Rubbers entropic origin

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... 

Similarly, one can derive the mean-square end-to-end distance =3l2lff. In the following, we will apply the Gaussian chain above to interpret the entropic origin of high elasticity of rubbers. 

Because of the entropic origin, the above property is called the entropy elasticity. It is not limited to Ganssian chains. Any chain that has a finite size, inclnding ideal chains and real chains, has this elasticity. By the same reason, a rubber is elastic. A rubber is a cross-linked polymer. A partial chain between two cross-links behaves elastically, giving rise to the elasticity of the material as a whole. 

To demonstrate the entropic origin of the elastic response of the rubber state, we consider the thermodynamics of this state, in which changes in the internal energy E are given by... 

X-ray diffraction and IR dichroism studies suggest that the long-range elasticity of wool is related to a reversible molecular transformation of the alpha-keratin to an extended beta form (66). No convincing evidence supports this mechanism in stratum corneum viscoelasticity. In fact, the available evidence suggests that the elastic behavior of corneum is primarily entropic in origin. At low deformations, the mechanical properties of hydrated stratum corneum is best described as the behavior of a lightly-crosslinked rubber. 

A rubber-like solid is unique in that its physical properties resemble those of solids, liquids, and gases in various respects. It is solidlike in that it maintains dimensional stability, and its elastic response at small strains (<5%) is essentially Hookean. It behaves like a liquid because its coefficient of thermal expansion and isothermal compressibility are of the same order of magnitude as those of liquids. The implication of this is that the intermolecular forces in an elastomer are similar to those in liquids. It resembles gases in the sense that the stress in a deformed elastomer increases with increasing temperature, much as the pressure in a compressed gas increases with increasing temperature. This gas-like behavior was, in fact, what first provided the hint that rubbery stresses are entropic in origin. 

Finally, we note that glassy polymers close to the glass-transition temperature exhibit a strain-hardening response that is nearly purely entropic in origin and can be dealt with by employing the tools of rubber elasticity. 

A molecular interpretation of the fact that rubber-like elasticity is primarily entropic in origin had to await H. Staudinger s much more recent demonstration, in the 1920s, that polymers were covalently bonded molecules, rather than being some type of association complex best studied by the colloid chemists [1]. In 1932, W. Kuhn used this observed constancy in volume to point out that the changes in entropy must therefore involve changes in orientations or spatial configurations of the network chains. These basic qualitative ideas are shown in the sketch in Fig. 1.5 [9], where the arrows represent some typical end-to-end vectors of the network chains. 

Thermodynamic analysis of rubber elasticity enables one to resolve the elastic force into entropic and energetic components, thereby elucidating the origin of rubberlike elasticity in general. It also indicates that intermolecular interactions do not affect the force and must be independent of the extent of the deformation and thus of the spatial configurations of the chains. Since the spatial configurations are independent of intermolecular interactions, the amorphous chains must be in random unordered configurations, the dimensions of which should be the unperturbed values. ... 

---