Molecular Beam Epitaxy

Electromigration

Oscillatory driving

Dewetting of solid films

Knots and entanglements

Wetting on nano-patterns

Membrane adhesion on nano-patterns

Macroscopic fluctuations

Knots and entanglements are common features observed on filaments or fibrous materials. The difficulty of writing down an entanglement constraint comes from its global character. This subject is the source of a large literature in mathematics and in physics.

We have analyzed he properties of stiff knots, i.e. knotted strings whose shape is dictated by the bending curvature energy. Stiff knots, such as loose knots with nylon strings, are ordinary objects in everyday life. An upsurge of interest in stiff knots recently came  from studies which pertain to  biology and nano-technologies, such as knots with actin filaments, nanotubes, nanotube fibers, and silica wires. These studies point out the wide relevance of stiff knots for the experimental determination of the bending rigidity, knot induced polymer and filament break-up, or  nano-manipulation.

Furthermore, knots may also be seen as elementary entanglements which capture some basic mechanical and geometrical properties of complex entangled structures. Stiff knots thus provide insights for the curvature energy dominated behavior of tightly entangled semi-flexible polymers and other fibrous materials.

Using some relevant topolgical knot invariant, we have obtained rigourous inequalitites for the knot energy. In the limit of thin string widths, we have identified a general phenomenon called braid localization. These results were confirmed with the help of Monte Carlo Simulations.