Force-Extension Curve of an Entangled Polymer Chain: A Superspace Approach

The statistical mechanics of an ideal polymer chain entangled with static topological constraints is studied using a superspace approach, in which the probability distribution of the polymer is obtained as solutions of the Fokker-Planck equation in a superspace with an inner structure characterized by the n-generator free group. The theory predicts that the force-extension curve of the polymer under the topological constraints has the generic form F=kl+Z/l, where l is an effective extension. Aside from the elastic term that is linear in l, the force-extension curve contains a universal term of the form Z/l. The magnitude of this topological term is determined by the topological charge number Z, which characterizes the topological nature of the static constraints. The theoretical results are further verified by a scaling analysis based on a blob model of the chain conformations.