Elastomers statistical thermodynamics

According to the statistical thermodynamic approach to be developed below, each conformation that a network chain segment may take is equally probable. The number of such conformations depends on the end-to-end distance, r, of the chain, reaching a rather sharp maximum at tq. The retractive force of an elastomer is developed by the thermal motions of the chains, statistically driven toward their most probable end-to-end distance, Tq. 

When an amorphous cross-linked polymer above Tg is deformed and released, it snaps back with rubbery characteristics. The dependence of the stress necessary to deform the elastomer depends on the cross-link density, elongation, and temperature in a way defined by statistical thermodynamics. 

The properties of elastomeric materials are controlled by their molecular structure which has been discussed earlier (Section 4.5). They are basically all amorphous polymers above their glass transition and normally crosslinked. Their unique deformation behaviour has fascinated scientists for many years and there are even reports of investigations into the deformation of natural rubber from the beginning of the nineteeth century. Elastomer deformation is particularly amenable to analysis using thermodynamics, as an elastomer behaves essentially as an entropy spring . It is even possible to derive the form of the basic stress-strain relationship from first principles by considering the statistical thermodynamic behaviour of the molecular network. 

A network is an ensemble of macromolecules linked together, each of them rearranging its configurations by Brownian motion. Classical thermodynamics explains the behavior of elastomers with regard to force, temperatme, pressme, and volume, but does not give the relationship between the molecular structme of the network and elastic quantities such as the moduli. Therefore, statistical mechanics was introduced in the 1940s (16,86,87,107-109), and its theoretical predictions were tested (110-112). Because of the complexity of network structures, two models based on affine and phantom networks were studied. The cross-linking points... 

---