Force-Extension Curve of a Polymer Chain Entangled with a Static Ring-Shaped Obstacle

The way to theoretically approach dynamic and static topological constraints of polymer entanglements still presents a great challenge in polymer physics. So far, only the problem of static entanglement with multiple simple objects has been solved in theory by a superspace approach in our previous work. This work is devoted to extending the superspace approach to study a polymer chain entangled with a relatively complicated object-a ring-shaped object with genus one. Taking advantage of the axial symmetry of the model setup, the 3D diffusion equations in the superspace can be numerically solved within the 2D coordinates using a specially designed alternating-direction implicit (ADI) scheme. A series of numerical calculations reveal that the topological entanglement effect of the ring will exert a topological entropy attractive force on the linear chain, which can be used to explain the viscosity-increase phenomenon observed in recent simulations and experiments. Furthermore, the influences of the ring size and the entangling modes on the topological entropy force are also investigated by examining the corresponding force-extension curves. This work, together with our previous work, might pave the path toward the complete formulation of static topological constraints.