## 3 Flotation Behavior
These studies typically utilize etching or abrasion/polishing methods, ensuring that the  
Figure 3: Effect of surface roughness on flotation recovery of different minerals: (**(a)** stainless steel disk-milled (Ra = 0.28 nm) and ceramic ball-milled (Ra = 1.59 nm) magnesite [3]; (**b**) stainless steel disk-milled (Ra = 0.52 nm) and ceramic ball-milled (Ra = 1.16 nm) quartz [41]; (c) smooth (stainless steel) and rough (ceramic ball) cassiterite [68]; and (**d**) sphalerite particles [31]). particle sizes and shapes remain constant. The findings consistently indicate that the surface roughness effectively enhances flotation recovery, although the focus of current research has primarily been on relatively small particles (mainly glass beads and quartz). [PERSON] et al. (1991) and [PERSON] et al. (2002) contributed the improved flotation recovery of ground glass spheres to an increased surface roughness. Similarly, [PERSON] et al. (2005) and [PERSON] et al. (2006) reported a maximum difference of approximately 12% in the flotation recovery between rougher and smoother glass beads (90-150 \(\upmu\)m) as the surface roughness increased. Furthermore, [PERSON] et al. (2005) observed that the shape factor had a more predominant influence on flotation behavior than surface roughness. [PERSON] et al. (2005) concluded that angular and rougher glass beads exhibited a better flotation performance compared to spherical and smoother ones. [PERSON] et al. (2009, 2009, 2010) observed similar results, that the surface nano-roughness of methylated glass particles enhanced flotation. Specifically, the flotation recovery was 23.6% for the spherical particles, whereas it increased to 31.3% for ground particles and 39.4% for abraded particles at a collector concentration of 1 \(\times\) 10\({}^{-6}\) M and a natural pH of 7.3 ([PERSON] et al., 2005). They also found that the flotation recovery of quartz particles increased from 34.1% to 38.1% as the surface roughness value increased from 3.12 \(\upmu\)m to 4.02 \(\upmu\)m with the addition of collectors ([PERSON] et al., 2005). The effects of surface roughness on flotation recovery have been extensively studied using methods such as grinding, acid etching, and abrasion/polishing. However, most research has primarily concentrated on the impact of surface roughness on the floatability of individual minerals, with limited studies addressing its effect on the flotation separation of different minerals. [PERSON] et al. (2009) found that the surface roughness enhanced the flotation separation of hematite from quartz, achieving a higher separation index (Figure 4a,b). [PERSON] et al. (2009) observed that rough particles tended to float preferentially from a 1:1 mixture of smooth and rough particles, while the tailings mainly consisted of smooth-surfaces particles, with only 17-22% being rough. [PERSON] et al. (2009) reported that rougher chalcopyrite particles floated preferentially from a 1:1 mixture of chalcopyrite and pyrite, with the selectivity index values increasing from 1.58 to 2.65 as the root mean square roughness (Rq) value decreased from 74.80 nm to 9.02 nm (Figure 4c,d). In addition to flotation recovery rate, the flotation rate is another important measurement index in the flotation process. It is typically calculated by the ratio of the cumulative flotation recovery to the flotation time ([PERSON] et al., 2009). For the same final flotation recovery rate, a higher flotation rate indicates that a shorter time is required for flotation, which is more conducive to flotation. Research has been conducted on the effects of different surface roughness levels on flotation kinetics, and the results demonstrate that an increased surface roughness effectively increases the flotation rate. [PERSON] et al. (2009, 2010) showed that larger flotation rate constants were obtained for magnesite particles with greater nano-asperities, which increased from 1.09 to 1.43 as the surface roughness increased from 8.3 to 8.7 (calculated by BET methods). Similarly, [PERSON] et al.