## 2 Measurement and Roughening Methods  
### Measurement Methods  
The size of the surface roughness is typically within the micro or even nanometer scale, necessitating the use of precise instruments to minimize measurement errors. Various methods and equipment have been employed to measure the roughness of particle surfaces, utilizing different techniques. These methods include the fitting ellipse method, optical profilometer, surface roughness measurer, Brunauer-Emmett-Teller (BET) method, and atomic force microscope (AFM) (Figure 1). And the measurement accuracy of various measurement methods of the surface roughness is shown in Table 1.  
Figure 1: The measurement methods of the surface roughness of particles: (**a**) the surface roughness measurer [24]; (**b**) the BET method; (**c**) the optical profilometer [25]; (**d**) the fitting ellipse method; and (**e**) the AFM [26]. The fitting ellipse method (Figure 1d) is a two-dimensional technique used to characterize the surface roughness of particles. This method involves collecting the contours of representative particles via optical microscopy. The fitted major axis (L) and minor axis (W) of the particles are then determined using analysis tools such as the Image J pro plus 6.0 software. The roughness is quantified by the ratio of the actual perimeter of the particles (P) to the perimeter of the fitted ellipse (Pe) (calculated by Equation (1)), which is shown in Equation (2) [27]. \[\mathrm{P_{e}}=\frac{\pi}{2}\Bigg{[}\frac{3}{2}(\mathrm{L+W})-\sqrt{\mathrm{LW }}\Bigg{]} \tag{1}\]  
\[\mathrm{Roughness}=\frac{\mathrm{P}}{\mathrm{P_{e}}} \tag{2}\]  
An optical profilometer (Figure 1c) is a device that scans the surface of particles with a microscope, records a series of scanning signals, and converts them into digital images [28; 29; 30; 31]. The roughness evaluation is conducted based on the threshold of these images after capturing the particles through the microscope at various magnifications. The analysis relies on the differences in the profile heights across each selected area on the particles. The surface roughness measurer (Figure 1(a1)) utilizes mechanical sensing to evaluate the surface topography. This process involves the electronic amplification of the acquired signal, alongside measurements of the area (A), perimeter (P), major axis (Length, L), and minor axis (Width, W) [32; 33]. Mechanical sensing involves moving a probe across the mineral surface, which is then converted into an electronically amplified signal. The roughness measurements of mineral surfaces are recorded along the work line, from which the final arithmetic average (Ra) value of surface roughness is derived. This method offers a highly accurate means of rapidly analyzing the surface topography without causing damage, with the main advantage being its direct mechanical nature [24; 34]. The AFM [35; 36; 37; 38; 39; 40; 41; 42] was initially developed to image the surface topography at a nanometer resolution (Figure 1e). A sharp probe, mounted near the end of a cantilever, is raster-scanned across the sample surface. The height of the sample is plotted against the lateral position, resulting in a three-dimensional topographic image created by moving the sample up and down to maintain a constant force between the tip and the sample. A mean line is calculated by the AFM using all measured points along the cantilever path. Surface roughness is determined by the difference between the probing path and the average line [43], and it is usually reported as the arithmetic average roughness (Ra) value and root mean square roughness (Rq or RMS) value to increase the accuracy, which are calculated based on Equations (3) and (4) [43].