# Natural Stones with a Self-Cleaning Surface via Self-Assembled Monolayers
## 4 Discussion
### Mechanism of the Liquid Wettability  
Aiming to explain the impact of surface roughness on the apparent CA of liquid droplets on solid surfaces, [PERSON] and [PERSON] developed two models, respectively. The [PERSON] model suggests full contact at the liquid/solid interface in which surface roughness increases the actual contact area. As a consequence, the wettability of the solid surface is amplified, and the apparent CA can be expressed as follows [31; 32; 33; 34]:  
\[\text{cos}\theta=r\text{cos}\theta_{0} \tag{1}\]  
where \(\theta\) is the apparent CA of droplets on the rough surface, and \(r\) is the surface roughness factor (\(r>1\)), which is the ratio of actual contact area of the solid surface with the liquid to the geometric area in contact with the droplet. In Equation (1), \(\theta_{0}\) is the equilibrium CA of the droplet on an ideal smooth surface of the same material, which is given by [PERSON]'s equation, \(\text{cos}\theta_{0}=(\gamma_{sv}-\gamma_{st})/\gamma_{lv}\), where \(\gamma\) is the interfacial tension; and the subscripts \(s\), \(v\), and \(l\) are attribute to the solid, vapor, and liquid phases, respectively.  
The Cassie model suggests an incomplete contact at the liquid/solid interface in which air bubbles are trapped in micro-pockets formed on a hydrophobic rough surface, leading to a composite interface that gives the apparent CA as follows [35; 36; 37]:  
\[\text{cos}\theta=f(\text{cos}\theta_{0}+1)-1 \tag{2}\]  
where \(f\) is the ratio of the solid area in contact with the liquid to the geometric area in contact with the droplet (\(f<1\)).  
According to the Wenzel model, the micro-pockets at the solid-liquid interface will be filled with the liquid. As a consequence, the droplets can hardly slide, or the sliding angle is large on the surfaces. Hence, the Wenzel model cannot fulfill the sliding requirement of self-cleaning surfaces; that is, the sliding angle is less than \(5^{\circ}\) (or the HA is less than \(20^{\circ}\)). Based on the discussion above, from the sliding requirement point of view, constructing a Cassie state is suitable for self-cleaning surfaces. In order to achieve a Cassie state, both a hierarchical micro-nanometer scale roughness and low-surface-energy material are necessary.  
When the Al\({}_{2}\)O\({}_{3}\) emulsion and FPA solution were mixed, FPA monolayers were self-assembled on the surfaces of Al\({}_{2}\)O\({}_{3}\) particles to form Al\({}_{2}\)O\({}_{3}\)@FPA microparticles (in the cases of nano-agglomerates) and nanoparticles. The FPA molecules were bonded with Al\({}_{2}\)O\({}_{3}\) via P-O-Al covalent bonds. Water/oil repellency of Al\({}_{2}\)O\({}_{3}\)@FPA particles come from the CF\({}_{3}\) terminating groups and the -CF\({}_{2}\)-CF\({}_{2}\)- chains of the FPA molecules [32].  
After being treated with PDA and Al\({}_{2}\)O\({}_{3}\)@FPA, the stone surfaces presented hierarchical structures, including micro- and nanoscale roughness (Figure 5). For the as-prepared self-cleaning stone surfaces, the Cassie state was constructed when they made contact with liquid droplets, and the interstices and holes on the stone surfaces acted as micro-pockets, which attract air at the interfaces of stone and liquid droplets. As a consequence, the actualsolid-liquid contact area is very small. In this work, the \(f\) can be estimated to be \(\sim\)2% according to Equation (2). The result agrees will with those reported in the literature [21; 22].