An application of linking probability to topological effects of polymer systems : rubber elasticity(Knots and soft-matter physics: Topology of polymers and related topics in physics, mathematics and biology)

Abstract

本論文では,2本の環状ランダム鎖が絡み合う確率(絡み目確率)をゴム弾性の非線形効果の問題に用いた例を示す。ネットワークのトポロジーが変形によって変化しないというトポロジー制限を考慮に入れると,ゴムのエントロピーは古典論で導かれる項とトポロジー制限による項の2項に分解される。このトポロジー制限から導かれた2項目には絡み目確率が自然と導入されており,Moonly-Rivlin則のC_2項と同じ様な振る舞いをする。

We offer an application of linking probability, which is defined by the probability that two random polygons are entangled each other, to the problem of the nonlinear behavior in rubber elasticity. By taking account of topological constraint that the topological state of the network does not vary under deformation, we prove that the total entropy of the network is decomposed into two terms: the entropy of the classical theory and that due to the topological constraint. We show that the linking probability is naturally introduced in the entropy due to the topological constraint and the entropic force derived from the topological entropy behaves like the C_2 term of the Mooney-Rivlin equation.