Structure of an ideal rubber

The ideal, classical, rubbery properties are displayed by polymers cross-linked by valence bonds (main-chain bonds or sulphur bridges, see Introduction, Fig. 0.1) these are termed chemical cross-links. Physical cross-links are also important in many useful rubbery materials. In physical cross-linking the chains are not chemically attached one to another, but are effectively pinned together in one of three ways  

The understanding of these physically cross-linked materials stems from the theory of rubber elasticity, which is based on a chemically cross-linked polymer, and it is to this simple idealized model that we confine attention. 

An ideal rubber consists of flexible cross-linked polymer chains undergoing violent liquid-like motions. No matter what the type of cross-linking, physical or chemical, the elastomers all have this in common the macromolecules between cross-links undergo extremely rapid molecular movement. 

The behaviour of the specimen as a whole follows from an understanding of the behaviour of a representative chain. Imagine it detached from the network with one end placed at the origin of a coordinate system (see Fig. 3.2). It will be assumed that its randomness is exactly reproduced by the Gaussian theory ( 2.9). That is to say, with the passage of time the molecule changes shape in a truly random manner. It follows from eqn (2.N.7.1) that 

For this detached Gaussian chain, the mean-square end-to-end distance averaged over time is (see eqn (2.N.7.5)) 

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