Free energy of entanglement-condensed systems

Computer simulations of entanglement-condensed systems are performed to study the interaction potential among local-knots (LKs), which are proposed by Iwata and Edwards as basic units of entanglement. By performing hypothetical element-exchange reactions between the system and an external element-bath, chemical potential Deltamu(chi) of chain-elements, measured from its topological equilibrium value, is computed numerically as a function of condensation ratio chi of LKs. Deltamu(chi) is transformed into free energy DeltaF(chi) of the system, which takes a minimum in the topological equilibrium state (chi = 1) and increases rapidly with increasing chi. It is argued that DeltaF(chi) comes mainly from the topological repulsive potentials among LKs, because DeltaF(chi) computed by the simulation is much larger than that predicted by the slip-link model in which the repulsive potential is neglected. To see farther evidences for the repulsive potential among LKs, average length (L) over bar (a)(chi) of each chain a, a = 1,2,..., is computed for various chi, and it is found that (L) over bar (a)(chi) changes inversely proportional toy and roughly proportional to m(a), the number of LKs trapped in chain a; these results are naturally explained by the existence of the repulsive potential among LKs. By tracing motion of LKs along chains using the local Gauss integral introduced in the previous work, it is found that (1) there are many kinds of LKs which have different volumes in the chains according to their complexity but (2) ca. 70 vol% of LKs are the simplest LK2,(1) which is composed of two stems and has the Gauss integral equal to +/- 1. From these results, it is concluded that the validity of LK model is sufficiently proved by the present work. Deltamu(chi) obtained here is applied to crystalline polymers in the next paper. (C) 2002 Published by Elsevier Science Ltd.