## 3 Flotation Behavior
(2009, 2010) discovered that nano-sized asperities improved the flotation kinetics rate from 0.9 min\({}^{-1}\) to 2.3 min\({}^{-1}\) as the roughness increased from 1 to 8 nm to 28-170 nm (Figure 5a,b). [PERSON] et al. (2009, 2010) reported that quartz particles with higher surface roughness and elongation ratios exhibited faster flotation kinetics constants. Furthermore, they found that the effect of surface roughness on flotation kinetics was more significant than that of shape parameters. They concluded that the empirical relationship between the flotation rate constant (k) and surface roughness (\(\lambda\)) could be expressed as k = a + b\(\lambda\). [PERSON] et al. (2009) confirmed their findings and established an empirical relationship between the surface roughness and flotation kinetics constant of quartz particles, as k = 3.9312 + 0.0856A. [PERSON] et al. (2009) reported similar results for low-ash cooking by determining the relationship as k = 32.02 \(\lambda\) - 7.07. However, [PERSON] et al. (2009) showed a slightly different perspective, that the relationship between the surface rough ness in calcite particles and the flotation rate (k) could be quantitatively expressed as k = 0.0039\(\lambda^{2}\) - 0.0589\(\lambda\) + 2.6097 for the ball mill and k = 0.1251\(\lambda\) + 0.8184 for the rod mill. Figure 4: Flotation separation of hematite from quartz ground under different (**a**) NaOL concentrations and (**b**) ratio of hematite to quartz (RHQ: rod-milled hematite (Ra = 2.1 nm) particles mixed with quartz; BHQ: ball-milled hematite (Ra = 8.5 nm) particles mixed with quartz [39] and flotation separation of chalcopyrite from pyrite ground by (**c**) ceramic ball (Rq = 9.02 nm) and (**d**) cast iron ball (Rq = 74.80 nm) under different sodium butyl xanthate concentrations [73]. Figure 5: Flotation kinetics of smooth (Ra = 1–8 nm), abraded (Ra = 6–18 nm), and etched (Ra = 28–170 nm) glass particles methylated in (**a**) 10\({}^{-5}\) M (\(\theta\) = \(\sim\)39\({}^{\circ}\)) and (**b**) 10\({}^{-5}\) M (\(\theta\) = \(\sim\)73\({}^{\circ}\)) trimethylchlorosilane solutions [57]. Numerous studies have been conducted to investigate the effect of surface roughness on the flotation behavior of various minerals or materials, but some conclusions are contradictory. Some scholars concluded that an increased surface roughness improves the mineral flotation behavior, whereas others believe it deteriorates the mineral flotation behavior. The contradictory conclusions can be explained by several reasons. Firstly, some scholars do not focus on narrow particle size ranges as their subjects [33; 62], ignoring the effect of the particle size on flotation behavior. Secondly, some studies ignore the influence of the particle shape on flotation behavior, even for particles with the same size fraction. These reasons lead to non-negligible errors in understanding the relationship between surface roughness and flotation behavior, resulting in controversial results. Furthermore, certain scholars combine surface roughness and particle shapes to investigate the role of surface roughness on the flotation behavior of minerals. To accurately study how surface roughness influences mineral flotation behavior, it is essential to select particles that have similar particle shapes and divide them into narrow sizes ranges. Additionally, some scholars have explored the effect of surface roughness through methods such as the abrasion, polishing, or etching methods without changing the particle shapes and sizes. These methods eliminate the potential influence of variations in particle sizes and shapes on the relationship between surface roughness and flotation behavior, resulting in more accurate and reliable findings. However, research using these methods has primarily focused on a limited number of materials, mainly glass beads and quartz particles. There is a need for further study into how surface roughness affects the flotation behavior across various minerals and materials.