Rubber elasticity internal energy

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). 

When a themally expansible homogeneous isotropic elastic continuum is deformed iso thermally the mechanical work done on the body is partly stored as internal energy partly converted to heat. In the range of strain up to about 10 percent (which is typical of natural rubber) the internal energy storage exceeds the work done so that heat must be added to the sample. Beyond the strain of 10 percent heat is increasingly liberated. An equivalent statement conveys the information that if the sample be stretched adiabatically there is an initial temperature drop then subsequent rise. 

Rubber elasticity, which is a unique characteristic of polymers, is due to the presence of long chains existing in a temperature range between the Tg and the Tm. The requirements for rubbery elasticity are (1) a network polymer with low cross-link density, (2) flexible segments which can rotate freely in the polymer chain, and (3) no volume or internal energy change during reversible deformation. 

To understand robber elasticity we have to revisit some simple thermodynamics (the horror. the horror ). Let s start with the Helmholtz free energy of our piece of rubber, by which we mean that we are considering the free energy at constant temperature and volume (go to the review at the start of Chapter 10 if you ve also forgotten this stuff). If E is the internal energy (the sum of the potential and kinetic energies of all the particles in the system) and 5 the entropy, then (Equation 13-26) ... 

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. 

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. 

In Fig. 2 and 3 the dependences a on generalized stress for studied rubbers, corresponding to the equations (13) and (14), are shown. As can be seen, in case of composites the linearity of these dependences is violated, i.e., at least, the filled rubbers behaviour does not corresponded to high-elasticity classical theory, that is assumed above. Differently speaking, filled rubbers are impossible to consider as ideal, for which internal energy change MJ is equal to zero in deformation process. 

The theory of rubber elasticity is largely based on thermodynamic considerations. It will be briefly discussed as an example of how thermodynamics can be applied in polymer science. Eor more detailed information the reader is referred to the various textbooks [10-13]. It is assumed that there is a three-dimensional network of chains, that the chain units are flexible and that individual chain segments rotate freely, that no volume change occurs upon deformation, and that the process is reversible (i.e., true elastic behavior). Another usual assumption is that the internal energy U of the system does not change with deformation. Eor this system the first law of thermodynamics can be written as ... 

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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