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In addition polymerisation, a double bond in the monomer is broken and a long chain of polymers is formed. How is this bond broken (where is the energy input) and how is the long chain formed?

(something to do with free bonds?)

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During radical polymerization, we use molecules called initiators $I$ for forming free radicals. The molecules of initiators contain at least one fragile bond that can easily dissociate upon heating, and yield free radicals. $$I\rightarrow2A^.$$ The as-formed radicals have enough energy to attack the monomer molecule $M$ and transfer the radical to the monomer molecule:$$A^.+ M\rightarrow AM^.$$ More precisely, one electron of the double bond in the monomer forms a covalent bond with the free electron of the radical $A^.$. The other electron of the double bond is now free. We have a radical on the monomer $AM^.$.

The radical monomer attacks another monomer molecule, and the polymerization reaction propagates: $$ AM^. + M \rightarrow AMM^. $$ In the above reaction, one electron of the double bond in the monomer molecule forms a covalent bond with the free electron of the radical $AM^.$.The other electron of the double bond is now free $AMM^. $. and so on..

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Addition polymerisation is usually classed as chain growth. Whilst free radicals are usual and the mechanism is by homolytic fission, cationic and anionic initiation are both valid where propagation is by heterolytic fission. For example, the polymerisation of cyanoacrylate to give superglue is usually initiated by trace water and would generate an anionic chain in the process. The chain would only grow by encountering and reacting with monomer. Step growth polymerisation e.g. condensation type, have multiple reactive oligomers or chains which can combine without the reaction terminating. This cannot happen in chain growth radical polymerisation as termination would occur and simply wouldn't happen for ionic polymerisation mechanisms.

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