Rubber elasticity energy contribution

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... 

Hie total elastic free energy, or the Helmholtz free energy, of a network consists of the sum of the free energies of the individual chains and the contributions coming from intermolecular correlations. Hie elementary theories of rubber elasticity ignore contributions from intermolecular effects. Improvements in the elementary models have been made by different researchers in different ways. In this section most of the models will be covered. Since the single-chain elasticity forms the basis of rubber elasticity, it will be discussed first in some detail. 

As will be amplified below, contributions to R arise from the energy actually required to create the new surface (a quantity related to the surface tension), the orientation of chains near the surface, the breaking of chains that spanned the cracking region, and rubber elasticity energy storage effects. 

TTie elastic energy contribution Uei describes the rubber elasticity of the cross-linked polymer chains, and is proportional to the cross-link density which is the number density of elastic strands in the undeformed polymer network. We use the Flory model [35] to specify U ... 

We can express the free energy of gel in terms of the above quantities, and calculate from it the osmotic pressure 7t acting on the network. For a gel to be in equilibrium with the outer solvent, n must be zero. Several different mechanisms are known to contribute to 7t, the mixing process, the rubber elasticity, and counter ions, etc. The osmotic pressure due to mixing is expressed as... 

In non-polymeric materials the entropy change on deformation is minimal so that the intrinsic and stored elastic energies are the same at least for rapidly occurring events - but in polymers not only may the entropy contribution predominate but for large strains in rubbers the internal energy term is nearly negligible (but not at small strains where it may amount to 20% of the free energy). 

The dominant contribution to the free energy of lengthy (rubbery) polymer chains is entropy. This is known to accoimt for rubber elasticity, which can be satisfactorily modelled by the entropy of the cross-linked pol3rmer chains alone. A simple illustrative model of copolymer self-assembly can be developed by extending rubber elasticity theory to include bending as well as stretching deformations, to calculate chain entropy as a function of interfacial curvatures in diblock aggregates. 

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. 

Thus, from equation (6-27) one should be able to compute the relative importance of contributions from energy and entropy to rubber elasticity. Once again, the difficulty of measuring accurate values of the partial (dVfdL)r,p renders this separation difficult at best. 

Table 6-1 shows the values of fe f for several elastomers. We see that in general we cannot expect the contribution of energy to rubber elasticity to be zero. Rather, a fraction of the stress is attributable to energy, the rest to entropy. However, since this fraction is a constant as a function of strain, the general shape of the stress-strain curve is still unaffected. Thus the neglect of... 

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. 

Rubber reinforcement can be described as an additional accumulation of elastic energy by the system, due to the contribution of a fraction of network chains, self-assembled to gradient interlayer around active filler particles. 

Finally, the simple treatment of rubber elasticity given above makes two assumptions that require further consideration. First, it has been assumed that the internal energy contribution is negligible, which implies that different molecular conformations of the chains have identical internal energies. Secondly, the thermo-d30iamic formulae that have been derived are, strictly, only applicable to measurements at constant volume, whereas most experimental results are obtained at constant pressure. For comprehensive elementary accounts of these complications, the reader is referred to the textbooks by Treloar [1] and Ward [9]. 

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