Many commercially important polymers that you may use in your daily lives, such as epoxy glue, rely on ring opening polymerization. Ring opening polymerization is characterized by using a monomer that is cyclic (a ring) and the mechanism can proceed by radical, anionic, or cationic pathways depending on the specific monomer and initiator. During polymerization, the ring opens and creates a linear polymer. In many ways, ROP is familiar to our other mechanisms, and we can use it to make many of the same polymers (and some new ones). It still requires initiation, it still goes through propagation and termination, it still follows the same rules for electron donating and electron withdrawing groups as we learned for cationic and anionic polymerization.

Introduction to Polymers, Third Edition, CRC Press, 2011.
So why use a cyclic monomer instead of a non-cyclic one? Well, one important thing going for cyclic monomers (with small rings) is that they have ring strain, which makes them somewhat less stable (and hence more reactive) than non-cyclic monomers. For larger rings, they have less ring strain, but more steric repulsion between the ring substituents, which also makes them unstable.
Let's first look at a cationic ring opening polymerization of epoxides, which are very common cyclic monomers, and are used to produce polyethers. We can use the same initiator molecules. Note the use of double-headed arrows for moving pairs of electrons. We see some equilibrium structures here as well that are new - the cationic intermediates are in equilibrium between the ring open and ring closed state. Termination can occur by rearrangement, just as we saw before in cationic polymerization.

Introduction to Polymers, Third Edition, CRC Press, 2011.
We can also consider ring opening polymerization of epoxides by anionic polymerization, as shown below.

Introduction to Polymers, Third Edition, CRC Press, 2011.
Notice that again we see the same kinds of initiators as was typical for anionic polymerization. We have added an "R" group onto the epoxide to help stabilize the anion, and because of this substitution, the nucleophilic attack will occur on the least hindered carbon. We also see no inherent termination process, unlike cationic polymerization.
In addition to epoxides, another common class of cyclic monomers are lactones:

Introduction to Polymers, Third Edition, CRC Press, 2011.
PROBLEM 1
Of the monomers shown below, which would yield a polyether when polymerized by ring opening polymerization?

- Isotactic polybutadiene
- Syndiotactic polybutadiene
- Atactic polybutadiene
ANSWER 1
A. Isotactic polybutadiene
Isotactic polybutadiene would yield a polyether

PROBLEM 2
Of the monomers shown below, which one would you choose to make this polymer?


ANSWER 2
D.
