Small-Angle Neutron Scattering

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.

Thermoelasticity

The important postulate that intermolecular interactions are independent of extent of deformation leads directly to the conclusion that such interactions cannot contribute to an energy of elastic deformation ΔEel at constant volume. In the earliest theories of rubberlike elasticity, it was additionally assumed that, intramolecular contributions to ΔEel were likewise nil. In this idealization that the total ΔEel is zero, the elastic retractive force exhibited by a deformed polymer network would be entirely entropic in origin. At the molecular level, this would correspond, of course, to assuming all configurations of a network chain to be of exactly the same conformational energy and thus the average configuration to be independent of temperature. Under these circumstances, the dependence of stress on temperature is strikingly simple, as shown, for example, by the equation . . . f* = υkT/V (〈r2〉i/〈r2〉0)(α – α-2) . . . . . . (9.1) . . . that characterizes a polymer network in elongation where, it should be recalled, 〈r2〉i3/2 is proportional to the volume of the network. This additional assumption that 〈r2〉0 is independent of temperature would lead to the prediction that the elastic stress determined at constant volume and elongation α is directly proportional to the absolute temperature. Such network chains would be akin to the particles of an ideal gas, which would obey the equation of state p = nRT(1/V) and thus exhibit a pressure at constant deformation (1/V) likewise directly proportional to the temperature.

Calculations and Simulations

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.

Intermolecular Effects: II. Constraints along Network Chains

In the constrained-junction model presented in chapter 3, intermolecular correlations were assumed to suppress the fluctuations of junctions. According to this model, the elastic free energy of a network varies between the free energies of the phantom and the affine networks. In a second group of models, to be introduced here, there is a constraining action of entanglements along the chains that may further contribute to the elastic free energy, as if they were additional (albeit temporary) junctions. Consequently, the upper bound of the elastic free energy of such networks may exceed that of an affine network. Since the entanglements along the chain contour are explicitly taken into account in the models, they are referred to as the constrained-chain models. The idea of constrained-chain theories originates from the trapped-entanglement concept of Langley, and Graessley, stating that some fraction of the entanglements which are present in the bulk polymer before cross-linking become permanently trapped by the cross-linking and act as additional cross-links. These trapped entanglements, unlike the chemical cross-links, have some freedom, and the two chains forming the entanglement may slide relative to one other. The two chains may therefore be regarded as being attached to each other by means of a fictitious “slip-link,” as is illustrated schematically in figure 4.1. The entangled system of chains representing the real situation is shown in part (a), and the representation of two entangled chains in this system joined together by a sliplink is shown in part (b). The slip-link may move along the chains by a distance a, which is inversely proportional to the severity of the entanglements. A model based on this picture of slip-links was first proposed by Graessley, and a more rigorous treatment of the slip-link model was given by Ball et al. and subsequently simplified by Edwards and Vilgis; section 4.1 describes this latter treatment in detail. In section 4.2, we present the extension of the Flory constrained junction model to the constrained-chain model by including the effects of constraints along chains, following Erman and Monnerie. One of the newest approaches, the diffused-constraints model, is then described briefly in section 4.3.

Intermolecular Effects: I. The Constrained-Junction Model

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.

Networks Having Multimodal Chain-Length Distributions

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.

Model Elastomers

Until quite recently, there was relatively little reliable quantitative information on the relationship of stress to structure, primarily because of the uncontrolled manner in which elastomeric networks were generally prepared. Segments close together in space were linked irrespective of their locations along the chain trajectories, thus resulting in a highly random network structure in which the number and locations of the cross-links were essentially unknown. Such a structure is shown in figure 10.1. New synthetic techniques are now available, however, for the preparation of “model” polymer networks of known structure. More specifically, if networks are formed by end linking functionally terminated chains instead of haphazardly joining chain segments at random, then the nature of this very specific chemical reaction provides the desired structural information. Thus, the functionality of the cross links is the same as that of the end-linking agent, and the molecular weight Mc between cross-links and the molecular weight distribution are the same as those of the starting chains prior to their being end-linked. An example is the reaction shown in figure 10.2, in which hydroxyl-terminated chains of poly(dimethylsiloxane) (PDMS) are end-linked using tetraethyl orthosilicate. Characterizing the un-cross-linked chains with respect to molecular weight Mn and molecular weight distribution, and then carrying out the specified reaction to completion, gives elastomers in which the network chains have these characteristics; in particular, a molecular weight Mc between cross-links equal to Mn, a network chain-length distribution equal to that of the starting chains, and cross-links having the functionality of the end-linking agent. It is also possible to use chains having a known number of potential cross-linking sites placed as side chains along the polymer backbone, so long as their distribution is known as well. Because of their known structures, such model elastomers are now the preferred materials for the quantitative characterization of rubberlike elasticity. Such very specific cross-linking reactions have also been shown to be useful in the preparation of liquid-crystalline elastomers. Trifunctional and tetrafunctional PDMS networks prepared in this way have been used to test the molecular theories of rubber elasticity with regard to the increase in non-affineness of the network deformation with increasing elongation.

Relationships between Stress and Strain

In the first section of this chapter, the relationships between the Helmholtz free energy, the stress tensor, and the deformation tensor are given for uniaxial stress. These relations follow from the general discussion of stress and strain given in appendix C, and the notation and approach closely follow the classic treatment of Flory. The detailed forms of the stress-strain relations in simple tension (or compression) are given in the remaining sections of the chapter for the (1) phantom network, (2) affine network, (3) constrained-junction model, and (4) slip-link model. Results of theory are then compared with experiment. The effects of swelling on the stress-strain relations are also included in the discussion. It is to be noted that the stress-strain relations in this chapter are obtained by treating the swollen networks as closed systems. The conditions for such systems are fulfilled if solvent does not move in and out of the network during deformation. A network swollen with a nonvolatile solvent and subject to simple tension in air is an example of a closed system. The same network at swelling equilibrium and subjected to compression will exude some of the solvent under increased internal pressure, and is therefore not a closed system. For semiopen systems, such as those under compression, or, in general, networks stressed while immersed in solvent, a more general thermodynamic treatment is required. This situation will be taken up in the following chapter.

Bioelastomers

There are a variety of biopolymeric materials which exhibit rubberlike elasticity. This is perhaps to be expected when one recalls that most biopolymers are randomly coiled chains with considerable flexibility, and that they are frequently covalently cross-linked or have sufficient numbers of aggregated units to exist in network structures. One very large group of plant materials, the polysaccharides, are in this category, and they do require some elastomeric properties in their functioning. In many of these cases, however, the cross-linking is there primarily for a secondary purpose, such as preventing solubility. When swollen with water or aqueous solutions, such polysaccharides form gels which do exhibit the high deformability and recoverability that are the hallmarks of rubberlike elasticity. Not surprisingly, however, relatively few mechanical property measurements have been carried out to characterize the structures of these gels. The bioelastomers occurring in animals, including vertebrates and mammals, however, are there specifically for their rubberlike elasticity. They are vital, for example, for the functioning of skin, arteries and veins, and much of the lung and heart tissue. Since they are produced by the ribosome “factories” in the body, they are proteins. Thus, the major focus of this chapter is on those proteins specifically designed to function as bioelastomers. It is useful to summarize some general information on bioelastomers that is presented elsewhere. Even with the temporary restriction to bioelastomers which are proteins, there is an almost staggering variety of interesting materials. For example, there is elastin in vertebrates (including mammals) resilin in insects abductin in mollusks, arterial elastomer in octopuses, circulatory and locomotional proteins in cephalopods, and viscid silk in spider webs. Since they are mammals, polymer scientists and engineers who are interested in bioelastomers have focused heavily on elastin! Any materials of this type, however, are worth studying in their own right, to learn more about rubberlike elasticity and biological function. Such studies should also provide guidance on how Nature might be mimicked by synthetic chemists, to produce better nonbiological elastomers.

Segmental Orientation

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.