approx a_{n} \tag{13}\]  
Substituting Equations (12) and (13) into Equation (9),  
\[A_{\rm sl}\approx 2\pi\sum\limits_{n=0}^{N-1}2a_{n}+2\pi a_{n}=2\pi\left( \sum\limits_{n=0}^{N-1}2a_{n}+a_{n}\right) \tag{14}\]  
Based on the definition of \(r^{\prime}\), Equations (7) and (14),  
\[r^{\prime}=\frac{A_{\rm sl}}{A_{\rm p}}=\frac{2\pi\left(\sum\limits_{n=0}^{N- 1}2a_{n}+a_{n}\right)}{\pi R^{2}}=\frac{2}{R^{2}}\left(\sum\limits_{n=0}^{N-1} 2a_{n}+a_{n}\right) \tag{15}\]  
\[ \begin{array}{l}a_{n}=a_{n-1}=\cdots=a_{0}=\int\limits_{0}^{m_{1}}S\left[1+ \left(\frac{dh}{dS}\right)^{2}\right]^{1/2}dS\\ =\int\limits_{0}^{m_{1}S_{0}}S\left[1+\frac{4\pi^{2}h_{0}^{2}}{S_{0}^{2}}\sin^ {2}\frac{2\pi S_{0}}{S_{0}}\right]^{1/2}dS\end{array}  \tag{16}\]  
Therefore, Equation (16) shows that the larger the value of \(h_{0}S_{0}\) is, the larger the \(a_{n}\), and the larger the value of \(r^{\prime}\) as well (see Equation (15)).