Yes, copper makes a better heatsink. In order from lowest to highest it is Al > Cu > Ag > diamond.
Yes Al will dissipate its heat quicker than Cu, but Cu will pick up more than Al (a gross oversimplication). So when you first turn on the PC, an Al heatsink will heat up quicker and start moving heat away sooner, but as the system approaches its thermal equalibrium (roughly about the time that Windows starts booting) the Cu heatsink will be moving more heat.
That's why some manufacturers make heat sinks with a Cu core and Al fins. In theory the Cu core will pick up the heat better and then the Al fins will disperse it quicker. In pratice it doesn't work this way, but I won't go into details here.
Silver (Ag) is even better than Cu. Diamond is the best of all. It is many times more efficient than Ag. Problem is that both of these materials cost alot, and Ag isn't that much more efficient that the extra cost is worth it.
Past diamond, there's graphite, but that will only conduct heat along one plane, plus it's too soft to use in a heatsink. After that its a certain form of liquid helium that only exists within a few degrees of absolute 0. After that there's nothing.
So realistically the best is silver. Pratically the best is copper. Al, Cu and Ag aren't that far apart though. When you start getting empirical, other factors greatly outweigh the construction material. It's better to have more/thinner fins, a greater surface area and a faster fan that it is to chose Cu over Al. If everything else is the same, you might see a 1 or possibly 2 degree difference between Al and Ag.
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