# Is thermal conductivity correlated with speed of sound in metals?

Right off the bat I should say: thermal conductivity and electrical conductivity are definitely correlated in metals.

But my question: is thermal conductivity correlated with speed of sound in metals? It seemed to me like it should be because-- according to highly rated answers to another question-- the speed of sound is like the "speed of information" about macroscopic deformations. So intuitively I expected faster speed of sound in a metal = faster conduction of heat in that same metal.

I used data from Engineering Toolbox's pages Speed of Sound in common Solids and Thermal Conductivity of Metals. The speed of sound dataset was the limiting set, so I was only able to get data on both for 11 metals Aluminum, Beryllium, Brass, Copper, Gold, Iron, Lead, Silver, Steel (low carbon steel), Stainless Steel, and Titanium:

Initially looks like there is no trend. However-- there are two outlier points. Lead and Beryllium are the metals marked in orange instead of blue. The other 9 metals had sound speeds between 3 and 7 thousand m/s. I removed those two from the data:

Here's where I get confused. My assumption was: faster speed of "information" through the metal equals higher thermal conductivity. It looks (to me) like there is a trend and the trend shows the opposite of what I assumed. What am I missing?

• Concerning the "speed of information" that you mention: if you want to send information to the other end of a piece of metal, it'll get there a lot faster if you apply an electric field to your end than if you vibrate that end. In some sense, the "maximum speed of information" in a metal is determined by its electrical properties, not by its mechanical properties like the speed of sound. – Michael Seifert Oct 21 '18 at 13:53
• I have edited my answer to try to clear a possible doubt you may have. If you think that the thermal conductivity is related to the speed of heat, then it is wrong. – AccidentalBismuthTransform Oct 25 '18 at 18:51

In metals, there is the Wiedemann-Franz law because in metals heat is carried mostly by electrons.

Alloying reduces the electron mean free path, but it does not affect the speed of sound that much. For example compare copper and brass (a Cu-Zn alloy) or iron and different steel alloys.

In insulators, thermal conductivity is mediated by phonons. Both speed of sound and mean free path are important there. But that was not what the question was about.

• the final graph in my post: is that trend real or just random? – pentane Oct 25 '18 at 11:59
• @pentane Difficult to say when points are not labeled. – user137289 Oct 25 '18 at 12:09

Generally in solids (be it metals, semiconductors or insulators) the speed of sound has to do with accoustic phonons. The case is more complex for the thermal conductivity.

In metals, heat conduction is mostly due to electrons, rather than phonons. As such, I would say there should not be a high correlation between the speed of sound and the thermal conductivity.

The situation is completely different for insulators where both the speed both of sound and the thermal conductivity have mostly to do with accoustic phonons. In that case the correlation between $$\kappa$$ and $$v_s$$ should be high, or much higher than in metals if you prefer.

In semiconductors the situation is in between the one of metals and insulators, with a strong dependence on the doping level. As a general trend, the more heavily they are doped, the more like a metal they behave. And the less they are doped, the more they behave like an insulator.

Note: Brass, steel and stainless steel are alloys.

Edit: Rereading your question, it seems to me like you confused thermal conductivity with the speed of heat propagation. If that's the case, then this is wrong. In a metal, the "speed of heat" (as in real speed, i.e. with units of distance divided by time), is of the order of Fermi speed, so about $$10^6 \mathrm{m}/\mathrm{s}$$ near room temperature. Again, this is so because it's the speed of heat and charge carriers.

$$\kappa$$ is related to the rate of heat transfer. The bigger the $$\kappa$$, the bigger the heat flux through a surface in the solid.

Also, note that the common heat equation $$\nabla \cdot (\kappa \nabla T)=C_p\frac{\partial T}{\partial t}$$ is a parabolic PDE, meaning it displays an infinite speed of propagation. So it cannot represent perfectly "reality". One has to modify it and transform it to a hyperbolic heat equation in order to account for a finite speed of propagation, as is usually done in relativistic situations.

• Doping will reduce the phonon mean free path of an insulator. Thermal conductivity might go down. – user137289 Oct 24 '18 at 21:40
• the final graph in my post: is that trend real or just random? – pentane Oct 25 '18 at 11:59
• I would agree with Pieter, it's hard to say when points are not labeled. And more points would be better. – AccidentalBismuthTransform Oct 25 '18 at 18:06
• The point are Aluminum, Beryllium, Brass, Copper, Gold, Iron, Lead, Silver, Steel (low carbon steel), Stainless Steel, and Titanium. Why do you need to know which is which? – pentane Oct 25 '18 at 21:46
• @pentane because the question is about metals, so all alloys should be removed. This leaves the question of which points are remaining? Are they enough to establish a correlation (only six points)? – AccidentalBismuthTransform Oct 26 '18 at 7:55