## 4 Wettability  
### Wettability Model for Rough Surface  
Wettability is commonly used to assess floatability in mineral processing, which is normally characterized by the contact angle. Generally, a more hydrophobic surface has a larger contact angle, which results in the improved floatability of the particles. The contact angle of a smooth surface normally can be calculated using [PERSON]'s Equation [104]. However, since mineral surfaces are usually rough and heterogeneous, the contact angle predicted by [PERSON]'s equation often differs from the actual contact angle observed on these surfaces [104]. To better understand and explain the wettability behavior of real, rough, and heterogeneous surfaces, the theories proposed by [PERSON] and [PERSON] have been widely adopted. [PERSON] proposed that the surface roughness increases the liquid-solid contact area. He assumed that the microgrooves and pores on the surface are completely filled with water. In this scenario, the actual surface area of a rough surface is r times greater than that of an ideal smooth surface. This results in the energy required to form the solid-liquid interface, being r(\(\gamma_{\rm SG}-\gamma_{\rm SL}\)), while the energy for forming the liquid-gas interface remains the same as that for an ideal surface (\(\gamma_{\rm SG}\)). [PERSON]'s model was put forward in 1936 [105], which considered that the apparent contact angle (\(\theta_{\rm m}\)) of the rough surface and its corresponding ideal contact angle (\(\theta_{\rm r}\)) of the perfectly smooth surface existed in the relationship of Equation (6) [105]. \[\cos\theta_{\rm m}=\frac{\rm r(\gamma_{\rm SG}-\gamma_{\rm SL})}{\gamma_{\rm LG }}=\rm r\cdot\cos\theta_{\rm r} \tag{6}\]where r is the ratio of a rough surface to an ideal smooth surface, \(\gamma_{\rm SG}\) is the wettability between the solid and gas, \(\gamma_{\rm SL}\) is the wettability between the solid and liquid, and \(\gamma_{\rm LG}\) is the wettability between the liquid and gas. According to the Wenzel model, a rougher surface results in a larger r value. For hydrophilic surfaces (\(\theta_{\rm r}<90^{\circ}\)), increasing the surface roughness leads to a decrease in the corresponding \(\theta_{\rm m}\), which indicates stronger hydrophilicity. Conversely, for hydrophobic surfaces (\(\theta_{\rm r}>90^{\circ}\)), an increased roughness results in a larger \(\theta_{\rm m}\), indicating stronger hydrophobicity. The Wenzel model also assumed that the surface features of the substrate are insignificant compared to the dimensions of the droplet and that the geometry of the droplet does not impact the surface area. If this assumption is disregarded, applying the Wenzel equation may result in incorrect conclusions and errors in measuring contact angles [104; 106]. [PERSON] and [PERSON] advanced the Wenzel model from a rough surface to a heterogeneous surface. They proposed that the liquid cannot fill the \"valleys\" on a rough hydrophobic surface, allowing air to be trapped and creating a composite surface. On a heterogeneous surface, the energy required to form the solid-liquid interface is expressed as \(r_{1}(\gamma_{\rm S,G}-\gamma_{\rm S,L})+r_{2}(\gamma_{\rm S,G}-\gamma_{\rm S,L})\), while the energy for forming the liquid-gas interface remains unchanged. Consequently, the contact angle could be calculated using the Cassie equation [107], which is shown in Equation (7). When one of the two phases is air, the Cassie equation can be simplified to the Cassie-Baxter model, which is shown in Equation (8) [104].