# introduction to thermal conductivity and noise management

## basics

Most people understand, to some extent, the principle of Fourier's law - that heat transfer is proportional to temperature difference. Most people reading this probably also understand triple-glazed windows, fiberglass insulation, and vacuum flasks, but:

When a window with 2 panes of glass has an extra layer added in the middle, the gas inside the window is divided into 2 regions. Each region of gas has about the same friction, but each has half the thermal gradient driving gas circulation, so gas circulation speed decreases, reducing thermal conductivity.

When insulation is made of fine fibers instead of large fibers, the effect is the same as adding layers to windows: gas regions become smaller, and circulation becomes slower because thermal gradients are smaller relative to friction of gas flow.

By removing all the gas, heat transfer from circulation can be eliminated, but radiative heat transfer still happens, so the inside of vacuum flasks should be reflective. Adding more layers of reflectors further reduces heat transfer, and [that approach](https://en.wikipedia.org/wiki/Multi-layer_insulation) is used in high-performance insulation for some spacecraft and cryogenic devices.




## complications

Different materials have widely varying thermal conductivity. One popular conception of thermal conductivity is that:
> liquids and plastics have low thermal conductivity because they mostly have weak hydrogen bonds, while solids with covalent or metallic bonding have stronger interactions and thus higher thermal conductivity.

However, if we look at a [list of material thermal conductivities](https://en.wikipedia.org/wiki/List_of_thermal_conductivities), that explanation doesn't hold up very well:
- Across crystals having the same categories of bonds (metallic, covalent, etc), thermal conductivity can vary widely. For example, copper has ~50x the thermal conductivity of bismuth, and aluminum nitride has ~11x that of aluminum oxide.
- With the exact same bonds in a different structure, thermal conductivity can vary widely. For example, ice has ~3.5x the thermal conductivity of water, and HDPE plastic has ~2x the thermal conductivity of LDPE.
- With the exact same material at a different temperature, thermal conductivity can very greatly. Notably, ultra-pure aluminum reaches a peak of ~10^5 W/mK at ~3 K, ~400x its normal value.

To understand those facts, we must consider [phonons](https://en.wikipedia.org/wiki/Phonon).


## phonon scattering

High-purity aluminum at low temperatures also has low electrical resistance, because electrons in it can travel ballistically across macroscopic distances. At higher temperatures, vibrating aluminum atoms collide with those traveling electrons, greatly increasing resistance.

In cold aluminum, heat is mostly conducted by electrons, but it can also be conducted by atomic vibrations. Some patterns of vibrations, called phonons, can travel continuously through a crystal. Like electrons moving through aluminum, the travel of those phonons can be disrupted by random vibrations and by irregularities in a crystal structure, decreasing the distance they travel and thus thermal conductivity.

Electrical resistance and thermal conductivity can both be considered analogous to optical transparency. Even a small amount of additives can make glass go from mostly-transparent to mostly-opaque. As for thermal conductivity, in diamond, even 3 ppm nitrogen impurity [noticeably affects](https://pubs.aip.org/aip/jap/article-abstract/133/2/025102/2869266/Thermal-conductivity-of-pink-CVD-diamond-Influence?redirectedFrom=fulltext) it.

Because of phonon scattering, thermal conductivity can decrease with temperature, but it can also increase with temperature, because at higher temperature, more vibrational modes are possible. So, crystals have some temperature at which their thermal conductivity peaks.

With this understanding, we'd expect amorphous materials to have low thermal conductivity, even if they have a 3d network of strong covalent bonds. And indeed, typical window glass has a relatively low thermal conductivity, ~1/30th that of aluminum oxide, and only ~2x that of HDPE plastic.


## noise management

Thermal phonon transmission is also analogous to sound transmission, and choosing things to block or absord sound is probably more relevant to most people than designing materials for thermal conductivity. To block sound, we want something amorphous or with lots of defects on the scale of sound wavelengths.

Considering human hearing sensitivity, 3cm to 2m wavelengths are relevant, and especially 9cm. So, we'd expect that adding lots of objects 1-10 inches wide to a room would reduce sound transmission significantly.

As far as practical implications, well, that implies that planting trees around freeways would be an effective way to block noise despite their low density. I'm sure that using trees to block sound is something nobody has ever thought of.

How about reducing noise levels in busy restaurants? That's different from reducing average *transmission distance*. What you want to absorb sound is materials with low density on their surface on a sound wavelength scale, so they don't reflect sound.

What would be something with appropriate scale and density, that's cheap and not too bad aesthetically? Nets of thin cord seem reasonable given those criteria, but another criteria is sufficient **coupling** to air movement; enough of the surface area needs to be covered. So, typical good materials for sound absorption are fine fibers and foam. Acoustic panels often have spikes on the scale of wavelengths to provide a density/reflectivity gradient, instead of a single continuous surface that reflects sound; [anechoic chambers for radar testing](https://en.wikipedia.org/wiki/Anechoic_chamber) use the same principle.

In that case, "nets" of ~0.02mm thick film with ~9cm spacing would be reasonable, but it's easier to take thin plastic sheet and cut holes in it. And indeed, perforated plastic sheets are sometimes sold for sound-proofing, but they're used as facing for objects (perforated polymer film acoustic facing) because they're not self-supporting.

But what's something that could be done cheaply with commonly-available materials? Here's my proposal:

- Hang up a net with perhaps 6" spacing.
- Get some plastic shopping bags with thin plastic.
- Attach the plastic bags to the net intersection points, perhaps by tying 1 handle with string or by poking a paperclip through the bottom of the bag. Attach bags to about half the net intersections. It's better for the bags to be somewhat crumpled, and upside-down ones would probably work a bit better.
- For better aesthetics, you can get multiple bag colors and arrange them in a pattern.

It's hard to do something much cheaper than that. As for the aesthetics, well, no comment.


## polymer design

This post is brought to you by me designing novel high-thermal-conductivity polymers for axial-flux electric motors.

If you're trying to engineer a plastic for higher thermal conductivity, an obvious approach is adding some filler with high aspect ratio, such as carbon nanotubes or graphene nanoplatelets, or perhaps something cheaper like talc, or a compromise like hexagonal boron nitride. On a larger scale, carbon fiber also has quite high thermal conductivity. But of course, the matrix properties are still important.

If you look up papers on epoxy resins with higher thermal conductivity, you can find people trying to add side-chain liquid crystal polymers to epoxy monomers, but of course, resin monomers have to flow well and cross-linking tends to disrupt liquid crystal structures. Well, it's just polymer physical chemistry; you just visualize the dipole moments and hydrogen bonding and crystallization and crystal properties.