I have my own theory on the "gives up heat" thing. One aspect is the misperception that thermal capacity has an effect on steady state heat transfer. In reality, all this means is that a block with lower thermal capacity will cool more quickly when you turn off your PC. Whoopey!
More importantly, this is a true statement for air-cooled heat sinks, but not for the reason most people think. A heat sink needs to "handle" three heat transfers. It needs to receive thermal energy from the chip, transfer through its own material, then convect it to air. Convection by nature is much less efficient than conduction through most metals. This means that surface area tends to matter a lot more than anything else.
By virtue of its lower density, you can create a lot more surface area with a given mass of aluminum than copper. That's pretty simple to understand. What's not obvious, and what takes a little thinking to understand, is that on a fixed weight basis the conduction of aluminum is better than copper. Just as an approximation, say copper beats aluminum by 2:1 on conductivity. In contrast, aluminum beats copper by over 3:1 in density. For the same weight, you can have more than three times the cross sectional area for conduction. Since delta_T is a function of both conductivity and section area, a given weight of aluminum will have a lower delta_T in conduction than the same weight in copper.
If you understand all that, you'll realize that aluminum really does "get rid of heat better than copper" for air-cooled heat sinks.
If you're extra-special smart, you'll further realize that the die represents a fixed area for heat transfer. This means that you can't take advantage of additional section area when touching the die. This is why early high performance air heat sinks migrated toward copper bases with aluminum fins or pins and why as a general rule all copper tends to beat all aluminum.
When it comes to water, convection gets much more efficient than air. The benefit of low density largely evaporates (pun intended) and all copper offers better performance than all aluminum.
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