Topological rubber elasticity theory. I. Basic theory

Abstract

This theory is based on an assumption that topological states of a network are determined by a set T 0 of the Gauss linking coefficient (GLC) defined for all the possible pairs of loops in the network. Since elements of T 0 are not linearly independent, a complete basic subset T 1 of T 0 is chosen as a set of independent topological invariants of the system. An equilibrium distribution function Φ( r , ϑ), in regard to r, a set of the all coordinates of junction points, and θ a set of the all GLC for strand pairs in the network, are computed first using the topological moment method presented in a previous paper [K. Iwata and T. Kimura, J. Chem. Phys. 74, 2039 (1981)], and it is transformed into Ψ(r, τ), a distribution function in regard to r and τ, a value of T 1 . In the calculation of free energies, it is assumed that (1) the network is composed of identical cells, each of which contains sufficiently large number of strands and junction points; (2) the network is periodic in regard to the cells; and (3) relative displacement of weight centers of the cells under macroscopic deformation is exactly affinelike. The free energy of the system under macroscopic deformation F̃(λ, r 0 ) is defined by F(λ, τ) = kT ln∫Ψ(r, τ)d r , F̃(λ, r 0 ) = Σ τ F(λ, τ)Ψ(τ,‖r 0 ), where λ is a set of parameters for macroscopic deformation, r 0 is the position of the junction points at the time of network formation, and Ψ(τ‖ r 0 ) is a distribution function of τ under the condition that r is fixed at r 0 ; the integration in regard to r is performed with restrictions (1), (2), and (3) stated above so that F̃(λ, r 0) becomes a function of λ. The free energy F̃(λ, r 0 ) consists of F̃ e , an elasticfree energy of the network, and ΔF m , an excess free energy of mixing. F̃ e is composed of three terms F̃ 0 , F̃ 1 , and F̃ 2 . The zeroth term F̃ 0 is an elasticfree energy of a network in which the all junction points deform affinley and θ is fixed at its initial value θ 0 at the time of network formation. The first term F̃ 1 is a correction due to nonaffine displacement of the junction points. The second term F̃ 2 is a correction due to fluctuation of θ. The zeroth term F̃ 0 is further divided into F 0,ph , a term coming from the entropic force acting between ends of the strands and F̃ 0, top , a term coming from the topological interaction among the strands. It will be shown in the following paper (part II), where the present theory will be applied to SCL networks, that the principal term of the elasticfree energy is F̃ 0, top .