But I concluded that you can't let third-party inks dry out in a cartridge and then refill without problems. In fact, I found it almost impossible to purge cartridges that had refilled ink dried out in them for a year or so. Purging old ink is not a problem either.
By the time I came up with the Freedom Refill Method, I was starting to use these old previously refilled cartridges and I started having problems with foaming which prohibits a successful refill. I think the most important thing is to not let your cartridges dry out.
On the contrary, I regard Freedom filling as almost the only method that can guarantee proper ink flow through the sponge, and that's why I still use it. People might be using the term "foaming" to refer either to foam in the syringe or to air in the sponge. If you remove the air correctly, you
will not have a problem with air in the sponge or anywhere else.
Let's consider how to get rid of the air. If you have correctly removed it from the ink chamber, then the only air left in the cartridge will be in the sponge and above the sponge. This too is reduced in stages by multiple strokes. Consider just the last stroke. Assume there is, let's say, 3 mL of air left in the sponge chamber (and none in the ink chamber because you have removed it), if you expand that 3 mL to 30 mL in the syringe and replace the vacuum with liquid ink, that will leave at most 10% air in the sponge (i.e., 3/30 = 10%). Actually, it will be less than 10%, because most of that remaining air should go to the space above the sponge. That may even be better than new.
But now that I think about it, the last stroke may be the most important one. If you only expand the syringe from, say, 50 to 60 mL on the last stroke, you are likely to leave a bit more air in the sponge. Maybe not too much, but that might be an area of concern, and it might be possible to make a mistake here. It might be better to empty the syringe of most of the air before the last stroke.
I am quite convinced that if people are having trouble with air in the sponge, it is because they have a leak. I
know how difficult it is to seal around the vent. If you tape over the label, air is likely to leak around the edge of the label, and this air will go into the sponge. (Note added later: PeterBJ says aluminum tape works well.) In addition, the label itself may leak where you don't have it covered. I first realized this when I noticed that my cartridges would not hold a vacuum when I pulled back on the syringe.
I think some people are also using an adapter that leaks at the fill port. This won't directly cause air in the sponge, but it will reduce the efficiency of the syringe.
Regarding old, dried cartridges, dye ink is quite soluble. I just dissolve it with Pharmacist's solution and drain the cartridge. Except in special cases* I don't worry about thorough flushing. To avoid diluting the ink, I usually give it another rinse with ink by filling the sponge with ink and draining it again before refilling.
With pigment ink you probably can't remove all the residue, although I have not experienced any problem with pigment black. For my HP564 cartridges, pigment black is the only bomb-proof cartridge. It's very easy to fill, and I've never had any problem. I avoid InkTec pigment black as you suggested, but IS and OCP seem fine.
* The yellow ink in some Canon cartridges will gel under some conditions, and that is a special case in which you have to remove
all traces of old ink from the cartridge, and probably from the head.
