# What is the physical limit of thermal insulation?

If we consider a container (at very low) temperature $T_0$ surrounded by some passive structure, in turn surrounded by an environment at temperature $T_1>T_0$, there will be an inward heat flow leaking through the structure. What is the physical limit of minimal heat leakage? Or, conversely, what is the limiting thermal insulation structure?

Insulation

An obvious choice of the structure is some insulating material. In this case the flow is $$q \approx kA_0\frac{T_1-T_0}{d}$$ where $k$ is the thermal conductivity, $A_0$ the area of the inner compartment, and $d$ the thickness (this ignores geometric effects, hence the approximation). Under standard conditions silica aerogel and polyurethane foam reaches 0.020 W/Km, but most of that is air thermal conductivity. By reducing pressure it can be reduced: at $10^{-7}$ atmospheres it is down to 0.000012 W/Km.

(The thermal conductivity $k$ of metals is temperature dependent as $k=k_0T$ at low temperature, so the formula above needs to be adjusted a bit. I will ignore this as follows, but it is worth noting. Also, a pre-cooled insulator can hence get lower conductivity; however this will not be stable as ambient heat diffuses in and changes it and the thermal capacity. Again, I will ignore this time-dependent nonlinearity since it does not appear to dominate over radiation.)

Were we to instead just have a perfect vacuum between the container and environment they would couple by blackbody radiation. The flow would be $$q=\epsilon A_0\sigma (T_1^4-T_0^4)$$ where $\epsilon$ is the emissivity and $\sigma$ the Stefan-Boltzman constant. There is no dependency on $d$ since the container only "sees" the environment in all directions. The minimum emissivity in standard tables is polished silver, 0.02.

If we for example assume $A_0=1$ m$^2$, $T_0=10^{-29}$ K, $T_1=3$ K then using $10^{-7}$ atm air insulation the flow is $q=3.6\cdot 10^{-5}/d$ Watts. Using vacuum and polished silver the flow is $9.186\cdot 10^{-8}$ W. In theory we could get a smaller flow by having a more than 391 meter thick insulation, but this implicitly presupposes that the entire mass of insulation has been cooled to an appropriate temperature beforehand.

Edit: There is multi-layer insulation that can reduce the thermal flow even in the radiative case. If we insert $N$ layers the heat transfer coefficient will be $4\sigma T^3/(N(2/\epsilon - 1)+1)$, which can be made arbitrarily small by increasing $N$.

The limit of the number of layers is when the distance become a few wavelengths since near-field effects will start coupling them. From the Wien displacement law we get $\lambda=b/T$, so for 3K this $\lambda \approx 1$ mm, while down at $T=10^{-29}$ K $\lambda\approx 9$ Gpc. For a given $d$ the number of layers that can fit in scales as $d^{3/4}$.

Discussion

So, refining the question a bit to avoid "cheating" by having infinitely thick pre-cooled insulation: for a fixed $d$, what sets the physical limit of heat flow? For conduction there are clearly many physical effects, but getting rid of the material gets rid of the phonons, only leaving photon modes in blackbody radiation. Other fields are too short-range to matter or couple too weakly (there is presumably some gravitational wave heat transfer, but it is vastly overshadowed by the electromagnetic contribution).

Closely related sub-questions: Is there any principle that gives a lower bound on emissivity? Does quantization effects hinder radiative heat transfer at very low temperatures (I am interested in the $\sim 10^{-29}$ K range)? Is there some thermodynamic principle that implies that the heat flow must be nonzero?

• I have used either this multilayered insulation or a very similar product. The data sheet claims a hard vacuum limit of a few microwatts per meter per kelvin at a mean temperature of 80K. – rob Aug 24 '18 at 0:18
• Did you intend to type $10^{-29}$ kelvin? The record low temperatures achieved with microscopic masses of Bose-Einstein condensates are many orders of magnitude warmer than this. Macroscopic passive insulation probably stops being important around the microkelvin range; the isolation of the dilute gas that makes the condensate uses a very different method. – rob Aug 24 '18 at 0:26
• @rob - Yes, the temperature is intentional. It is around the de Sitter temperature of the universe. In my research, I am interested in extreme low-temperature physics of the far future. – Anders Sandberg Aug 24 '18 at 1:07