## 5 Bubble-Particle Interaction
### Bubble-Particle Attachment  
A successful particle-bubble attachment involves three key microprocess: the rupture of the liquid film, the formation of a three-phase wetting peripheral, and the formation of a stable wetting perimeter (Figure 10) [21; 142; 143]. The particle slides around the bubble's surface when it meets the bubble in the slurry. The liquid film between the bubbles and the particles subsequently thins to the critical thickness and breaks to form the three-phase contact (TPC) line. This contact line continues to expand until a stable wetting perimeter is formed [144; 141]. It is essential for all these sub-processes to occur in a brief contact time between a bubble and a solid surface, particularly under quite turbulent conditions [145]. The bubble-particle attachment time (t\({}_{\text{at}}\)) can be calculated by summing the times required to complete each of the three attachment processes, as described by the following relationship (Equation (10)) [146; 147]:  
\[\text{t}_{\text{at}}=\text{t}_{\text{i}}+\text{t}_{\text{r}}+\text{t}_{\text{ prec}} \tag{10}\]  
where t\({}_{\text{i}}\) (induction time) refers to the time required for the liquid film to thin to a critical film thickness, t\({}_{\text{r}}\) is the time required for the liquid film to rupture and form the three-phase contact nucleus, and t\({}_{\text{prec}}\) is the time for the TPC line to expand from the critical nucleus radius to establish a stable wetting perimeter [146; 148]. The bubble-particle attachment time is closely related to the flotation recovery, and a shorter attachment time normally corresponds to the stronger hydrophobicity and better floatability of the minerals. If the bubble-particle contact time exceeds the attachment time, attachment will occur [149]. Numerous studies have examined the effect of surface roughness on bubble-particle attachment, finding that it effectively affects the collision process, the rupture of the liquid film, and the formation of the TPC line. [PERSON] et al. [145] confirmed that the bubble visibly rebounded from the surface when its kinetic energy surpassed the viscous energy dissipation in the surrounding liquid film. Conversely, the bubble did not rebound when insufficient inertia failed to overcome the energy dissipation. They reported that the roughness significantly affected the collision process and the energy dissipation within the liquid film. They observed that the time required for the rupture of the wetting film and the formation of the TPC line could be dramatically shortened due to fewer bouncing collisions and quicker attachments of air bubbles to rough solid surfaces. Similar results were reported by [PERSON] et al. [150], [PERSON] et al. [151], and [PERSON] et al. [152; 153; 154], who found that the energy dissipation on rough particles occurred significantly faster than on smooth particles. Additionally, [PERSON] et al. [127; 128; 155; 156] observed that the bubbles attached to rough surfaces (roughness over 50 \(\upmu\)m) without any bouncing during their first collision, whereas those colliding with smooth surfaces (roughness below 1 \(\upmu\)m) experienced four to five \"approach-bounce\" cycles before attachment (Figure 11). [PERSON] et al. [157] observed that surface roughness significantly influences the attachment probability, as shown through CFD simulations. [PERSON] et al. [55] demonstrated that increasing the particle roughness could enhance the bubble-particle attachment. Specifically, rough glass bead particles exhibited a stronger adhesion to the bubble surface compared to smooth glass beads due to an increased contact area. [PERSON] [158; 159] reported similar results, that the attachment time between smooth sulfur surfaces and bubbles was longer than that between rough sulfur surfaces and bubbles. [PERSON] et al. [146]  
Figure 11: (**a**) Experimental setup for monitoring bubble collision process [127]; (**b**) snapshots of bubble collisions and three-phase contact spreading on Teflon plates with different roughness [155].