## 5 Bubble-Particle Interaction
### Bubble-Particle Interaction Energy  
In addition to directly investigating how surface roughness impacts the attachment time and adhesion force between air bubbles or water droplets and particle surfaces, many researchers have also simulated these effects using various theoretical models. [PERSON] (DLVO)'s work and the extended DLVO theory have been widely used in the study of bubble-particle interactions, which posits that most electrostatic double-layer forces and van der Waals forces in the adhesion process of bubbles on particles are repulsive forces. The particle must overcome the energy barrier formed by the electrostatic double-layer forces and van der Waals forces for the successful adhesion of bubbles to particle surfaces [169]. According to the extended DLVO theory [170], the barrier energy between a particle and a bubble is calculated by Equation (11) [171; 172]. \[\mathrm{E_{e.b.}}=\mathrm{E_{e.}}+\mathrm{E_{d.}}+\mathrm{E_{h.}} \tag{11}\]  
where \(\mathrm{E_{e.b. }}\) is the energy barrier; \(\mathrm{E_{e.}}\), \(\mathrm{E_{d.}}\), and \(\mathrm{E_{h.}}\) are the electrostatic, Van der Waals, and hydrophobic energy during the particle-bubble interaction. A lower energy barrier suggests a lower energy required for the attachment of the particle to the gas bubble, improving the flotation recovery and flotation rate constant [37; 40]. Some researchers have quantified and analyzed how the surface roughness affects the interaction energy between particles and bubbles. Their results indicate that an increased surface roughness significantly lowers the energy barrier, making it easier for bubbles to adhere to the surfaces of the particles. [PERSON] and [PERSON] [173] found that both the size of asperities and their distribution on surfaces significantly affect the interaction energy. Specifically, they observed that the interaction energy decreased with an increase in the average surface roughness on simulated spherical particles of varying sizes (Figure 15a,b). [PERSON] et al. [174] concluded that surface roughness decreased the energy barrier in a quadratic manner. They determined that the influence of heterogeneities on the interaction energy barrier follows this order: surface contact angle heterogeneity < surface roughness < surface charge heterogeneity. They indicated that relatively low surface roughness fractions (\(\mathrm{f_{SRs}}\)) and heights (\(\mathrm{h_{SRs}}\)) of just a few hundred nanometers can significantly lower the energy barrier, even in the presence of strong electrostatic repulsion. For a given particle diameter and \(\mathrm{h_{SR}}\), the energy barrier value decreased from a maximum at a \(\mathrm{f_{SR}}\) = 1 to a minimum as the \(\mathrm{f_{SR}}\) decreased to low values (Figure 15c,d). [PERSON] et al. [42; 57] were the first to suggest that the size of nano-sized hydrophobic asperities on spherical microscopic particles influences the height of the energetic barrier. They discovered that spherical glass particles with a nanoscale roughness could overcome lower energy barriers to adhere to bubbles more effectively than those with smooth surfaces, as explained by the extended DLVO theoretical model. Hydrophobic asperities with a radius as small as 10-70 nm could reduce the energy barrier against the particle attachment to a gas bubble by one to two orders of magnitude, accelerating the attachment of rough particles to gas bubbles in flotation (Figure 16b) [57]. They also observed that increasing the roughness of coal particle surfaces from 0 to 23 nm and 57 nm, decreased the energy barrier significantly from 2.79 \(\times\) 10\({}^{-16}\) to 6.51 \(\times\) 10\({}^{-17}\) and 1.06 \(\times\) 10\({}^{-17}\) J/m\({}^{2}\). This led to the conclusion that a greater surface roughness correlates with enhanced flotation kinetics and a lower energy barrier (Figure 16f) [42]. [PERSON] et al.