Entropic contributions to rubber

ENTHALPIC AND ENTROPIC CONTRIBUTIONS TO RUBBER ELASTICITY THE FORCE-TEMPERATURE RELATIONS... 

VII. Enthalpic and Entropic Contributions to Rubber Elasticity Force-Temperature Relations Vtn. Direct Determination of Molecular Dimensions IX. Single-Molecule Elasticity References... 

In typical crystalline solids, such as metals, the energetic contribution dominates the force because the internal energy increases when the crystalline lattice spacings are distorted from their equilibrium positions. In rubbers, the entropic contribution to the force is more important than the energetic one. In ideal networks there is no energetic contribution to elasticity, so /e = 0. 

This is identical with equation 3.26 if R = R. [However, Fiory, Hoeve and Ciferri [6] have pointed out that the inversion process between equation 3.35 and 3.36 is valid only at large n values. For n <10, the error is large.] The entropic origin of force was not assumed in the derivation of equation 3.37 so that this result is generally valid and not restricted to ideal rubbers. The energy contribution to rubber elasticity may be calculated from equation 3.16... 

The elastic contribution to Eq. (5) is a restraining force which opposes tendencies to swell. This constraint is entropic in nature the number of configurations which can accommodate a given extension are reduced as the extension is increased the minimum entropy state would be a fully extended chain, which has only a single configuration. While this picture of rubber elasticity is well established, the best model for use with swollen gels is not. Perhaps the most familiar model is still Flory s model for a network of freely jointed, random-walk chains, cross-linked in the bulk state by connecting four chains at a point [47] ... 

Before developing the entropic, or statistical, theory of rubber elasticity in a quantitative way, it is important to be sure that this really is the most important contribution, i.e. to be sure that any contribution to the elasticity due to changes in the internal energy on stretching is very small compared with the contribution due to changes of entropy. This is shown to be so in the following section. 

Fig. 11. Plot of the energetic (/ e) and entropic (fs) contributions to the stress at 20 C for sulfur-vulcanized rubber (99). N/mm = MPa to convert MPa to psi, multiply by 145. Courtesy of the American Chemical Society.

In this chapter, wc introduce two more methods of thermodynamics. The first is Maxwell relations. With Maxwell relations you can find the entropic and energetic contributions to the stretching of rubber, the expansion of a surface or a film, or the compression of a bulk material. Second, we use the mathematics of homogeneous functions to develop the Gibbs-Duhem relationship. This is useful for finding the temperature and pressure dependences of chemical equilibria. 

In order to determine the nature of the force generated by a polymer gel, one must consider all the relevant interactions that contribute to the force or displacement. We learned from the thermodynamics of rubber elasticity that the molecular mechanism of force generation in the network chains is made up of two different contributions. In general, energetic and entropic effects must be taken into account ... 

As noted, TPEs are either block copolymers or combinations of a rubber-dispersed phase and a plastic continuous matrix. The attribute contributed by the rubbery phase - such as butadiene or ethylenebutylene in an S-E-S or SEB-S styrenic block copolymer, or the completely vulcanized EPDM rubber particles in a polypropylene (PP)/EPDM EA - is classical elastomeric performance. The elastic properties of a rubber result from long, flexible molecules that are coiled in a random manner. When the molecules are stretched, they uncoil and have a more specific geometry than the coiled molecules. The uncoiled molecules have lower entropy because of the more restricted geometry and, since the natural tendency is an increase in entropy, the entropic driving force is for the molecules to retract. 

Here./g is called the energetic elasticity, and/ is called the entropic elasticity. For instance, the spring exhibits a high elasticity mainly contributed by the energetic elasticity due to the metallic bonds for the strong interactions of iron atoms, while the rubber exhibits a high elasticity mainly contributed by the entropic elasticity due to the chain conformations for the large deformations of polymers. 

In addition to timescale shifts with temperature, the magnitude of the compliance or modulus can change. The kinetic theory of rubber-like elasticity suggests that the entropically based contribution of the modulus to the viscoelastic response should increase in direct proportion to the absolute temperature. Correspondingly, the reciprocal of the steady-state recoverable compliance should be directly proportional to the absolute temperature. This is true at temperatures that are greater than 2Tg, but, between 1.2Tg and 2Tg, the steady-state recoverable compliance Js is essentially independent of temperature. At still lower temperatures a strong decrease of Js, is seen [51]. 

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