Ions are soluble in water and not in organics such as styrene. To get organics together with water soluble molecules, we run reactions in mixtures of water and organics called emulsions. Polymerization in these mixtures is called emulsion polymerization. You have shaken styrene with water on several occasions, and while you are shaking, the styrene is broken up into tiny spheres that have a large surface area at which styrene and water are in contact. When you stop shaking, the tiny spheres coalesce and become large enough for gravity to exert more force on them than molecular motion.
Gravity pulls the more dense water under the bulk styrene.
When this is shaken with styrene and water, the ionic part is held more strongly by the water, so it does not go into the styrere spheres. The 12-C hydrocarbon part has no dipoles, so it can't hold onto waters as strongly as they hold each other. The total attraction between the hydrocarbon part and styrene is quite large, so the hydrocarbon part is down in the styrere spheres with the ionic part on the surface of them. Like charges repel, so the spheres can't come together to coalesce.