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Would it be a problem to use a lead solder paste to solder lead-free BGA package parts? Are there contamination/compatibility issues? Does it affect the physical connection integrity between the board-component? How about using a lead-free solder paste for soldering components that contain lead? Which of the two is more reliable?

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Would it be a problem to use a lead solder paste to solder lead-free BGA package parts?> Are there contamination/compatibility issues?

Obviously there is a contamination issue, because your assembly is no longer lead-free, and couldn't be sold in jurisdictions that restrict lead content.

Does it affect the physical connection integrity between the board-component?

Conceivably you could also have a reliability issue, because the lead-free and tin-lead solder will not mix evenly during the reflow process. This will produce various alloys in different parts of the solder joint. Very likely there will be residual stress in the cooled joint, and stress is associated with tin whisker formation (and other issues) in the lead-free part of the joint. Whether this is a real issue or just paranoia, I'm not sure, but it's not something I would bet my job on.

Another issue is that in order to heat the assembly hot enough for the lead-free solder to melt, you will have to heat the tin-lead portion hotter than it's designed for. This can make the flux components of the solder vaporize too quickly, causing voids in the joint.

How about using a lead-free solder paste for soldering components that contain lead?

If the part is not designed for the higher temperature of lead-free reflow, it could cause reliability problems.

Also the mix of materials could still cause the stress issues, and mismatched temperature profile issues I mentioned before.

Which of the two is more reliable?

I would rather not do either.

If you can afford it, it's possible to have a BGA re-balled with new solder balls that match the assembly process you're going to use.

If you have no choice, or if you're doing a short-life project where reliability isn't critical, I'd probably do lead-free balls in leaded paste, because a greater percentage of the solder in the joint is designed for the temperature profile you're going to use.

Further Reading:

You can find more information about mixed-material soldering in the book chapter, "Backward and Forward Compatibility" by Jianbiao Pan, Jasbir Bath, Xiang Zhou, and Dennis Willie in Lead-Free Soldering, 2007. 173-197. Also available online.

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Aside from @ThePhoton 's thorough answer, the materials science concept of the eutectic point should be considered. Solder alloys for electronics exhibit behavior that is close to eutectic, meaning that as temperature rises the phase transition is very close from all solid to all liquid, without going through a slurry in between. Solders for other purposes might not have this sudden transition.

Subtle variations in alloys can have larger impacts on this behavior than one would think. You change something a bit, and the whole liquification point can rise, maybe even high enough that standard temp profiles don't work

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