# Why does thermal resistance go down as temperature goes up?

Here is the thermal resistance data for three speaker coils disengaged from the speaker cone. Any ideas? I would think it would be a horizontal line.

http://en.wikipedia.org/wiki/Thermal_resistance Absolute thermal resistance of the part, K/W

edit: I also want to point out that the same trend occurred with the loudspeaker functioning as a normal speaker (before I tore out the voice coils to test separately). The reason for this test was that I thought the drop in thermal resistance was due to unwanted cooling effects (actually desirable, but not at the moment) from diaphragm movement, but the drop in thermal resistance vs temperature is still there.

edit: I am going to run the same experiment using a 1/4W resistor today and report my findings.

For x_C = 40, 60, 80:

1. Heat oven to x_C
2. Measure electrical resistance
3. put .2W of power into resistor
4. Let the resistance measurement settle (thermal time constants)
5. Measure increased electrical resistance
6. Calculate resulting change in temperature of resistor
7. Calculate resulting thermal resistance (dT/dR)
• That is a good question since usually resistance in metal rises with temperature. – mwengler Jul 24 '12 at 23:26
• I should clarify that it is a plot of thermal resistance (K/W) vs Temp. Edited the question – MaxRunFast Jul 24 '12 at 23:31
• It is not clear what you are asking about. You write that you measured electrical resistance ($\Omega$) then calculate "thermal resistance" in $K/\Omega$. Presumably this is actually the change in electrical resistance with temperature. However your graph of "thermal resistance" is in different units again ($K/W$). So it is not clear what you have done in your experiment. – sammy gerbil Mar 31 at 21:53

As you increase the temperature, the electrons, which are a cold Fermi gas, get more excited. Only a thin skin of electrons around the Fermi energy, of width kT, can do anything at all, like conduct heat. As the temperature increases, the electrons can conduct more heat.

The heat conductivity of electrons is greater than the heat conductivity of phonons, and together these account for the entire thermal conductivity. What is going on is that as the temperature goes up, the higher thermal conductivity electrons are carrying a larger fraction of the heat, and this makes the thermal conductivity go down.

In the absence of electronic thermal conductivity, for an insulating material, the thermal conductivity would go down with temperature, and this is also true for the thermal conductivity of just the phonons in the metal. But the electron contribution leads to this otherwise paradoxical effect.

It always helps to remember that a metal is never classical, the electrons are always quantum. A metal is a like a gigantic chemical bond involving all the atoms in a metal nonlocally, this is the conduction band, and the shared electrons have classically paradoxical properties. This is why the Drude model is wrong and the Fermi model is right.

Hmmm. This really is an interesting case, because according to the equation that @mwengler posted a link to, the thermal resistivity is positively correlated to temperature. However, I wonder if what you're observing has something to do with the fact that at higher temperatures, the molecules begin to move much quicker as they are gaining kinetic energy. This increased molecule motion is sure to cause more bumping and collisions between all the molecules.

I would imagine that as the number of collisions between the molecules increases, the molecules in the metal will begin to be more spread out, thus decreasing the density per unit mass of molecules in the metal. The lower the density per unit mass of molecules, the less resistance there will be. That's just my educated guess.

• This isn't about resistance, it is about thermal resistance, and classical pictures like the ones you give are completely wrong for both. – Ron Maimon Jul 25 '12 at 4:30
• Then please, what do you think it is? – Ryan Christman Jul 25 '12 at 4:33
• Thermal resistance doesn't have to go up with temperature, like electrical resistance usually does. Electrical stuff is only carried by electrons, heat is carried by both phonons and electrons. My initial guess was that at higher temperatures, you change the proportion of heat that is carried by electrons vs. the heat carried by phonons, so that electrons carry a larger fraction of the heat, and the electrons just conduct the heat more. I need to verify it with data and estimation before I post it as an answer. Both the phonon and electron density change with temperature with different laws. – Ron Maimon Jul 25 '12 at 4:44
• I see. Well, excuse my elementary understanding of thermal resistivity as compared to yours, but my area of expertise is in atmospheric physics...I'm just trying to apply theories I've learned. – Ryan Christman Jul 25 '12 at 4:52
• I'm not an expert on this, this is just folklore. Please don't feel insulted, I just was pointing out an error, not trying to say you were somehow faulty. You had a classical picture, it's just not right--- the classical motion is probably not what is going on, because I think heat conduction is described by phonon collisions at these low temperatures, not by non-quantum atomic collisions. The heat expansion argument in the second paragraph is what I was objecting to the most--- the density change is not associated with less resistance, neither heat nor electrical. – Ron Maimon Jul 25 '12 at 5:30

I'm not an expert on it so the explanation below is my own. Electrical resistance of metals rises with temperature. But electrical resistance arises as a result of "collisions" between electrons and lattice ions. For electric current it means turning of electric current into heat.

But for heat conduction it is just like turning heat current into heat. Collisions between ions and electrons assist heat conduction. That's why thermal resistance decreases with temperature.