How do ethylene molecules add repeatedly to a growing polyethylene chain?

In the context of free-radical polymerization, the growth of a polyethylene chain occurs through a series of chain-propagation steps initiated by a reactive species, such as a peroxide radical.12

The process follows these stages:

• Initiation: An initiator radical ($R·$) reacts with an ethylene monomer ($C{H}_2=C{H}_2$). This breaks the double bond and creates a new radical with an unpaired electron on a carbon atom.12
• Propagation: This new radical species then adds to a second ethylene molecule. This process repeats successively, with each new ethylene molecule adding to the growing chain, maintaining the radical at the end of the chain.12
• Termination: The chain continues to grow until the monomer is depleted or the reaction is terminated, typically through the combination of two chains or a disproportionation reaction involving the transfer of a hydrogen atom.1

Because the growing chain is flexible, the radical end can sometimes abstract a hydrogen atom from the middle of its own chain, a process known as "backbiting" or chain transfer, which results in the branched structure characteristic of low-density polyethylene (LDPE).1