This paper deals with radiation’s contribution to thermal insulation. The mechanism by which a stack of absorbers limits radiative heat transfer is examined in detail both for black-body shields and grey-body shields. It shows that radiation energy transfer rates should be much faster than conduction rates. It demonstrates that, for opaque screens, increased reflectivity will dramatically reduce the rate of heat transfer, improving thermal insulation. This simple model is thought to contribute to the understanding of how animal furs, human clothes, rockwool insulators, thermo-protective containers, and many other passive energy-saving devices operate.
© 2014 Optical Society of America
Fur or pelage is unique to mammals. In addition to thermal insulation, it serves to protect the skin from abrasion or harmful ultraviolet radiation, to communicate with other species – signalling or concealing – or to gain information about its environment by sensing its surroundings. Mammalian hair can vary in form from straight to curly: there is a correlation between this geometry and the form (circular to flattened) seen in the hair’s cross-section. For aquatic mammals, the hair shape will allow an air layer to be maintained within the fur during submersion, increasing its water-repelling function . It should be noted that furs generally contain two kinds of hair: ground hair (shorter, denser and often curly) and guard hair (longer, straighter and coloured). A similar distinction is found in birds’ feathers.
The colouration of a mammal’s guard hair is usually cryptic, matching its surrounding landscape (Fig. 1). In the form of specialised vibrissae, guard hair can also function as a tactile sensor. Taken as a whole, hair makes up a complex system; it grows from pits (follicles) in the skin. A sebaceous gland next to the hair root delivers a conditioning substance and tiny muscles can temporarily modify their implantation angle .
The most important purpose of the pelt is to thermally insulate the body in high or low temperatures and improve thermoregulation. Heat conservation is the chief function of ground hair [3, 4, 5]. This hair grows continuously forming a woolly layer. They are shed in a process called moulting (continuous or seasonal) and is periodically replaced. Thermal insulation may not have been its original function in mammals, as only full-grown dense fur insulates effectively. Sensing the environment may have been the earliest evolutionary advantage it provided. Hair is usually not preserved in fossils, but its presence on a therapsid (mammal’s ancestor) from the Cretaceous period may be inferred from the presence of characteristic pits, usually associated with vibrissae, on the rostrum of this fossilised animal . It is highly probable that the earliest mammals already had hair: their very small size indicates a high surface-to-volume ratio, which would result in a high risk of heat loss . It is difficult to imagine that such small animals would not have had the protection of a substantial layer of hair.
Surprisingly little is known about the detailed physical mechanism of thermal insulation in structured material, such as criss-crossed fibres. Several studies have, albeit experimentally, investigated thermal insulation in penguin feathers [8, 9, 10]. Their shape has evolved into a very short, stiff and lance-shaped form with a long afterfeather, giving a general two-layer fibrous layout resembling a mammal’s hair. Moreover, the packing of penguin’s feathers on the skin is densely and evenly distributed, increasing thermal insulation .
Thermal insulation most often refers to thermal conduction by air trapped between the hairs or feathers – and standard measurement techniques often favour this point of view – [9, 10]. Nevertheless, it is a fact that the thermal conduction can (and often must) also be driven by convection and radiation [11, 12]. Thermal insulation mainly depends on fibre density and length. Measurements on the two-layer fur of rock squirrels (Spermophilus variegatus) show an abrupt decrease in transmission through the inner, dense coat in comparison to the outer coat . Insulation is further improved by skin coloration underneath . However no precise mechanism is proposed for thermal insulation. Studying this mechanism in order to understand how warm-blooded animals keep their heat in cold or hot environments is interesting, but in terms of biomimetism, the importance of such a study brings up further questions on thermal insulation in other biological areas  and in material science, which have implications for energy-saving habitation design, textiles for clothes, automobile and more.
In this paper, we address the question of the contribution of far-infrared radiation to the thermal conductance of what we call an “interfacial material”, by which we mean a solid material made of at least two different media separated by interfaces distributed throughout its volume. In such materials, the physical properties are closely linked to the configuration of these internal surfaces. In this case, the distance travelled by the radiation exchanged between the interfaces is greater than its wavelength. At temperatures on most locations on the planet, this means, in application of Wien’s law , distances larger than 9 μm (+50°C) or 13 μm (−50°C). In this paper, we only consider simple one-dimensional models, leaving forthcoming publications to examine more complete three-dimensional transcriptions, as well as experimental comparisons for more specific insulating systems.
2. Importance of radiation
When two bodies at different temperatures, say Ta > Tb, are separated by still air, the heat transfer from a to b takes place via two mechanisms: conduction and radiation. In this instance, there is no convection. Conduction refers to energy transfer through the boundary between two bodies and radiation concerns energy transferred by means of electromagnetic waves. As the rate of heat transferred is directly linked to insulation capability, it makes sense to compare which process will result in the most rapid losses.
The rate of energy transfer (W, in joule units) by conduction can easily be estimated by using the expression16],
Let us now look at radiation losses. The emissivity of the skin of a polar bear in the mid- and far-infrared is close to unity, as it is for most non-metallic surfaces. The atmosphere is assumed not to reradiate the emitted power, so we consider that the skin and the atmosphere are black bodies exchanging thermal energy. The net heat transfer rate is given by Stefan-Boltzmann’s law,
3. Black-body shields
In order to introduce a “radiative” insulator, we will consider a fibrous material which fills the space between two black-body thermostats at different temperatures. The aim of this study is to determine the energy exchange rate as a function of the number of radiative shields. The idea is that each thermostat emits and absorbs heat, which will be absorbed and re-emitted by interposed absorbing reflectors. Even if three-dimensional absorbing scatterers are more realistic, one-dimensional reflectors will be considered, because they lead to exact analytic results and contain the necessary physical components. Fig. 2a shows the geometry of the simple model we will study first.
In this geometry, two semi-infinite black-body media, at temperatures Ta > Tb, are considered. Their surfaces are parallel and normal to the horizontal (z) direction. When considering radiative energy transfer, the distance between the surfaces is unimportant and the rate of heat transfer per unit surface is:
Our first model radiative insulator is a series of n equidistant sheets, parallel to the thermostat surfaces. The sheets are black bodies, with an emissivity ε = 1, which corresponds to an absorptivity a = 1. Each sheet – for example sheet number i (i = 1, 2,...n) – absorbs and emits energy to and from neighbouring sheets i − 1 and i + 1 and equilibrates at a temperature of Ti. For convenience, the high-temperature thermostat at a temperature of Ta can be viewed as sheet number 0 and the low-temperature thermostat at a temperature of Tb can be viewed as sheet number n + 1. As shown in Fig. 2(b), the power per unit area transferred from sheet i to sheet i + 1 is denoted by and the power per unit area transferred in the opposite direction, from sheet i +1 to sheet i is denoted by . For convenience, the power emitted from sheet i in one direction will be written:
With all black bodies, these energy transfers depend only on the temperature of the sheets, as follows:Eq. 11 also means that: Fig. 2(b) expresses that the net heat energy exchange between two successive sheets is constant and equal to the net global energy exchanged between the thermostats.
A practical solution to this very simple equilibrium problem is given by the following system of n linear equations, with n unknowns, obtained by rewriting equation 11 in the following form: for i = 2, 3, 4,...n − 1Fig. 3(b). The change of temperature in the thickness of the “fur” is almost, but not exactly, linear.
The purpose of this analysis is also to determine the effect of interposing black-body sheets on the global energy transfer between the thermostats. Quantitatively, this can be determined by calculating the ratio η(n) of the energy transfer in presence of n black-body sheets to the corresponding energy transfer R0 (see Eq. 6 and previous discussion)
The result of this calculation is shown in Fig. 3(b). n = 0 is the reference system, so that the energy exchange factor is η(0) = 1. The energy exchange decreases rapidly as new sheets are added. With about 10 sheets, the energy transfer is reduced by one order of magnitude while, with a hundred sheets, we gain two orders of magnitude. The rapid decrease in heat transfer rate with the increasing number of intermediate absorbers means that the radiative exchange results in an efficient thermal insulation mechanism. In the next section, we use a slightly more elaborate model, using grey-body shields, to show that black bodies may not be the best conceivable absorber sheets.
4. Grey-body shields
In this second model, we keep exactly the same geometry as in the previous section, with full half-space thermostats at temperatures Ta and Tb, but we replace black-body planar shielding sheets by grey-body sheets. With grey bodies, Stefan’s radiation law must be modified by introducing an emissivity ε which has been shown by thermodynamics to be equal to the absorptivity a = 1 − t − r of the emittor. The absorptivity a, the reflectivity r and the transmittivity t are new parameters which characterise the grey shield. Even if this one-dimensional geometry is kept simple, the problem is now made more complicated by the fact that the sheets have some degree of transparency, so that the light emitted by any sheet can be reabsorbed by any other, and long-range coupling can take place, not just nearest neighbour coupling as in the case of black-body shields. An incoherent multiple-reflection/transmission picture is needed.
Fortunately, this can be examined analytically. In fact, the scheme of energy exchange depicted in Fig. 2(b) can be maintained, with the following changes to the energy transfers,Eq. 11, with the forward and backward energy transfers given above, which leave us with a set of n linear equations for the n unknown temperatures. At this stage, this approach does not restrict the value of shield transparency, as all terms in a multiple incoherent-scattering expansion have been taken into account.
Even if the following case is not realistic, it is interesting to consider: if the transmission becomes perfect (t = 1), we force the reflectivity and the absorptivity to vanish and the energy attenuation factor takes the maximum value η(n) = 1. In this case, the equilibrium equation leads to the values wi = 0 for all shields, which means that all shield temperatures are zero: the shields are effectively discoupled from the thermostats, being unable to absorb energy from them. This is not advisable if we seek a strong thermal insulation effect, so, for the purposes of illustration, we will turn to the opposite case: a weak transmission value. The energy touching one of the screens is distributed in the reflected and absorbed parts and we can describe the effect for varying absorptivities, from a ≈ 0 to a ≈ 1.
The case t = 0 leads to a remarkably simple situation, as second nearest neighbors sheets are optically shielded by the first nearest neighbors. Indeed, we see that
The result, for increasing reflectances, is shown in Fig. 4. The distribution of shields’ temperatures is identical for all values of reflectances and, as expected, corresponds to the black-body case. The energy attenuation factor is shown in Fig. 4(b). The curve which corresponds to the highest value has been calculated for a reflectance r = 0 – and then a = 1 – which corresponds to the use of black-body shields, as in section 3. With increasing reflectivity, the rate of transfer is dramatically reduced for any number of shields. It is clear that the use of a very small absorption with high reflectances improves overall thermal insulation. Photonic crystals which produce, without absorption, a total reflection in the far infrared (more specifically with wavelengths around 10 μm) would be ideal in the present model. Other very good insulation candidates involve the stack of metallic layers, in vacuum.
5. Three-dimensional thermal inhomogeneous insulators
One of the important points underlined by the above one-dimensional models is that the thermal transparency of the structure decreases rapidly when more shields are interposed between the thermostats. The exact distance between the shields is not important. In the case of a complex three-dimensional structure filling the space, the number of multiple reflections can increase very rapidly when the internal surface per unit volume becomes larger than the inverse of the distance between the thermostats. The decrease in transferred energy with the number of shields can be interpreted as a decrease if the distance between the thermostats is a given constant. This decrease with depth can be understood in approximate terms as a series of black-body shields between a hot “transmitter” thermostat and a zero-temperature “receiver”. The first shield, close to the hot thermostat, fully absorbs the energy provided but, at equilibrium, re-emits it in the form of radiation in both directions. For one half, back to the hot thermostat and for the other half, to the next shield. Neglecting multiple re-emission, a second shield will further halve the transmitted energy, and so on for the third and other shields. The resulting exponential decrease is only approximately observed in the decays shown in Fig. 3(b) for a few thermostats, because the multiple re-emission tends to slow down the decrease. The observed decay is typical of a tendency to enhanced retro-diffusion. Such a classic weak light localisation [17, 18] can arguably explain the large backscattering of light from white diffusing structures: the base of thick clouds can be quite dark, in spite of the fact they contain few absorbing materials. The top of the same clouds redistribute nearly all the incidental light they receive to higher altitudes, providing a high albedo value. The decay of the light with the depth towards the cloud base is caused, in this case, by multiple scattering on spherical droplets.
Given that the distance between the diffusing elements is often large, structured layers providing a visible white appearance and efficient thermal insulation in the far infrared can often be found. This is particularly useful to animals, such as mammals and birds which live in snowy areas. White color matching that of the snow, is a camouflage useful both to prey and predators. But, at the same time, the same snowy areas are obviously very cold environments, from which the warm-blooded animals need protection. The structure of polar bear or snow fox fur is actually multifunctional, providing both visual camouflage and good thermal insulation.
From this multifunctional point of view, the white peacock is an interesting example of the absence of melanin in an organism that should have inherited a coloring structure. The white peacock is assumed to result from a genetic mutation of a recent ancestor of the Indian blue peacock Pavo cristatus  which migrated to the North of Europe, where it forms a stable variation, easily reproduced in captivity. This type of peacock and peahen is entirely white and their offspring are white peafowls. This variation is usually referred to as Pavo cristatus mut. alba. They should not be confused with albino birds: in the alba variation, pigments are systematically found in the eyes and on the feet. The white peacock mutant can still be bred with Pavo cristatus, resulting in a pied bird perfectly able to reproduce.
Feathers are composed of a keratin hollow shaft, the rachis, which support a number of side branches called barbs. The barbs themselves support finer sub-branches called barbules. In the white peacock, the barbs contain a three-dimensional array of large polyhedral cells of about 20 μm extent (see Fig. 6), with thin keratin membranes (about 300 nm width) which can diffuse both visible and thermal radiation. In its blue ancestor, the same polyhedral structure is found but walls and cortex are filled with melanin, absorbing considerable amounts of light and decreasing the white coloration . White peacock barbules have evolved to lose the coloring structure of its ancestor [20, 21] and modify its outer structure. They appear as a flat, bulky blade, and, in contrast to the blue peacock, with longitudinal segmentation and long appendages (Fig. 5) which add to its ability to diffuse visible and far-infrared radiation. Lacking melanin, the whole white peacock feather system produces a bright white color, mainly coming from barbs, as seen in Fig. 6. A dense fleece of down fluffy feathers with segmented barbules close to the skin may provide, like for penguins, very effective thermal insulation.The genetic mutations that lead to the loss of all structural coloration of Pavo cristatus does not appear to be a complex morphological modification, as it results from the loss of melanin . The change of barbule’s form does, however, have an impact on thermal regulation and, probably, increases a discoloration which happens to be also very beneficial, making it inconspicuous in the Scandinavian snowy winter. A detailed study taking into account the full 3D barb foam geometry would need to be further investigated in order to determine the influence of melanin on thermal insulation.
The considerations here are also important for artificial insulating layers, which are needed for passive thermal regulation of future human habitats. Insulation can help to avoid energy consumption, both in summer and winter. It is already known that woolly materials such as rock or glass wool can be very good thermal insulators when their layer is thick enough (modern specifications tend to recommend a thickness of about 50 cm for roof insulation in Northern European countries). This thickness requirement is directly correlated to the increase of the number of interfaces for multiple scattering of far infrared.
Consideration of heat transfer by radiation is mandatory in certain cases, where vacuum layers are involved. We know of structures such as Dewar bottles  where highly reflective metal-coated glass, separated by evacuated space, act to slow down heat transfer. These are a special case of the structures studied here. They have important applications in science, medicine and everyday use.
Other important areas where radiative thermal insulation is vital are spatial applications, outside the Earth’s atmosphere. No conduction or convection through air can occur there, and the only cooling or heating mechanism uses radiation. Satellites would be a good place to use the physical principles examined in the present work. Here, there are difficult challenges posed by some instruments, firstly, because the temperature differences between the parts exposed to the Sun and those in the dark are extremely high. In some cases, the active parts must be maintained at a temperature of 0.1 K, while the rest of the instrument is exposed to high and fluctuating temperatures . The design of a challenging thermal shield protecting the radiometers was crucial: the designers came up with multiple V-grooved reflector able to provide passive cooling to the level of 50 K.
Because of their efficient scattering power, criss-crossed fibrous materials are effective localisers of thermal radiation. Starting from the one-dimensional models presented here, new tracks for functional optimization can certainly be found. Guided by the the results from Fig. 4, a stack of thin metal layers, separated by thin homogeneous or fibrous spacing layers might provide an insulating system that can be as efficient or even outperform actual systems, while keeping the thickness to a much more manageable minimum . Reports have been published showing that insulating layers with such a structure can be 3 to 5 times thinner than standard woolly layers, with similar performances .
Natural selection means that organisms gradually change when exposed to extreme conditions. Polar bear fur contains several different sizes of hair, with a large density of interfaces. This produces the scattering needed to retrodiffuse heat. A polar bear (Ursus maritimus) does not produce a strong thermal image against snow. The same structure also gives it its white color because the illuminating daylight is effectively scattered over the fur. The slight greenish coloration of these bears is actually related to the presence of marine algae that develop in the fur and absorb blue light as most plant forms. The dromedary (Camelus dromedarius) has to deal with a different challenge: keeping heat out of his body. Here again, a double layer of hair, with a strong melanin concentration reduction, provides the necessary multiple scattering of solar radiation, from the outside, to help in blocking heat absorption. It should be observed here that the scattered radiation corresponds to the solar spectrum, which only contains visible, some UV and near infrared rays. The thermal insulation here is then directly related to the visual appearance of the animal and the elimination of thermal absorption is directly visible in the luminosity and the desaturation of the color of the dromedary fur.
This work, which wants to contribute to the understanding of thermal insulation in living systems, suggests that radiative thermal transfer provides a better view of thermal insulation than the more frequent approach based on the low thermal conductance of air. Structures like furs and arrays of feathers do slow down air convection, but radiation seems to be much faster than conductance to provide thermal leaks. So, the control of thermal radiation seems to be the dominant factor for biological adaptation in organisms like mammals and birds living in very high or cold environments and which need to control heat exchange efficiently.
P. Simonis acknowledges S. Upton for helpful comments on the manuscript. The project was partly funded by the “Actions Concertées” (ARC), Grant No 10/15-033 from the Belgian Federation Wallonia-Brussels. This research used resources of the ”Plateforme Technologique de Calcul Intensif (PTCI)” (http://www.ptci.unamur.be) located at the University of Namur, Belgium, which is supported by the F.R.S.-FNRS. The PTCI is member of the ”Consortium des Equipements de Calcul Intensif (CECI)”.
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