You said "giant" sunflowers... But they're in pots? How big are the pots? ( and yes, as you found out, they definitely need drainage)
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Ants on the sunflowers indicate a pest insect. Not the ants themselves -- ants herd them and harvest sweet excretion from them. Aphids are the most commonly known ant "herdbeasts" but on sunflowers I would look for Lacebugs. Mine are also being attacked by Treehoppers.
Lacebugs occur in swarms, lay eggs and soon becomes an infestation. You'll start seeing tiny white spots on the surface but they will usually be on the underside of the leaves. Initially, just brushing them off with wet hands and squishing the ones that get stuck to your hand is sufficient. SOAPY water spray can be used but you must be sure to use soap and not detergent. Sudsing a bar soap in a bowl of water is an easy way to make some that your children might enjoy
( but don't let them make it too concentrated, just enough to suds is sufficient!).
-- I keep thinking of things to add to this post!
* Yes, you can wash off the bugs stuck to your hand rather than squishing them Use the bowl of sudsy water.
* Fatty soap water clogs insects' breathing orifices which are normally hydrophobic and remain open even when wet. Also, as it dries, the salts dehydrate soft bodied insects.