# Application of Areal Topography Parameters in Surface Characterization
## 3 Assessment of the Quality of Measurement Results Obtained via an Atomic Force Microscope
### Influence of Surface Profile Geometry on the Parameters in AFM Measurements: A Monte Carlo Approach  
The motivation for this analysis stems from the results of an interlaboratory comparison, which revealed significant discrepancies in the measured values of the \(Sa\) parameter between laboratories. The primary suspected cause was the uncalibrated or improperly calibrated lateral component of the measurement system. This raised the question of which surface parameters are particularly sensitive to such lateral deviations and which remain stable regardless of them. To address this issue, five areal surface texture parameters defined in ISO 25178-2, \(Sa\), \(Sz\), \(Sq\), \(Sdq\), and \(Sdr\) were selected for detailed analysis. These parameters collectively provide a representative and functionally relevant description of surface topography: \(Sa\), \(Sz\), and \(Sq\) represent amplitude characteristics, while \(Sdq\) and \(Sdr\) reflect slope and surface complexity, respectively. \(Sa\) (arithmetical mean height) and \(Sz\) (maximum height) are the most commonly used parameters in industrial practice and technical standards, and are often the only parameters specified in technical drawings. However, the inconsistency in \(Sa\) across laboratories, unlike the relatively stable \(Sz\), suggested that some parameters may be more susceptible to errors in lateral calibration. To better understand this sensitivity and to support more reliable surface characterization, the analysis was expanded to include \(Sq\) (root mean square height), \(Sdq\) (root mean square gradient), and \(Sdr\) (developed interfacial area ratio). The parameter \(Sdq\) was selected because it captures the average surface slope, which is critical for understanding functional properties such as friction and adhesion. Its sensitivity to micro-slopes, rotation, and feature density makes it particularly useful for analyzing surface functionality. Finally, \(Sdr\) was included as a measure of the increase in actual surface area compared to its projected area, effectively quantifying surface complexity. This is especially relevant in applications dependent on contact area, such as hydrophobicity, capillary effects, and surface engineering. All parameters were computed from simulated surface data based on a predefined topographical profile (e.g., rectangular). Surface points (\(x_{i}\), \(y_{i}\)) were randomly sampled over a defined area, and height values \(z_{i}\) were assigned according to the surface geometry. \begin{table}
\begin{tabular}{c c c c c} \hline  & **Lab A** & **Lab B** & **Lab C** & **Lab D** \\ \hline \(\overline{h}/\)nm & 98.9 & 101.3 & 100.2 & 100.8 \\ \(u(h)/\)nm & 1.45 & 2.55 & 1.75 & 2.80 \\ \(En(h)\) & 0.40 & 0.70 & 0.69 & 0.55 \\ \hline \(Sa/\)nm & 15.60 & 29.68 & 25.45 & 16.76 \\ \(En(Sa)\) & 3.32 & 4.14 & 1.89 & 2.71 \\ \hline \(Sz/\)nm & 116.9 & 122.8 & 117.8 & 122.1 \\ \(En(Sz)\) & 0.98 & 0.95 & 0.69 & 0.73 \\ \hline \end{tabular}
\end{table} 
Table 3: Results of interlaboratory comparative measurements for step height \(h\). In the simulations, parameters were evaluated under various conditions, including added lateral noise in \(x\) and \(y\), variability in step width and step pitch, and the inclusion of surface slope or rotational misalignment Table 4.  
The results of simulated surface parameters are presented in Table 5.  
The simulations clearly showed the following:  
The simulations demonstrate that parameters \(Sa\) and \(Sq\) are sensitive to the distribution and density of structures (e.g., step pitch and step width). Even small geometric variations can lead to several nanometers of difference in computed values, despite the total step height remaining constant. Parameter \(Sz\) remains consistent across all simulations, as it is solely based on the extreme values (\(z_{\max}-z_{\min}\)). For this reason, it is stable even under variations in slope, rotation, or noise, making it useful for detecting depth/deformation, though in-sufficient for describing overall texture.