Hi Snapper,
There are lots of "secret" formulae out there (Gaddy's for example) but it is more the prep of the barrel and how the acid solution is applied than the solutions themselves. Almost any good browning solution will work. I use Wahkon Bay but have also used TOW True Brown. I dilute the solution almost 25% with distilled water and an additional 10-15% with isopropyl alcohol. I polish the barrel with 600 to 1200 grit wet sand paper then burnish it with 0000 steel wool. I take care in that process not to lose any markings. I degrease the barrel and suspend it by the ends. With a cotton swab I apply the diluted browning sparingly making only a single pass. Then I let the barrel rust for no more than 6 hours depending on temp and humidity. I just want a very fine coat of rust not big scales. I card the barrel using a soft stainless steel wheel by Grobet turned on my lathe at no more than 600 RPM. I don't want any heat build up. Then I coat the barrel again and let it rust for no more than 4 hours before carding. This is a very slow process and may take 20 or more cycles to achieve a good translucent brown that highlights the pattern. That process produces the usual dark plum colored brown. However, I usually want a redder color so after a couple of cycles I card the barrel and then pour scalding (NOT BOILING) water over it , let it dry, and apply the next coat of brown. That produces a redder color. To finish the process, I wash the barrel, dry it and warm it with a heat gun. When warm I apply Sutherland-Welles tung oil mixed with beeswax. I let that dry and buff it.
dave