Jan Smrek (Vienna): Topological constraints of polymers in- and out-of-equilibrium

I will give an overview of how we use simulations, polymer theory together with computational geometry and topology to study the impact of topological constraints on the properties of polymer materials and biological systems.
Topological constraints arise from the fact that polymer chains cannot cross themselves, or each other, on relevant time scales. These inherently nonlocal constraints restrict the conformational space and relaxation modes of dense polymer solutions and thereby define mechanical properties of these complex liquids in equilibrium. For example, ring polymers adopt self-similar structures akin to a peculiar class of space-filling fractal curves of Hilbert, and to those of chromosomes of higher eukaryotes. The structural scale invariance is imprinted in their dynamics and scale-free response functions. Out of equilibrium, topological constraints can be used to tune the material response. Active-passive copolymer rings can generate 'active topological glass', a disordered solid state of polymer matter distinct from known polymer or active glasses. In contrast, topological constraints in such linear copolymers can rectify polymer dynamics, generate directional motion and speed up their relaxation. Similarly, tunable properties can arise from topological constraints related to torsional degrees of freedom controlling the supercoiling of DNA-based materials.
The discussed examples show that geometry and topology lie at the heart of polymer matter physics, with implications for our understanding of biological systems and the design of future active materials.