The application of patina to solder and other metals consists of three stages, cleaning, application, protection.
Clean the solder bead
Both flux residues and contaminants are left after soldering. Cleaning can be a combination of cleaning with abrasives and also with alkaline materials. Baking soda – sodium carbonate – can be used to neutralise any of the acidic flux left. It bubbles in the presence of acid, so gives a visual check on the amount of acid left. Since it is an alkaline substance, it will react with the acid of the patina to weaken the reaction. It is important to rinse the baking soda off with plenty of clean water.
Abrasive cleaning with materials such as steel or copper wool can be used. Some advise against this as metal residues can be left. However the abrasive scrubber used in scrubbing pots can be used to finish the cleaning without leaving metallic residues.
Apply the Patina
As patinas work by chemical reactions, they are specific to various metals. Those designed for solder work by reaction with the tin mainly, those for lead obviously react with lead, and those for zinc react with zinc - while each of these may work with other metals, they work best with the metal they are designed for. Too much rubbing while the patina is wet simply interferes with the action of the acid. Application of enough patina to allow it to "soak" - actually time to react with the metal - is much better than rubbing. To repeat, it is a chemical reaction, not a buffing process.
Once the patina is dry, you can use the baking soda solution again to neutralise the remaining acid on the metal. Rinse well. rub gently with a dry cloth to assist the drying, do not scrub. When dry again you can wax the metal.
Protect the Patina
Some allow the waxed piece to sit for 24 hours and then re-wax it. The applicator should be a soft object such as a cotton bud or soft cloth. Do not polish it until it no longer is showing black. Rather allow it to sit undisturbed for several days. The patina will get darker and smoother over time. Don't worry too much about getting it pitch black, or birght copper the first day.