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The classical theories of rubber elasticity are based on the Gaussian chain model. The only molecular parameter that enters these theories is the mean-square end-to-end separation of the chains constituting the network. However, there are various areas of interest that require characterization of molecular quantities beyond the Gaussian description. Examples are segmental orientation, birefringence, rotational isomerization, and finite extensibility, and we will address these properties in the following chapters. One often needs a more realistic distribution function for the end-to-end vector, as well as for averages of the products of several vectorial quantities, as will be evident in these chapters. The foundations for such characterizations, and several examples of their applications, are given in this chapter. Several aspects of rubber elasticity (such as the dependence of the elastic free energy on network topology, number of effective junctions, and contributions from entanglements) are successfully explained by theories based on the freely jointed chain and the Gaussian approximation. Details of the real chemical structure are not required at the length scales describing these phenomena. On the other hand, studies of birefringence, thermoelasticity, rotational isomerization upon stretching, strain dichroism, local segmental orientation and mobility, and characterization of networks with short chains require the use of more realistic network chain models. In this section, properties of rotational isomeric state models for the chains are discussed. The notation is based largely on the Flory book, Statistical Mechanics of Chain Molecules. More recent information is readily found in the literature. Due to the simplicity of its structure, a polyethylene-like chain serves as a convenient model for discussing the statistical properties of real chains. This simplicity can be seen in figure 8.1, which shows the planar form of a small portion of a polyethylene chain. Bond lengths and bond angles may be regarded as fixed in the study of rubber elasticity because their rapid fluctuations are usually in the range of only ±0.05 A and ±5°, respectively. The chain changes its configuration only through torsional rotations about the backbone bonds, shown, for example, by the angle for the ith bond in figure 8.1.

As was mentioned in chapter 10, end-linking reactions can be used to make networks of known structures, including those having unusual chain-length distributions. One of the uses of networks having a bimodal distribution is to clarify the dependence of ultimate properties on non-Gaussian effects arising from limited-chain extensibility, as was already pointed out. The following chapter provides more detail on this application, and others. In fact, the effect of network chain-length distribution, is one aspect of rubberlike elasticity that has not been studied very much until recently, because of two primary reasons. On the experimental side, the cross-linking techniques traditionally used to prepare the network structures required for rubberlike elasticity have been random, uncontrolled processes, as was mentioned in chapter 10. Examples are vulcanization (addition of sulfur), peroxide thermolysis (free-radical couplings), and high-energy radiation (free-radical and ionic reactions). All of these techniques are random in the sense that the number of cross-links thus introduced is not known directly, and two units close together in space are joined irrespective of their locations along the chain trajectories. The resulting network chain-length distribution is unimodal and probably very broad. On the theoretical side, it has turned out to be convenient, and even necessary, to assume a distribution of chain lengths that is not only unimodal, but monodisperse! There are a number of reasons for developing techniques to determine or, even better, control network chain-length distributions. One is to check the “weakest link” theory for elastomer rupture, which states that a typical elastomeric network consists of chains with a broad distribution of lengths, and that the shortest of these chains are the “culprits” in causing rupture. This is attributed to the very limited extensibility associated with their shortness that is thought to cause them to break at relatively small deformations and then act as rupture nuclei. Another reason is to determine whether control of chain-length distribution can be used to maximize the ultimate properties of an elastomer. As was described in chapter 10, a variety of model networks can be prepared using the new synthetic techniques that closely control the placements of crosslinks in a network structure.

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.

Classical theories of rubber elasticity are based on models of flexible polymer chains that are sufficiently long to exhibit Gaussian behavior as described in chapter 1 and in appendix F. Additionally, the chains are phantomlike in the sense that they do not interact with one other along their contours. The theories described in chapter 2 were based on this picture of the individual chain. In this chapter, we describe the elasticity of networks that depart substantially from those addressed in the classical theories. The departures may result from two sources: (1) the chains may be only semiftexible, as a result of which the segments of neighboring chains compete for space in the deformed network, and choose preferentially oriented configurations, and (2) the chains may form crystallites, upon deformation, as a result of which the homogeneous structure of the classical network model may be transformed into a nonhomogeneous one having microphases of crystalline and amorphous regions. The subject of crystallization under deformation, for networks in general, is relatively old, and has been treated in some detail in previous studies. For this reason, crystallization and some of its effects will be reviewed only briefly at the end of this chapter. The main emphasis will be given to networks with semiflexible chains. Examples of networks with semiflexible chains are those in which the chains have rodlike segments separated by flexible spacers, or those where the chains have bond angles appreciably larger than tetrahedral. Incorporation of these chains into a network structure results in materials that exhibit segmental orientations significantly larger than those shown by classical networks. Specific examples would include networks prepared from aromatic polyamide chains or from chains containing liquid-crystalline sequences along the direction of the backbone. Because of their unique chain structures, these networks are easily orientable, at the molecular level, by macroscopic deformations. The orientational transitions may easily be controlled by application and removal of anisotropic strains, and are therefore of great technological interest for use in optical devices. Other examples of networks with easily orientable chains are those with rigid sequences in the side groups.

The classical theories of rubber elasticity presented in chapter 2 are based on a hypothetical chain which may pass freely through its neighbors as well as through itself. In a real chain, however, the volume of a segment is excluded to other segments belonging either to the same chain or to others in the network. Consequently, the uncrossability of chain contours by those occupying the same volume becomes an important factor. This chapter and the following one describe theoretical models treating departures from phantom-like behavior arising from the effect of entanglements, which result from this uncrossability of network chains. The chains in the un-cross-linked bulk polymer are highly entangled. These entanglements are permanently fixed once the chains are joined during formation of the network. The degree of entanglement, or degree of interpenetration, in a network is proportional to the number of chains sharing the volume occupied by a given chain. This is quite important, since the observed differences between experimental results on real networks and predictions of the phantom network theory may frequently be attributed to the effects of entanglements. The decrease in network modulus with increasing tensile strain or swelling is the best-known effect arising from deformation-dependent contributions from entanglements. The constrained-junction model presented in this chapter and the slip-link model presented in chapter 4 are both based on the postulate that, upon stretching, the space available to a chain along the direction of stretch is increased, thus resulting in an increase in the freedom of the chain to fluctuate. Similarly, swelling with a suitable diluent separates the chains from one another, decreasing their correlations with neighboring chains. Experimental data presented in figure 3.1 show that the modulus of a network does indeed decrease with both swelling and elongation, finally becoming independent of deformation, as should be the case for the modulus of a phantom network. Rigorous derivation of the modulus of a network from the elastic free energy for this case will be given in chapter 5. The starting point of the constrained-junction model presented in this chapter is the elastic free energy.

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.

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.

Segmental or molecular orientation refers to the anisotropic distribution of chain-segment orientations in space, due to the orienting effect of some external agent. In the case of uniaxially stretched rubbery networks, which will be the focus of this chapter, segmental orientation results from the distortion of the configurations of network chains when the network is macroscopically deformed. In the undistorted state, the orientations of chain segments are random, and hence the network is isotropic because the chain may undertake all possible configurations, without any bias. In the other hypothetically extreme case of infinite degree of stretching of the network, segments align exclusively along the direction of stretch. The mathematical description of segmental orientation at all levels of macroscopic deformation is the focus of this chapter. Segmental orientation in rubbery networks differs distinctly from that in crystalline or glassy polymers. Whereas the chains in glassy or crystalline solids are fully or partly frozen, those in an elastomeric network have the full freedom to go from one configuration to another, subject to the constraints imposed by the network connectivity. The orientation at the segmental level in glassy or crystalline networks is mostly induced by intermolecular coupling between closely packed neighboring molecules, while in the rubbery network intramolecular conformational distributions predominantly determine the degree of segmental orientation. The first section of this chapter describes the state of molecular deformation. In section 11.2, the simple theory of segmental orientation is outlined, followed by the more detailed treatment of Nagai and Flory. The chapter concludes with a discussion of infrared spectroscopy and the birefringence technique for measuring segmental orientation. For uniaxial deformation, the deformation tensor λ takes the form λ = diag(λ, λ-1/2, λ-1/2), where diag represents the diagonal of a square matrix, and λ is the ratio of the stretched length of the rubbery sample to its undeformed reference length. The first element along the diagonal of the matrix represents the extension ratio along the direction of stretch, which may be conveniently identified as the X axis of a laboratory-fixed frame XYZ. The other two elements refer to the deformation along two lateral directions, Y and Z.

In the preceding chapter, the stress-strain behavior of networks was described for the case where the swollen network was assumed to be a thermodynamically closed system (such that solvent molecules did not enter or leave the network during deformation). In this chapter, a more general thermodynamic analysis will be given for the case where the network-solvent system will be regarded as semiopen. In such systems, the solvent may enter or leave the network depending on the chemical potential of the solvent and the extent of deformation of the network. In the first section, we will describe general thermodynamic relations for network-solvent systems, followed by a discussion of isotropic swelling of networks in the second section. This topic has already been treated in classic books, and the reader is referred to them for background information. In this chapter, we discuss more recent improvements and approaches, and newer experiments in this field. In the third section of the chapter, we will describe the effects of an externally applied deformation on networks immersed in solvent. Again, the study of deformation of immersed networks goes back to approximately the mid-1940s, and only more recent developments will be included in this chapter. Also, we will consider in detail, in this chapter, the swelling of nonionic networks only. The effects of ionic groups on network chains are discussed in the following chapter in relation to critical phenomena and phase separation in swollen networks. We conclude with a discussion of sorption of linear and cyclic diluents into networks. Also covered is their extraction and, in the case of the cyclic molecules, their trapping within the network structure.