If you have a Lowes hardware store nearby, I would suggest you pick up a can or two of Dupont Teflon multi use chain protectant. I use it on all my chain drive bikes as well as my Max IV.
Here is a write up on it:DuPont Teflon Chain Saver - DuPont Teflon Chain Lube - webBikeWorld
BTW, it stays put on the chain and it will not mess up your tub.