The Molecular Kink Paradigm proceeds from the intuitive notion that molecular chains that make up a natural rubber (polyisoprene) network are constrained by surrounding chains to remain within a ‘tube’. Elastic forces produced in a chain, as a result of some applied strain, are propagated along the chain contour within this tube. Fig. 2 shows a representation of a four-carbon isoprene backbone unit with an extra carbon atom at each end to indicate its connections to adjacent units on a chain. It has three single C-C bonds and one double bond. It is principally by rotating about the C-C single bonds that a polyisoprene chain randomly explores its possible conformations. Sections of chain containing between two and three isoprene units have sufficient flexibility that they may be considered statistically de-correlated from one another. That is, there is no directional correlation along the chain for distances greater than this distance, referred to as a Kuhn length. These non-straight regions evoke the concept of ‘kinks’ and are in fact a manifestation of the random-walk nature of the chain. Since a kink is composed of several isoprene units, each having three carbon-carbon single bonds, there are many possible conformations available to a kink, each with a distinct energy and end-to-end distance. Over time scales of seconds to minutes, only these relatively short sections of the chain, i.e. kinks, have sufficient volume to move freely amongst their possible rotational conformations. The thermal interactions tend to keep the kinks in a state of constant flux, as they make transitions between all of their possible rotational conformations. Because the kinks are in thermal equilibrium, the probability that a kink resides in any rotational conformation is given by a Boltzmann distribution and we may associate an entropy with its end-to-end distance. The probability distribution for the end-to-end distance of a Kuhn length is approximately Gaussian and is determined by the Boltzmann probability factors for each state (rotational conformation). As a rubber network is stretched, some kinks are forced into a restricted number of more extended conformations having a greater end-to-end distance and it is the resulting decrease in entropy that produces an elastic force along the chain.
There are three distinct molecular mechanisms that produce these forces, two of which arise from changes in entropy that we shall refer to as low chain extension regime, Ia and moderate chain extension regime, Ib. The third mechanism occurs at high chain extension, as it is extended beyond its initial equilibrium contour length by the distortion of the chemical bonds along its backbone. In this case, the restoring force is spring-like and we shall refer to it as regime II. The three force mechanisms are found to roughly correspond to the three regions observed in tensile stress vs. strain experiments, shown in Fig. 1.
The initial morphology of the network, immediately after chemical cross-linking, is governed by two random processes: (1) The probability for a cross-link to occur at any isoprene unit and, (2) the random walk nature of the chain conformation. The end-to-end distance probability distribution for a fixed chain length, i.e. fixed number of isoprene units, is described by a random walk. It is the joint probability distribution of the network chain lengths and the end-to-end distances between their cross-link nodes that characterizes the network morphology. Because both the molecular physics mechanisms that produce the elastic forces and the complex morphology of the network must be treated simultaneously, simple analytic elasticity models are not possible; an explicit 3-dimensional numerical model is required to simulate the effects of strain on a representative volume element of a network.