Numerical and topological description of contact areas at different size scales for the contact of rough solid surfaces.

This study introduces a novel heuristic phenomenological model for analyzing the evolution of contact areas on rough surface. Contrasting with traditional methods, it employs a cut-off threshold approach to track numerical and topological metrics across different deformation stages. The model quantifies contact area distributions, nested sub-regions, and self-affine parameters, revealing universal trends across scales spanning nanometers to kilometers. Metrics for synthetically generated isotropic surfaces with Hurst exponents = 2.5 and 3.5 correlate closely with those from AFM and SEM experimental datasets, respectively. In addition, the model has been tested on NASA's SRTM datasets. Cross-correlation demonstrate significant similarities in numerical and topological metrics across diverse measurement techniques, surface types, and scales, highlighting the method's robustness and calibration-free scale invariance. This approach bridges gaps in multiscale tribological analysis, offering deeper insights into frictional transitions and surface interactions. Beyond tribology and materials science, this general approach enables fundamental characterization of surface morphology as such, making it applicable to diverse fields including geomorphology, biomimetics, and nanotechnology.