Greetings from Nebraska.
KenB259 is spot on, I usually turn my blanks as fast as my lathe will go typically 3600 RPM if one can believe the tachometer on the lathe. Also as already noted by hewnch, not all "Acrylics" i.e. Plastic Blanks are the same. Some are true Acrylic, but some are Polyester, Urethane, epoxy, etc. and they all have some unique characteristics, the most pertinent of which is their level of brittleness. The worst in my opinion is Inlace Acrylester as it is very brittle and "chippy" when it is being turned.
Try the suggestions, speed up your lathe, make sure you have a very sharp tool, and take light passes. I have had the best success using a cutter with a negative rake grind. Good luck finding the right combination as the Plastics can result in very beautiful pens.
Regards, Dave
PS A final word of caution, for plastics that are translucent, you will have a much more satisfying result if you paint the inside of the blank after drilling so that the brass tube doesn't show through - unless that is a look that you are going for of course.