Decoding chirality in circuit topology of a self entangled chain through braiding

Circuit topology employs fundamental units of entanglement, known as soft contacts, for constructing knots from the bottom up, utilising circuit topology relations, namely parallel, series, cross, and concerted relations. In this article, we further develop this approach to facilitate the analysis of chirality, which is a significant quantity in polymer chemistry. To achieve this, we translate the circuit topology approach to knot engineering into a braid-theoretic framework. This enables us to calculate the Jones polynomial for all possible binary combinations of contacts in cross or concerted relations and to show that, for series and parallel relations, the polynomial factorises. Our results demonstrate that the Jones polynomial provides a powerful tool for analysing the chirality of molecular knots constructed using circuit topology. The framework presented here can be used to design and engineer a wide range of entangled chain with desired chiral properties, with potential applications in fields such as materials science and nanotechnology.Comment: 8 pages, 5 figures, 3 table

Aggregation and structural phase transitions of semiflexible polymer bundles: a braided circuit topology approach

We present a braided circuit topology framework for investigating topology and structural phase transitions in aggregates of semiflexible polymers. In the conventional approach to circuit topology, which specifically applies to single isolated folded linear chains, the number and arrangement of contacts within the circuitry of a folded chain give rise to increasingly complex fold topologies. Another avenue for achieving complexity is through the interaction and entanglement of two or more folded linear chains. The braided circuit topology approach describes the topology of such multiple-chain systems and offers topological measures such as writhe, complexity, braid length, and isotopy class. This extension of circuit topology to multichains reveals the interplay between collapse, aggregation, and entanglement. We show that circuit topological motif fractions are ideally suited order parameters to characterise structural phase transitions in entangled systems.Comment: 8 pages, 8 figures + Supplemental Materia

Macrotribological Studies of Poly( L -lysine)- graft -Poly(ethylene glycol) in Aqueous Glycerol Mixtures

We have investigated the tribological properties of surfaces with adsorbed poly(L-lysine)-graft-poly(ethylene glycol) (PLL-g-PEG) sliding in aqueous glycerol solutions under different lubrication regimes. Glycerol is a polar, biocompatible liquid with a significantly higher viscosity than that of water. Macrotribological performance was investigated by means of pin-on-disk and mini-traction-machine measurements in glycerol-PLL-g-PEG-aqueous buffer mixtures of varying compositions. Adsorption studies of PLL-g-PEG from these mixtures were conducted with the quartz-crystal-microbalance technique. The enhanced viscosity of the glycerol-containing lubricant reduces the coefficient of friction due to increased hydrodynamic forces, leading to a more effective separation of the sliding partners, while the presence of hydrated polymer brushes at the interface leads to an entropically driven repulsion, which also helps mitigate direct asperity-asperity contact between the solid surfaces under boundary-lubrication conditions. The combination of polymer layers on surfaces with aqueous phases of enhanced viscosity thus enables the friction to be reduced by several orders of magnitude, compared to the behavior of pure water, over a large range of sliding speeds. The individual contributions of the polymer and the aqueous glycerol solutions in reducing the friction have been studied across different lubrication regime