Rubber elasticity: Solution of the James-Guth model

Abstract

The solution of the many-body statistical mechanical theory of elasticity formulated by James and Guth in the 1940s is presented. The remarkable aspect of the solution is that it gives an elastic free energy that is essentially equivalent to that developed by Flory over a period of several decades. IntroductionRubber elasticity is the first bulk property of polymers that yielded to theoretical analysis. The identification of the relation between the Gaussian distribution of the end-to-end distance of a random walk with the quadratic strain dependence of phenomenological stress-strain theory was key to this success. Of course, the underlying physics that makes this connection inevitable and viable is that a polymer chain in the bulk amorphous melt phase is unperturbed by intermolecular interactions.[1] This fact allows one to realize the force that a chain delivers to the cross-links that terminate it, and which tie it to other chains in the three-dimensional space filling random network, is determined solely by the chain's intramolecular potential. While high elasticity theory was the first to successfully predict bulk polymeric materials behavior, it remains one of the few, and perhaps the only, analytical theory of polymers to do so, computer simulations notwithstanding. This is reason enough to justify efforts to improve upon the theory. Given the history of the subject, rubber elasticity is one of the first soft materials to admit an atom-based theoretical analysis.The extent to which the elastic equation of state is determined by the interaction between network connectivity and chain statistics has been a point of contention from the earliest days of polymer theory. The theory initiated by Kuhn,[2] elaborated by Wall [3,4]and Flory and Rehner, [5,6] and discussed extensively in treatises, [1,7,8,9,10] is constructed by adding together the contributions to the stress from independent chains. This requires the so-called affine assumptionthe displacement of the ends of an average network chain is congruent to the macroscopic strain.This theory is relatively easy to execute, but by treating the chains as independent it incurred the criticism of James and Guth. [11,12,13] In their many-body theory the individual chains obey Gaussian statistics, just as in the independent chain theory, but James and Guth emphasized the fact that in tying the chains together with cross-links they become an indissoluble whole that must be treated as a single entity. This insight carried a heavy price -their many-body formulation was too difficult to be convincingly solved. To make progress, James and Guth [11] introduced the

Molecular Dynamics Study on the Validity of Miller–Macosko Theory for Entanglement and Crosslink Contributions to the Elastic Modulus of End-Linked Polymer Networks

“…However, to solve the underlying many-body problem, James and Guth had to introduce the unphysical concept of fixed (nonfluctuating) junctions that are displaced affinely with the macroscopic deformation. − This concept was strongly criticized by Flory on the grounds that the resulting value of the modulus depends on the choice of the fixed junctions. , But in his earlier works, Flory had used essentially the same eq to discuss the swelling of polymer networks. − …”

“…2−4 This concept was strongly criticized by Flory on the grounds that the resulting value of the modulus depends on the choice of the fixed junctions. 6,7 But in his earlier works, Flory had used essentially the same eq 1 to discuss the swelling of polymer networks. 8−10 The ANT formula 1 automatically accounts for any possible topological defects.…”

Odd–Even Effect of Junction Functionality on the Topology and Elasticity of Polymer Networks

“…As a result, quantitatively understanding the correlation between the topology and properties of polymer networks remains one of the largest outstanding challenges in polymer science. Our fundamental knowledge about polymer networks is built upon an assumption of homogeneous loop-free treelike structures that lies at the foundation of the affine and phantom network theories. ,− However, real polymer networks possess topological defects at different length scales, such as dangling ends, cyclic defects (loops of different order), and heterogeneity in cross-link density. − Although dangling ends and heterogeneity in cross-link density can be largely reduced by stoichiometric end-linking of polymer precursors with multifunctional junctions, loops inevitably exist in all polymer networks as illustrated in Figure . These cyclic defects have vital effects on the mechanical strength of polymer networks .…”

Manifestation of the silicate filler additives and electron beam irradiation on properties of SBR/devulcanized waste tire rubber composites for floor tiles applications

“…As indicated by the classical kinetic theory of rubber elasticity which was initially developed by Wall Flory, James and Guth. 28 They ascribed the high elasticity of a cross-linked rubber to the variation of the conformational entropy of long flexible molecular chains. The theory predicts the following relation in simple extension; Equation (3):…”