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This chapter describes three molecular theories of rubber elasticity. Section 2.1 outlines the elementary theory of Kuhn and Treloar, which is of particular importance since it presents the basic elements of rubberlike elasticity in a very transparent way. Section 2.2 presents the phantom network model developed by James and Guth, and section 2.3 presents the affine network model developed by Wall and Flory. Historical aspects of the theories have been given in an article by Guth and Mark, and in a book prepared as a memorial to Guth. Finally, the major features of both theories are briefly summarized in a review. Separately, the James-Guth theory has been reviewed by Guth and by Flory, and the phantom network model of section 2.2 is based on the Flory treatment. The affine network model has been described in detail in Flory’s 1953 book. This model is described in section 2.3 by generalizing the phantom network model (as was done in one of Flory’s subsequent studies). The simple, elementary statistical theory described in section 2.1 paved the way to the current understanding of rubber elasticity. Further progress in the understanding of rubberlike systems was possible, however, only as a result of the two more precise and accurate theories: the phantom network and the affine network theories. Despite their differences, these two theories and the corresponding molecular models have served as basic reference points in this area for more than four decades. They still serve this purpose for the interpretation and explanation of experimental data. The differences between the assumptions and the predictions of the two models have led to serious disagreements during their development, as may be seen from the original papers cited earlier. The main point of disagreement was the magnitude of the front factor that appeared in the expression for the elastic free energy and the stress. For tetrafunctional networks, the James-Guth phantom network theory predicts one-half the value of the front factor obtained by the Wall-Flory affine network theory.

Small-angle neutron scattering (SANS) experiments from networks were initiated by Benoit and collaborators in the mid-1970s. Currently, SANS is an important major technique used in studying network structure and behavior. Its importance lies in its being a direct method with which observations may be made at the molecular-length scale without the need for a theoretical model for interpreting the data. This feature makes neutron scattering a valuable tool for testing various molecular theories on which current understanding of elastomeric networks is based. The general features of the technique are explained in section 14.1, followed in section 14.2 by a review of relevant experimental work. Section 14.3 then describes different theories of neutron scattering from networks, and compares them with experimental results. The technique of neutron scattering and its application to polymers in the dilute and bulk states, to blends, and to networks are described in several review articles and a book. The reader is referred to this literature for a more comprehensive understanding of the technique and the underlying theory. The neutrons incident on a sample during a typical experiment are from a nuclear reactor. Neutrons leaving the source are first collimated so that they arrive at the sample in the form of plane waves. Figure 14.1 shows such an incident neutron wave on two scattering centers i and j. After interacting with the scattering centers, the neutrons move in various directions. In a neutron scattering experiment, the intensity of the scattered neutron wave is measured as a function of the angle θ shown in the figure, in which the vectors k0 and k are the wave propagation vectors for incident and scattered neutron rays, respectively. In general, the magnitudes of k0 and k differ if there is energy change upon scattering, and in this case the scattering is called inelastic. Inelastic scattering experiments are particularly useful in studying the dynamics of a system, such as relaxation or diffusion.

This chapter is a brief overview of the topics treated in the book. It is aimed, in particular, at providing some qualitative information on rubber elasticity theories and their relationships to experimental studies, and at putting this material into context. The following chapter describes in detail the classical theories of rubber elasticity, that is, the phantom and affine network theories. The network chains in the phantom model are assumed not to experience the effects of the surrounding chains and entanglements, and thus to move as “phantoms.” Although this seems to be a very severe approximation, many experimental results are not in startling disagreement with theories based on this highly idealized assumption. These theories associate the total Helmholtz free energy of a deformed network with the sum of the free energies of the individual chains—an important assumption adopted throughout the book. They treat the single chain in its maximum simplicity, as a Gaussian chain, which is a type of “structureless” chain (where the only chemical constitution specified is the number of bonds in the network chain). In this respect, the classical theories focus on ideal networks and, in fact, are also referred to as “kinetic” theories because of their resemblance to ideal gas theories. Chain flexibility and mobility are the essential features of these models, according to which the network chains can experience all possible conformations or spatial arrangements subject to the network’s connectivity. One of the predictions of the classical theories is that the elastic modulus of the network is independent of strain. This results from the assumption that only the entropy at the chain level contributes to the Helmholtz free energy. Experimental evidence, on the other hand, indicates that the modulus decreases significantly with increasing tension or compression, implicating interchain interactions, such as entanglements of some type or other. This has led to the more modern theories of rubber elasticity, such as the constrained-junction or the slip-link theories, which go beyond the single-chain length scale and introduce additional entropy to the Helmholtz free energy at the subchain level.