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We can form covalent bonds between the polymer molecules, and if we do this the material will become much more rigid because the chains are no longer free to move apart. The more cross-links between chains, the more rigid the rubber until eventually the polymer is so cross-linked that it is no longer rubbery because there is no flexibility of the chains between the cross-links. Goodyear’s vulcanization process produces a controlled amount of cross-linking. The sulfur reacts with the double bonds and forms sulfur bridges as cross-links between the chains, resulting in a huge three-dimensional network as you can see here.

(a) Unvulcanized natural rubber molecules have few if any cross-links.

(b) Vulcanized rubber has a network structure with cross-links.

(c) Vulcanized rubber on stretching.
The covalent cross-links survive the stretching and help the molecules to spring back once the tension has been relaxed. This type of network molecular structure lies behind the explanation of why rubber is rubbery.
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