Describing complex, non-recurring topographies is challenging and classical methods like Fast Fourier Transform and root mean square roughness (RMS) fall short in capturing their intricacies. However, the specific topography greatly influences physical, chemical, and biological properties. An accurate characterization and precise description of topographies are therefore essential. Numerous different nanostructures with specific topographies have been designed for various material properties, yet the lack of an intuitive, interdisciplinary accepted and comprehensive description of roughness persists. The team of scientists from experimental (bio)physics, theoretical physics and mathematics aims to fill this gap and has set itself the goal of developing the most suitable analysis and characterization methods for a range of very different samples used in nanotechnology. Atomic force microscopy is the main experimental method for recording topographies on the nanoscale. The mathematical tools include advanced shape descriptors, based on Minkowski functionals, and newly developed spatial models for (nano)rough surfaces and aggregation models using methods from fractal and stochastic geometry.

Surfaces with roughness on the nanoscale offer promising prospects for applications such as reducing or preventing pathogens from adhering to the surface of implants.
The proposed project will develop a rigorous morphometric analysis to unravel the complex interplay of the random geometries and physical properties, in particular bacterial adhesion.

In a direct collaboration of theory and experiment, we develop a precise description of surface roughness that captures global features and correlations while being suitable for daily laboratory use. This includes improved shape descriptors (based on Minkowski functionals) and measurement protocols, as well as the development of process-driven physical models of nanostructured surfaces (rigorously expressed as random fields), and statistical hypothesis tests.

The mathematical analysis is motivated by the experimental challenges of nanostructured surfaces and applied to AFM measurements to provide new standard techniques for characterizing nanostructured surfaces in experiments.