*Burak Erman and James E. Mark*

- Published in print:
- 1997
- Published Online:
- November 2020
- ISBN:
- 9780195082371
- eISBN:
- 9780197560433
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780195082371.003.0003
- Subject:
- Chemistry, Materials Chemistry

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 ...
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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.
Less

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.

*Burak Erman and James E. Mark*

- Published in print:
- 1997
- Published Online:
- November 2020
- ISBN:
- 9780195082371
- eISBN:
- 9780197560433
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780195082371.003.0006
- Subject:
- Chemistry, Materials Chemistry

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 ...
More

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.
Less

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.

*Burak Erman and James E. Mark*

- Published in print:
- 1997
- Published Online:
- November 2020
- ISBN:
- 9780195082371
- eISBN:
- 9780197560433
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780195082371.003.0007
- Subject:
- Chemistry, Materials Chemistry

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 ...
More

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.
Less

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.