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Polymerization is a very important chemical process that is used to produce all sorts of plastic products for everyday use... e.g.: nylon, paint, synthetic rubber and even chewing gum! The process really involves the joining of small molecules to form long chains that keep repeating.
PVA glue contains polyvinyl alcohol (also known as PVA), a chemical which is made up of carbon, hydrogen and oxygen atoms. When borax is dissolved in water, borate ions are released. Putting the two together, borate ions are able to cross-link the long PVA chains by forming weak ionic bonds between them. These bonds are not strong enough to form a complete solid, but just enough to thicken the mixture. This is called a hydrogel, and it is like a matrix, trapping water within its network. The more water you add to it the more it absorbs, even past its maximum capacity. At which point the weak ionic bonds can't hold on anymore, the network breaks down and all you're left with is a soupy mess.