Disclaimer: I don't even have an e-cig yet, just doing lots of research before I buy (hopefully next week!). So, all I know is what I've read and seen in videos.
Before my current career I was a chemist and have a MS in Analytical chemistry, but haven't' been using that for a while, so I've forgotten a lot. However, there's many a chemistry lab that resulted in burnt on black stuff on glassware and there's lots of ways to try to get it off, and different things work depending on what it is. Most of them have been tried on atomizers from what I can see, but here was the general order before giving up on getting it clean:
1. Water. Just wash it. Looks like limited use here. Mostly would dissolve salts and other stuff that is water soluble. The main problem seems more oily or solid.
2. Soap and water. Looks like it's had some success, and generally good for oils and things that dissolve in water.
3. Acetone. Nail polish remover. Will remove paint sometimes. Good for things that won't dissolve well in water, oils and heavier organic compounds. Nicely volatile and shouldn't leave a residue itself, though if it did dissolve something and wasn't washed away that will stay behind. Not much good for salts. Also seems to have had limited success.
4. Acid. The part that makes it acidic may react with the residue and make it something that will dissolve in water and/or the cation of the acid may react (possibly why phosphoric acid and trisodium phosphate coffee pot cleaner are both reporting success, and it's in most sodas as well). Good for some organics and many metals.
5. Acid with acetone. Combine reaction with better organic dissolving power. We usually used 10% acetone with 10% nitric acid in the lab, no idea if mixing acetone and vinegar or lemon
juice, or citric acid for canning would do the job.
6. Base. Same idea as the acid, but different reaction that may make the stuff more soluble in water. Baking soda, coffee pot cleaner, some drain cleaners (especially lye, but bad for aluminum) would all be bases readily available, and look like they've been tried too, with the usual mixed success.
7. Base with acetone. Same idea as acid with acetone, try to make it more soluble, but put some extra organic dissolving power in. Same proportions. I haven't seen this one tried and would suggest baking soda and hot water (near boiling) until no more soda will dissolve, add nearly equal amount of acetone, soak a few minutes, water rinse and dry. Lemme know if anyone tries it.
8. Lots of scrubbing. Just mechanically scrape the stuff off. For glassware, really part of every step. For atomizers, just not viable since the coil is pretty delicate and small and hard to get to.
9. Give up. Sometimes the time it takes to clean isn't worth the cost of a new one.
Just figured I'd toss that in the information pool.