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 Δμ(χ) of chain-elements, measured from its topological equilibrium value, is computed numerically as a function of condensation ratio χ of LKs. Δμ(χ) is transformed into free energy ΔF(χ) of the system, which takes a minimum in the topological equilibrium state (χ=1) and increases rapidly with increasing χ. It is argued that ΔF(χ) comes mainly from the topological repulsive potentials among LKs, because ΔF(χ) 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 L̄a(χ) of each chain a, a=1,2,…, is computed for various χ, and it is found that L̄a(χ) changes inversely proportional to χ and roughly proportional to ma, 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. Δμ(χ) obtained here is applied to crystalline polymers in the next paper (Polymer, 2002;43: (the following paper in this issue)).