## 2 Materials and Methods
### Surface Quality Measurements  
The surface quality assessment plays a fundamental role in evaluating the effectiveness of the ball burnishing process [34]. As a critical indicator of surface quality, surface roughness comprises two essential components: geometric roughness, primarily influenced by the tool geometry and processing parameters, and natural roughness, which results from inherent process variations and material characteristics [35]. These components collectively contribute to the processed material's surface topology and functional performance [36]. The quantitative surface quality evaluation was conducted through two primary roughness parameters: the arithmetic average roughness (\(R_{a}\)) and the mean roughness depth (\(R_{z}\)). The \(R_{a}\) value, which represents the arithmetic mean of the absolute ordinate values within the sampling length, is calculated using the following mathematical expression:  
\[R_{a}=\frac{1}{L}\int|y(x)|x, \tag{2}\]  
where \(L\) represents the sampling length, and \(y(x)\) denotes the profile height at position \(x\). Additionally, \(R_{z}\) is the arithmetic mean of the sum of the maximum peak height (\(R_{P}\) \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Factors**} & \multicolumn{5}{c}{**Levels**} \\ \cline{2-5}  & **Code** & **1** & **2** & **3** \\ \hline Environment & E & Dry & MQL & \\ \hline Force (N) & F & 50 & 100 & 200 \\ \hline Feed rate & \multirow{2}{*}{Fr} & \multirow{2}{*}{0.5} & \multirow{2}{*}{1} & \multirow{2}{*}{2} \\ (mm/min) & & & & \\ \hline Passes & P & 1 & 2 & 4 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Process parameters and their limits. and the maximum valley depth (\(R_{v}\)) within each sampling length, calculated over multiple sampling lengths. Mathematically, it is expressed as follows:  
\[R_{z}=\frac{\sum(R_{P}+R_{v})}{n}, \tag{3}\]  
\(R_{P}\) represents the maximum peak height within a sampling length, while \(R_{v}\) is the maximum valley depth within a sampling length. The parameter n refers to the number of sampling lengths used in the calculation (Figure 3) [36; 37]. Surface roughness measurements were performed using a calibrated Mitutoyo Surftest SJ-210 profilometer (Mitutoyo Corporation, Kawasaki, Japan) equipped with a diamond stylus having a 2 \(\mu\)m tip radius and 60\({}^{\circ}\) angle. The measurement protocol adhered to stringent standards, with the profilometer configured to operate with a detector force of 0.75 mN following JJS'01 guidelines. The instrument was set to utilize a cut-off length of 0.8 mm and an evaluation length of 4 mm while maintaining a consistent measurement speed of 0.75 mm/s with Gaussian filtering for optimal signal processing. All measurements were conducted under controlled environmental conditions at 22.5 \(\pm\) 1 \({}^{\circ}\)C to ensure measurement stability and repeatability. Seven independent measurements were conducted at carefully chosen locations on each specimen to achieve statistical reliability, with measurements performed perpendicular to the burnishing direction to capture true surface profile characteristics. The final surface roughness values were calculated as the arithmetic mean of these measurements, with standard deviation values recorded to assess measurement consistency. In addition, surface topography of the unburnished and burnished samples was examined using an optical stereo microscope (AOB-XJP6, Shanghai optical instrument factory, Shanghai, China) at 50\(\times\) magnification. The analysis highlights the impact of lubrication and process parameters on the surface morphology. This analysis enabled the quantification of microscale surface irregularities and the assessment of the burnishing process's effectiveness in reducing surface roughness. Furthermore, the microscope's high-resolution capabilities allowed for a detailed comparison of the morphological changes induced by the burnishing technique.