I'm a bit confused here. For the sake of argument, if both metals of same density and mass were in a controlled environment and both were required to dissipate heat, which would go faster? I understood that Aluminum would dissipate faster?
Interesting question. It is not very useful to compare Al and Cu and say they have the same density- it defeats the purpose of a comparison. Let's say they are both the same geometry. It is important to distinguish between short-term and long-term (steady-state) performance. It is also important to define "dissipate." Do you mean watts from the chip, or watts from the heatsink?
In the very short term, the Cu sink will probably take more watts from the chip because it has more heat capacity and better conduction. it is possible for the Al sink to dissipate more heat to the air at first, though, because it will heat up faster due to its low heat capacity.
In the long term, steady state solution heat capacity does not enter into the equation (it is like a mass in a physical system, the conductivity like a spring). Heatsinks usually have a maximum temperature determined by the processor they are cooling. The tradeoff is that the hotter the sink gets, the more heat it can dissipate. You can imagine then that an ideal heatsink will be uniformly hot, enabling maximum dissipation to the air across its entire geometry.
Cu will almost always work better because its high conductivity will keep the heatsink temperature more uniform, while the Al sink will tend to be cooler at its extremes. We choose Al over Cu when weight, cost, or corrosion is a concern.
to evsp341995: I have had a hard time machining copper- it is very gummy and you need to get the right alloy for it to be easy to work with. Still, if you have the experience and don't mind the extra few ounces, it is probably the superior material. I think a lot of manufacturers treat their Cu heatsinks with a thin film to protect against corrosion- you might want to look into this as well. No coatings are necessary for Al and as long as you choose a 6 or 7000 alloy it will machine nicely.
I think attaching the sink to the bottom, probably through a thermal interface pad, is a good idea, as long as your lap doesn't mind the extra heat. It is hard to measure spaces when you can't get a micrometer in there. I would put shims in between the heatsink and bottom and just keep adding them until you feel some resistance putting the bottom back on. Measure the shims and plan for that distance.