Copper or Aluminum Heatsink?

Copper has got better thermal conductivity.

Aluminium - \$ \mathrm{ 200 \frac {W} {m\cdot K} } \$
Copper - \$ \mathrm{ 400 \frac {W} {m\cdot K} } \$
(from here, also here)

But thermal conductivity within the solid material is only a part of the story. The rest of the story depends on where one wants to dump the heat into.

Liquid coolant

Copper heatsink (one may also call it heat transfer block) will perform better than aluminium.

Air with forced convection

In other words, there's a fan blowing onto the heatsink. Copper heatsink will perform better than aluminium.

Air with natural convection

I've saved best for last. It also looks like it's the O.P.'s case too.

With natural convection air, the copper heatsink perform only marginally¹ better (in °C/W) than aluminium. This is because the bottleneck isn't in the transfer withing metal. When you have air with natural convection, the bottleneck is in the transfer between metal and air, and it's the same for Al and Cu.

¹ _{I might add that the marginal increase is often not worth the cost of Cu.}

This curve demonstrates the non-linear relationship between heat transfer and material thermal conductivity. The curve is generic. It applies to any application having both conduction and convection components to the total heat transfer. [Radiation is typically small and is ignored in this calculation.] The shape of the curve is the same regardless of the application. The quantitative values on the axes are not shown because they depend on the power, part size and convective cooling conditions. They become fixed for any given application and set of conditions. It’s obvious from the shape of the curve that heat transfer depends on material thermal conductivity but there is also a point, a knee in the curve, where increasing thermal conductivity produces negligible improvement in the heat transfer.
(source, emphasis mine N.A.)

Phil have already linked an ECN article which compares aluminium and copper in air with natural convection. Here's another take at it: what if we compare aluminum to a material with lower thermal conductivity (unlike copper). There is a company that makes thermally conductive plastic. It has got \$ \mathrm{20 \frac {W} {m\cdot K} } \$ conductivity, but that's a lot for a plastic. They have benchmarked it against aluminium in air with natural convection.

E2 is the plastic (source)

It's a complex question, with many factors. Let's look at some physical properties:

• thermal conductivity (\$\mathrm{W \over m\cdot K}\$)
  • copper: 400
  • aluminium: 235
• volumetric heat capacity (\$\mathrm{J \over cm^3 \cdot K }\$)
  • copper: 3.45
  • aluminium: 2.42
• density (\$\mathrm{g \over cm^3}\$)
  • copper: 8.96
  • aluminium: 2.7
• anodic index (\$\mathrm V\$)
  • copper: -0.35
  • aluminium: -0.95

What do these properties mean? For all the comparisons that follow, consider two materials of identical geometry.

Copper's higher thermal conductivity means the temperature across the heatsink will be more uniform. This can be advantageous since the extremities of the heatsink will be warmer (and thus more effectively radiating), and the hot spot attached to the thermal load will be cooler.

Copper's higher volumetric heat capacity means it will take a larger quantity of energy to raise the heat sink's temperature. This means copper is able to "smooth out" the thermal load more effectively. That might mean brief periods of thermal load result in a lower peak temperature.

Copper's higher density makes it heavier, obviously.

The differing anodic index of the materials might make one material more favorable if galvanic corrosion is a concern. Which is more favorable will depend on what other metals are in contact with the heat sink.

Based on these physical properties, copper would seem to have superior thermal performance in every case. But how does this translate to real performance? We must take into account not only the heatsink material, but how this material interacts with the ambient environment. The interface between the heatsink and its surroundings (air, usually) is very significant. Furthermore, the particular geometry of the heatsink is significant as well. We must consider all these things.

A study by Michael Haskell, Comparing the Impact of Different Heat Sink Materials on Cooling Performance performed some empirical and computational tests on aluminium, copper, and graphite foam heatsinks of identical geometry. I can grossly simplify the findings: (and I'll ignore the graphite foam heatsink)

For the particular geometry tested, aluminium and copper had very similar performance, with copper being just a little bit better. To give you an idea, at a 1.5 m/s airflow, copper's thermal resistance from the heater to the air was 1.637 K/W, while aluminium was 1.677. These numbers are so close it would be difficult to justify the additional cost and weight of copper.

As the heatsink becomes large compared to the thing being cooled, copper gains an edge over aluminium due to its higher thermal conductivity. This is because the copper is able to maintain a more uniform heat distribution, drawing the heat out to the extremities more effectively, and more effectively utilizing the entire radiating area. The same study did a computational study for a large CPU cooler and calculated thermal resistances of 0.57 K/W for copper and 0.69 K/W for aluminium.

You have a lot of good information from the users above! Please consider my answer significant and important bit supplimental to the advice you already have:

The thermal interface material (TIM) can matter as much and easily even more than the material you choose for your heatsink! I say this from experience and personally testing dozens of types and varieties of interface material. Your budget, attachment methods, and other design perameters will probably narrow your choices a specific type of TIM. For example: a paste requires the heatsink to be mechanically secured and an adhesive does not. Some materials are messy and difficult to use but perform well and some stuff out there is almost worthless in its performance and may or may not be easy to use.

I would say with a lot of confidence that the TIM you use can easily matter way more than if you use copper or aluminium. Not in every case but the performance differences can be surprising.

Seeking popular and well reviewed materials for CPU/heatisinks can give you some good options to choose from.

Good luck!