Hi,
The formulas used to brown stub and Damascus barrels are less important than how they are applied. In general, you want any acid solution to be dilute even if you buy it commercially, dilute it with something like alcohol. To do a first rate job, the barrel should be very well polished. Degrease the barrel thoroughly and wear rubber gloves when handling the barrel. Acetone works well for that. Apply a thin coat of browning solution on the barrel with a swab but just swab once and evenly. Don't rub back and forth and put on a heavy coat. Let the barrel sit until it develops a thin coat of rust but no more than 4 hours unless you are in a very dry location. It is best to have a soft stainless steel buffing wheel such as those made by Grobet. These are very soft, more mop like than bristle like. Fine Scotch Bright pads will work in a pinch. If using the wheel, turn at very slow rpms so no heat develops and card the barrel (remove the loose rust) with the wheel. Then apply the next coat of dilute browning and repeat. However, just let the barrel rust for 2 hours or so and card again. You may have to repeat the cycles 20-30 times to create the color you desire but be patient. The goal is very fine grained rusting that leaves a smooth shiny translucent surface behind rather than a coarse, mat surface. After the final carding you can do several things. Wash the barrel in water and dry it and then simply heat the barrel a little and apply beeswax and buff it to a high polish. You can also wash the barrel with scalding (NOT BOILING) water, dry it and let it after rust a little, then a final carding. That will bring out deeper red colors and then heat and buff with beeswax. Finally, you can wash or sit the barrel in boiling water, remove, let dry and then gently rub it with fine Scotch Bright or 0000 steel wool, heat and buff it with beeswax. The boiling will turn the browning to a blue-black color.
Here is a barrel done by Pete Mazur, one of the best gun finishers around, that shows the fine, smooth, translucent finish found on high quality English guns.
dave