The thin foils that stop radiant heat

The liquid nitrogen in a laboratory flask stays liquid for days in a room two hundred degrees warmer. The flask is two glass walls with the air pumped out from between them, and the inner surfaces of the gap are silvered. With no air there is nothing to conduct or carry heat across the gap; what is left is radiation, the infrared glow every warm surface sends out and every cold one absorbs. The silvering is there for the radiation, and it is the larger part of the flask’s performance. Unsilvered, the same evacuated flask would lose its litre of nitrogen in a couple of hours.

Spacecraft are wrapped in blankets of the same idea: tens of layers of plastic film a few hundredths of a millimetre thick, each coated on both sides with a film of aluminium, separated by a fine netting so that they barely touch. A blanket a couple of centimetres thick, in the vacuum of space, insulates better than a metre of glass wool on Earth. The reason is not in the materials’ thermal conductivity at all. It is in the arithmetic of radiation between surfaces, and in the fact that a surface which absorbs little must also emit little.

Radiation across a gap #

The curve that would not come down found the spectrum of a black body’s glow, and its total, $\sigma T^4$ per square metre, rises steeply with temperature. Two facing black walls, one at 293 K and one at 77 K — a room and liquid nitrogen — exchange the difference, $\sigma(T_1^4 - T_2^4)$: 340 watts per square metre, flowing from the warm wall to the cold one.

Real surfaces are not black. A surface with emissivity $\varepsilon$ radiates that fraction of the black body’s glow, and by Kirchhoff’s law it absorbs the same fraction of what falls on it and reflects the rest; the glow that says nothing about the surface found a polished aluminium block reading as twenty-five degrees on a thermal camera when it was at a hundred, for exactly this reason. Between two large parallel walls, radiation bounces back and forth, partly absorbed at each bounce, and summing the bounces gives the net flux

$$q = \frac{\sigma(T_1^4 - T_2^4)}{1/\varepsilon_1 + 1/\varepsilon_2 - 1}.$$

The denominator is a resistance. Two aluminised walls, emissivity 0.03 each, have a resistance of 65.7 against the black walls’ 1.2, and pass 6.3 watts per square metre instead of 340.

A shield in the gap #

Now hang a thin sheet of the same material in the gap, touching neither wall. It receives radiation from the warm wall and from the cold one, and it settles at the temperature at which it sends away exactly as much as it receives. In that steady state the flux from the warm wall to the sheet equals the flux from the sheet to the cold wall, and each is the formula above with the sheet as one of the walls. The sheet has added a second gap in series with the first, and series resistances add: a shield with emissivity $\varepsilon_s$ on both faces adds $2/\varepsilon_s - 1$.

When the shields have the walls’ emissivity, each adds exactly the walls’ own resistance, and $N$ shields divide the flux by $N + 1$.

The figure is the whole design principle of a radiation shield. Put numbers on a tank of liquid nitrogen with ten square metres of surface, in a vacuum jacket whose outer wall is at room temperature: with both surfaces painted black, 3.4 kilowatts would cross the gap, boiling away about seventy-five litres of nitrogen an hour; with both aluminised, 63 watts; with thirty aluminised foils between, about 2 watts, a litre a day. The number of shields matters, and shininess matters much more. A black shield adds a resistance of 1.2; an aluminised one adds 65.7. One shiny foil in the gap does more than thirty black ones, and a blanket of thirty shiny foils is three thousand times better than an empty gap between black walls.

None of this needs the shield to be thick. A foil a few micrometres thick is opaque to infrared, and its job is only to absorb a little of what reaches it and emit a little to each side. Its thermal conductivity, the property that decides how good an ordinary insulator is, does not enter, because no heat has to be conducted through it — it is at one temperature throughout. That is why aluminised plastic film, which would be a poor insulator if it were used as a slab, is the best insulator there is in a vacuum.

Where the foils settle #

Each shield floats at a temperature fixed by its neighbours, and with identical shields the fourth powers of the temperatures are equally spaced from the cold wall’s to the warm wall’s. The temperatures themselves are not.

The first shield sits only seven degrees below the warm wall, the last eighty-six above the cold one, and most of the temperature drop happens across the last few gaps. At low temperature radiation is feeble — $\sigma T^4$ at 77 K is a two-hundredth of its value at 293 K — and a large temperature difference is needed to drive the same flux across a cold gap. The cold end of a stack does most of the insulating, which is why cryostats for liquid helium, at 4 K, surround the helium vessel with a shield cooled to around 77 K by liquid nitrogen or a refrigerator: the radiation reaching 4 K from 77 K is a two-hundredth of what would reach it from room temperature, and every watt that reaches 4 K costs hundreds of watts of refrigeration to remove.

Why a good conductor makes a good shield #

The shields work because aluminium’s emissivity in the infrared is a few per cent, and that is a fact about its electrons. A metal reflects light because its free electrons respond to the wave’s electric field and cancel it at the surface, sending the wave back. How far a field gets into metal found that a field oscillating at a given frequency penetrates a conductor only to a skin depth set by its resistivity, and the fraction of the wave absorbed in that skin, rather than reflected, is small when the skin is thin compared with the wavelength. The Hagen–Rubens relation, from 1903, makes this quantitative for long wavelengths: the emissivity is about $0.365\sqrt{\rho/\lambda}$, with the resistivity $\rho$ in ohm-metres and the wavelength $\lambda$ in metres.

The relation predicts 0.019 for aluminium at 10 micrometres, close to what a clean polished surface measures. The best conductors are the best reflectors, and the emissivity falls at longer wavelengths — the radiation of colder bodies is reflected even better. Stainless steel, chosen for cryostats because it conducts heat poorly and is strong, conducts electricity poorly too, and its emissivity is five times aluminium’s; a steel cryostat’s inner walls are covered with aluminium foil or tape for that reason. Gold, nearly as good a conductor as aluminium and immune to oxidation, is used where a surface must stay shiny for years, which is why the thermal blankets on some spacecraft and instruments are gold.

The relation also says why the shields cannot simply be made perfect. Emissivity falls only as the square root of the resistivity, and even silver, the best conductor, leaves a per cent and a half. Cooling helps — a metal’s resistivity falls with temperature — but at low temperatures the electrons’ mean free path grows longer than the skin depth and the simple relation fails, leaving a floor of a fraction of a per cent.

Packed too tight #

A blanket of thirty foils, each hung freely, would be thirty times better than one. Real blankets have to be wrapped round things, and their layers touch. Wherever two touch, heat conducts across the contact, and the more tightly the layers are packed, the more they touch.

Radiation falls as one over the number of layers; contact conduction, in the form fitted to measurements on real blankets, grows as the layer density to the power 2.63, divided among the layers. The two cross, and the total has a minimum: in the model drawn, about fourteen layers per centimetre. Pack more and the blanket gets worse, until at eighty per centimetre the contacts carry more heat than the radiation did with no blanket at all. Real blankets are interleaved with a fine net or a crinkled film to keep the contacts few and small, and are made with from about ten to thirty layers per centimetre depending on the spacer. The worst blankets are the ones compressed during installation, where a fold or a tight strap squeezes the layers together; the blanket’s performance at such a seam can be a tenth of its performance elsewhere, and the edges and penetrations of a blanket usually leak more heat than its whole area.

It is a version of the paradox the insulation that makes a wire lose more heat found for a thin wire in plastic: adding insulation is not always adding resistance, because the added material opens a new path for heat while it closes the old one. There, the plastic’s outer surface shed more heat than the bare wire could; here, the layers’ contacts conduct more than their radiation saves.

The same foil in a window #

The same physics is built into most new windows. A double-glazed window traps a layer of air or argon between two panes, and the gas is thin enough and still enough that it conducts little heat; but glass is nearly black in the thermal infrared, and the two panes radiate across the gap to each other as two black walls do. In an ordinary double-glazed unit, radiation carries about two-thirds of the heat crossing the gap. A coating of silver a few nanometres thick on one of the inner faces, too thin to dim visible light noticeably, has an emissivity in the infrared of a few per cent, for exactly the reason a silver shield has: the metal’s free electrons reflect the long-wavelength radiation while the film is too thin to absorb much of the short-wavelength light. The coating cuts the radiative part of the gap’s heat flow by an order of magnitude and the window’s total heat loss by roughly a third to a half. It is a radiation shield that is also transparent, which works only because sunlight and thermal radiation sit at wavelengths twenty times apart.

The same separation of wavelengths is what lets a surface facing a clear night sky cool below the air, as the cold reservoir overhead followed: the sky is a cold radiator in one band of the infrared, and a surface that emits strongly there while reflecting everything else sends its heat out through that window. A radiation shield and a radiative cooler are the same idea pointed in opposite directions, one suppressing emission and the other choosing where it goes.

The outer layer, in sunlight #

A spacecraft’s blanket has a last job its inner layers do not: its outermost surface faces the Sun. Its temperature there is set by the balance the height a planet is seen from used for a whole planet — absorbed sunlight against emitted infrared — and the two are governed by different properties: the absorptance for sunlight, at visible wavelengths, and the emissivity in the infrared. A bare aluminised surface absorbs about a tenth of the sunlight falling on it and emits only a few per cent in the infrared, a ratio of several, and in full sunlight it runs very hot. That is why the outer layer of a blanket is often not metal at all but a film with low solar absorptance and high infrared emissivity — white paint, or a metal coated on its back face behind transparent plastic, which emits from the plastic and reflects from the metal. The inner layers suppress radiation; the outer one manages it.

The flask #

James Dewar built his vacuum flask in 1892 to store liquefied gases, and he silvered it after finding that the evacuated gap alone was not enough. The arithmetic shows why.

Glass is nearly black in the thermal infrared, emissivity about 0.9; an evacuated but unsilvered flask holding liquid nitrogen would boil off a litre to radiation alone in a little over two hours. Silvered to an emissivity of 0.02, the same flask would take seven days. In practice a flask’s nitrogen lasts a day or two, and what limits it is no longer radiation but the heat conducted down the glass of its neck and the poorly evacuated gap after years of slow leakage — the silvering has moved the bottleneck elsewhere, which is the most that a single improvement can do.

The vacuum is essential, and not for a reason one might expect. The viscosity that does not care how much gas there is found that a gas’s conductivity of heat, like its viscosity, does not depend on its pressure — fewer molecules, each travelling further between collisions — so a partially evacuated gap conducts heat as well as one at atmospheric pressure. Only when the pressure is so low that molecules cross the gap without meeting each other does the gas conduction fall, in proportion to the pressure. The gap must be pumped to well below a thousandth of an atmosphere before radiation becomes the main path, and only then is silvering worth having.

What the figures leave out #

The figures treat infinite parallel surfaces, which is good for a narrow gap and approximate for a flask’s curved walls, where the geometry changes the formula slightly. The emissivities are taken as single numbers independent of wavelength and direction; real metals’ emissivities rise at grazing angles and change with temperature and surface condition — an aluminium film that has oxidised or been handled with bare fingers can have twice its clean emissivity. The Hagen–Rubens relation is a long-wavelength approximation and fails for good conductors at low temperature, as the text says. The blanket’s contact conduction is modelled with a fitted power law rather than derived, and real blankets’ performance depends on their spacers, their edges and how they were installed more than on the idealised layer count. The domain is radiative exchange between large surfaces in a good vacuum, at gaps much wider than the thermal wavelength.

Still open: radiation across a gap narrower than its wavelength #

Everything here assumes the gaps are wide compared with the wavelength of the thermal radiation, about ten micrometres at room temperature. When surfaces are brought closer than that, the evanescent fields that cling to every warm surface reach across the gap, and the heat they carry can exceed the black-body limit by orders of magnitude — the heat that crosses a gap too narrow for light followed that effect. It means that the layers of a compressed blanket, at gaps of a micrometre, exchange more radiation than the wide-gap formula says, and that near-field radiation could be used for the opposite purpose: thermal diodes and switches that pass heat one way more easily than the other, or cooling devices that pump heat across nanometre gaps. Whether such devices can be built at useful sizes, with gaps held uniform over square centimetres, is being worked on; the wide-gap physics, which every flask and spacecraft blanket relies on, is settled.

In a vacuum, radiation between surfaces adds resistances in series, $1/\varepsilon_1 + 1/\varepsilon_2 - 1$ per gap, so $N$ thin shields of the walls’ emissivity divide the flux by $N + 1$ — thirty aluminised foils bring 6.3 W/m² between 293 K and 77 K down to 204 mW/m² — and shininess matters more than number, since an aluminised shield adds 54 times a black one’s resistance; the foils float at temperatures whose fourth powers are evenly spaced, and packed too tightly, they conduct more through their contacts than they save in radiation. A foil stops radiant heat not by being thick but by being a poor radiator, and a stack of poor radiators is the best insulation there is.