Your arms won't get as big as they potentially can if you don't do direct arm work.
You should do all types of curls so you work the brachialis, the bicep, and the various muscles in the forearm.
Bicep: Suppinating Dumbbell Curls (you turn your wrist while you do them), Wide-Grip EZ-Bar Curls, Wide-Grip Rows, Wide-Grip Chins,
Brachialis: Hammer Curls (don't turn your wrist), Close-Grip EZ-Bar Curls, Close-Grip Rows, Close-Grip Chins
Forearm: Zottman Curls (Hard to explain on paper), Reverse-Grip Curls, anything where you use your grip strength.
A lot of people only do wide grip barbell curls because that's where they're strongest and only do suppinating Dumbbell Curls because that's the way they see everyone do them. You should do a variety of arm exercises in order to get the best results BUT if you're just trying to look like a badass and you want to work all the parts of your elbow flexors then do Hammer Curls and Wide-Grip Barbell Curls. Hammer Curls are cake. I can do pretty strict Hammer Curls with 65lb dumbbells and my arms are tiny.