Topological rubber elasticity theory. I. Basic theory

This theory is based on an assumption that topological states of a network are determined by a set T0 of the Gauss linking coefficient (GLC) defined for all the possible pairs of loops in the network. Since elements of T0 are not linearly independent, a complete basic subset T1 of T0 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 T1. 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, r0) is defined by F(λ, τ) = kT ln∫Ψ(r, τ)dr, F̃(λ, r0) = ΣτF(λ, τ)Ψ(τ,‖r0),where λ is a set of parameters for macroscopic deformation, r0 is the position of the junction points at the time of network formation, and Ψ(τ‖r0) is a distribution function of τ under the condition that r is fixed at r0; the integration in regard to r is performed with restrictions (1), (2), and (3) stated above so that F̃(λ, r0) becomes a function of λ. The free energy F̃(λ, r0) consists of F̃e, an elastic free energy of the network, and ΔFm, an excess free energy of mixing. F̃e is composed of three terms F̃0, F̃1, and F̃2. The zeroth term F̃0 is an elastic free energy of a network in which the all junction points deform affinley and θ is fixed at its initial value θ0 at the time of network formation. The first term F̃1 is a correction due to nonaffine displacement of the junction points. The second term F̃2 is a correction due to fluctuation of θ. The zeroth term F̃0 is further divided into F0,ph, a term coming from the entropic force acting between ends of the strands and F̃0,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 elastic free energy is F̃0,top.