Quantifying the Complexity of Rough Surfaces Using Multiscale Entropy: The Critical Role of Binning in Controlling Amplitude Effects

A salient feature of modern material surfaces used in cutting-edge technologies is their structural and spatial complexity, which endows them with novel properties and multifunctionality. The quantitative characterization of material complexity is a challenge that must be addressed to optimize their production and performance. While numerous metrics exist to quantify the complexity of spatial structures in various scientific domains, methods specifically tailored for characterizing the spatial complexity of material surface morphologies at the micro- and nanoscale are relatively scarce. In this paper, we utilize the concept of multiscale entropy to quantify the complexity of surface morphologies of rough surfaces across different scales and investigate the effects of amplitude fluctuations (i.e., surface height distribution) in both stepwise and smooth self-affine rough surfaces. The crucial role of the binning scheme in regulating amplitude effects on entropy and complexity measurements is highlighted and explained. Furthermore, by selecting an appropriate binning strategy, we analyze the impact of 2D imaging on the complexity of a rough surface and demonstrate that imaging can artificially introduce peaks in the relationship between complexity and surface amplitude. The results demonstrate that entropy-based spatial complexity effectively captures the scale-dependent heterogeneity of stepwise rough surfaces, providing valuable insights into their structural properties.