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Presenting the polymer capacitor by Nippon Chemi-Con Corporation. Link here (PDF)
Conductive polymer aluminum solid capacitors, which will be abbreviated to “polymer capacitors” in the following, have been recently extending in their applications. The polymer capacitors as well as conventional aluminum electrolytic capacitors are featured by large capacitance and excellent bias characteristics which multilayer ceramic capacitors can never compete with. In addition to these advantages that aluminum electrolytic capacitors have as well, the polymer capacitors have extremely low ESR characteristics. Regarding ESL, which it is determined by inside structure and terminal configuration of the capacitors, by making structural improvements the polymer capacitors have low ESL compared with the conventional aluminum electrolytic capacitors.
Also, concerning the dry-out of electrolyte in service life and the changes of characteristics at a range of low temperatures that have been regarded as disadvantages in aluminum electrolytic capacitors, the polymer capacitors have realized very high reliability and superior low temperature characteristics by using solid polymer materials as an electrolyte.
The polymer capacitors having these features have been demonstrating their capabilities in various fields. Their common uses are described in this application note.

Stability over time, low ESR. Hopefully this will be a viable alternative to tantalum which is a conflict mineral.
“Peter says: Stability over time, low ESR. Hopefully this will be a viable alternative to tantalum which is a conflict mineral.”
Short Answer: As long as you know what you’re doing, there’s a good chance you can find a drop-in solid aluminum-polymer replacement for tantalum-polymer or nobium-polymer capacitors. These days, you would be hard pressed to not be designing with solid Al-poly caps instead of wet Al-electrolytic caps. Replacing tantalum with Al-poly may not be so simple, especially in some signal-chain applications.
Longer Answer: Solid tantalum (tantalum-polymer) caps will have longer life than solid aluminum polymer caps (solid Al-Poly dielectric material has a couple of failure mechanisms caused by thermal degradation and/or material shrinkage due to age), however Al-poly caps have longer lives compared with wet-Al electrolytic and Al-poly-hybrid caps. Solid tantalum caps are often physically smaller than solid Al-poly for a given capacitance value. Tantalum capacitors have similar temperature stability compared with solid Al-poly. Tantalum capacitors have a self-healing mechanism. Tantalums can usually handle higher reverse voltages. Tantalums will have higher ESR compared with solid Al-poly. Solid Al-poly caps have higher ripple-current ratings. Al-poly have higher leakage voltage specs. Tantalum caps are more expensive than Al-poly caps. Al-poly caps can handle higher voltages. Both tantalum and Al-poly caps do not display microphonics. Gee, there are a lot more things to consider, but I can’t remember any more off the top of my head. (Use at your own risk – corrections and/or additions welcome.)
The way I see it, hobbyists don’t really need tantalums anymore, but some still use ’em, even one or two boards shown here on the DP blog used tantalums to keep the board size small I guess…
I wonder if the likes of Greenpeace are still monitoring coltan/tantalum and the industry’s effort to go non-conflict, because if you look at SSDs, many use tantalums (double-layer supercaps can dry up, multilayer ceramics can crack and fail, etc) and presumably manufacturers have a documentation chain to prove their stuff is non-conflict. Still I wonder if they are trustworthy, noting that Volkswagen’s Audi promo is “Truth in Engineering”…
The two places where I worry about caps are unlikely to ever switch to polymer caps. One is the PC power supply, even the best low ESR electrolytic caps will get to end-of-life and need to be changed. Maybe high-end PSUs have better caps, but their price is high-end too. The second is ADSL routers, the cheap ones often run plenty warm and can kill the electrolytic cap in the power section in a few years.