No, I don't.
Actually, I guess I kinda do.
Before I run the water for dishes, I move the dishes that have collected and clean the sink with dish soap and hot-tish water to get anything that may have been left on the dishes and dropped into the sink. But I don't use a special cleaner. I let that water drain then rinse the sink with hot water. Then plug it and run water for dishes. After dishes are done, I just let the water run out and rinse.