## 2 Measurement and Roughening Methods
### Measurement Methods
\[\mathrm{R_{a}}=\frac{1}{n}\underset{i=1}{\overset{n}{\underset{i=1}{\overset{n}{ \underset{i=1}{\overset{n}{\underset{i=1}{\overset{n}{\underset{i=1}{  
where n represents the number of measured points, and yi denotes the heights of these measured points. However, there is a significant associated error in the data due to the influence of the particle shape and size. The collected data reflect the spatial variations in the surface, which can have differing heights at various points. Generally, the finer the particle size, the greater the associated error. Therefore, measuring the surface roughness of samples with different shapes, such as spherical particles, should be conducted carefully to minimize potential errors. [PERSON] et al. (2016) proposed a methodology to correct the roughness measurements of spherical particles. The resulting measurement provides lateral surface data (XY coordinates) within the Z range of the cantilever oscillation.  
For flat materials, the quantitative analysis of roughness can be easily conducted using a surface roughness measurer or an AFM. However, for ground and irregular particles, surface roughness is often characterized indirectly through calculation equations. The BET method is commonly used to measure the surface roughness of particles ([PERSON] et al., 2016; [PERSON] et al., 2016; [PERSON] et al., 2016). In this method, gas molecules (e.g., N2) are adsorbed onto clean solid surfaces, forming adsorbed layers. The true surface area can be derived from the BET isotherm (Figure 1b). Surface roughness is defined as the ratio of the real surface area of a particle to the surface area of an imaginary sphere with the same diameter. This can be calculated using Equation (5) ([PERSON] et al., 2016; [PERSON] et al., 2016; [PERSON] et al., 2016):  
\[\lambda=\frac{\rho\text{DA}_{\text{BET}}}{6} \tag{5}\]  
where \(\rho\) is the grain density of the mineral (g/cm\({}^{3}\)), D is the average grain diameter of the particle (um), A\({}_{\text{BET}}\) is the specific surface area (m\({}^{2}\)/g), and \(\lambda\) is the calculated particle surface roughness (dimensionless). The BET equation is relatively straightforward to apply to a range of minerals with varying structural properties and provides reasonably consistent values for roughness. However, the surface roughness derived from this method is closely related to the sizes and shapes of the particles. Measuring surface roughness is important for understanding its influence on flotation behavior, wettability, and bubble-particle interactions. Nonetheless, these measurement methods come with their own advantages and disadvantages, along with potential errors. These discrepancies can arise from differences in the particle size and shape, measurement philosophy, and measurement accuracy (such as surface roughness measurers which typically operate at the micron level, while AFMs function at the nano level). These existing errors may result in significant differences or even contradictory conclusions regarding surface roughness when measured using different methods. To mitigate the influence of particle size on these measurements, it is advisable to divide particles into narrower size ranges. Furthermore, improving the accuracy of surface roughness measurers or combining multiple measurement techniques is essential to enhance the reliability of surface roughness assessments.