Speaker
Description
Chaos is traditionally defined as the sensitive dependence on initial conditions in the time evolution of nonlinear dynamical systems—phenomenon that unfold in the time domain. Complexity extends this perspective to spatiotemporal systems with many degrees of freedom, where structures emerge that balance order and randomness in unexpected yet functional combinations. But what happens when we shift our attention solely to the spatial domain? How can the core concepts of chaotic dynamics and complex systems be adapted to characterize and model spatial structures that lie between full order and complete randomness, assuming their form is frozen in time? And what is - if any - the fundamental connection between temporal and spatial complexity?
In this talk, we review recent work on developing theoretical frameworks and computational tools—rooted in chaos theory and complex systems—for quantifying the spatial complexity of nanostructured and nanotextured surfaces. Our overarching goal is to establish an effective language for describing the complexity of nanoworld geometry and to deepen our understanding of how nanoscale morphology underpins functional behavior. Specifically, we employ concepts such as chaotic mixing, multiscale Shannon entropy, and hierarchical organization, adapted for application to characterizing nanosurface morphologies and their microscopy images.
Our first approach treats the 2D nanosurface morphology as the phase space of a strongly chaotic map (e.g., the Arnold map), with pixels serving as initial conditions. Iterative application of the map reshuffles pixel positions through the stretching and folding mechanism of chaos, progressively degrading the recognizable morphology and texture. The rate of this degradation provides a quantitative measure of spatial complexity exhibiting maximum values at morphologies between full order and randomness/noise [1].
The second approach quantifies the information content of nanosurfaces using Shannon entropy. Given the presence of spatial correlations, we compute multiscale Shannon entropy at each pixel through local, scale dependent averaging. The mean multiscale entropy can be used as a measure of surface complexity, reflecting the distance of the morphology from full homogenization [2,3].
Finally, we examine hierarchical surfaces, widely used in nanotechnology for their excellent multifunctional performance. We present a theoretical framework for defining and classifying hierarchical morphologies, along with a suite of modeling and quantitative characterization tools [4].
The above methods are compared and critically discussed in the context of their ultimate goal: robust and meaningful nanoscale characterization. They are also applied to both synthesized and experimental surfaces—produced through plasma etching, laser texturing, deposition, and other nanotexturing techniques—to ensure validation and insight derived from real morphological data.
Keywords: nanotechnology, nanostructures, chaos, Shannon entropy, hierarchical surfaces,
rough surfaces
Acknowledgement: This work is supported by the project “plasmAI” of the program “AI-Aware Pathways to Sustainable Semiconductor Process and Manufacturing Technologies”, Intel Corporation & Merck KGaA.
References
[1] A Kondi, V Constantoudis, P Sarkiris, K Ellinas, E Gogolides Physical Review E 107 (1), 014206 (2023)
[2] A Arapis, V Constantoudis, D Kontziampasis, A Milionis, CWE Lam, et al., Materials Today: Proceedings 54, 63-72 (2022)
[3] A Kondi, V Constantoudis, P Sarkiris, E Gogolides Mathematics 13, 2325 (2025)
[4] G Papavieros, V Constantoudis, N Vouroutzis and E Gogolides, Nanotechnology 34 (40), 405702
(2023)