### Heat Transfer at Macro- and Nano- Scales

January 21, 2016

I was just nine years old when Earth's first artificial satellite, Sputnik 1, was launched. Sputnik marked the true start of space exploration, and it was the precursor to the many space observatories now in Earth orbit. While astronomical observations at infrared wavelengths are somewhat possible atop high mountains and in dry climates, infrared observations of faint sources are only possible using space observatories outside the atmosphere.

Infrared is one way to measure the temperature of a planet, but Pluto is at such a great distance from the Earth that it wasn't until 1987 that its temperature was measured. A measurement using the Infrared Astronomical Satellite (IRAS) showed an average temperature of about 45 kelvin (K),[1] which is half the temperature of liquid argon.

Pluto, photographed on July 14, 2015, by NASA's New Horizons spacecraft. This is a composite of images taken by the Ralph/Multispectral Visual Imaging Camera.

(NASA / Johns Hopkins University Applied Physics Laboratory / Southwest Research Institute image, via Wikimedia Commons.)

As a child I had a few books on planetary astronomy, and they listed the temperatures of the planets, including that of Pluto, before Sputnik's launch, and long before any temperature measurement of that planet (Yes, in those days, Pluto was a planet). Where did the value for Pluto's temperature come from?

Knowledge of a few physical laws will often lead to some rather accurate estimates of unknown quantities. As I wrote in an earlier article (Estimation, December 21, 2011), Enrico Fermi was a master of such back-of-the-envelope calculations. There's a subset of such problems called "Fermi problems," which are unusual estimates of quantities based on whatever information is at hand. One example of a Fermi problem that I've done is estimating how many people live in our county based on the number of supermarkets.

Since all the bodies in our Solar System are heated by the Sun, we estimate Pluto's temperature by considering the heat transfered to it from the Sun. To do this, we use the Stefan–Boltzmann law relating the energy of an ideal radiator, called a black body, with its temperature. We assume that all the energy at Pluto comes from the Sun, and we also assume that Pluto is in thermal equilibrium; that is, it radiates as much energy as it receives.

We could calculate the solar radiance received at Pluto, and start plugging numbers into the Stefan–Boltzmann equation, as follows,

where j is the black-body emissive power (equal to the solar radiance in our model), T is the absolute temperature, and σ, called the Stefan-Boltzmann constant, is expressed as a combination of some fundamental constants,

where k is the Boltzmann constant, h is Planck's constant, and c is the speed of light. The best known value for this constant is 5.67037 x 10−8 watt meter−2 kelvin−4.[2]

Instead of concerning ourselves with the absolute radiance at Pluto, it's easier to ratio its values with those of Earth, whose temperature we know quite well. The Sun's radiance diminishes with distance according to the usual inverse-square, 1/r2, law. We can get the ratio of the radiance at Pluto to that at Earth by taking the ratio of the squares of their distances from the Sun (see figure).

An astronomical unit (au), the average distance from the Earth to the Sun, is a convenient unit of distance in the Solar System.

The Earth is about 93 million miles (150 million kilometers) from the Sun.

(Created with Inkscape.)

Pluto has a highly elliptical orbit, with an aphelion of 49.32 astronomical units (au) and a perihelion of 29.66 au, and it's presently about 33 au from the Sun. When we look at the r2 ratio of the Earth and Pluto distances, we see that the level of the Sun's radiance at Pluto is just 0.092% of that at the Earth. That's the reason why the New Horizons spacecraft was powered by a radioisotope thermoelectric generator and not solar panels. I wrote about radioisotope thermoelectric generators in an earlier article (Radioactive Heat, July 26, 2011).

The fourth-root of the ratio of radiance gives us the ratio of the temperatures of Earth and Pluto. Assuming 300 K for Earth's temperature gives us about 52 K for Pluto's temperature. This is quite close to the 45 K value found in the IRAS measurement mentioned above. The agreement is surprising, since Earth and Pluto are not strictly black bodies, and this reaffirms our faith in educated guessing.

We need to be careful when we extend this method into the nanoscale, since the distance between objects becomes comparable to the wavelength of the radiation. A team of scientists, mathematicians, and engineers from the University of Michigan (Ann Arbor, Michigan), the Universidad Autónoma de Madrid (Madrid, Spain), the Massachusetts Institute of Technology (Cambridge, Massachusetts), and the Donostia International Physics Center (DIPC, San Sebastiá, Spain) has examined the heat transfer between nanoparticles, and they've discovered an interesting effect. The heat transfer happens 10,000 times faster than at the macroscopic scale.[3-4]

Ultra high vacuum scanning thermal microscope used to measure heat transfer at the nanoscale.

(University Michigan image by Joseph Xu.)

While the wavelength of visible light is about 500 nm, the scale investigated in this study was 10 nm, comparable to the diameter of DNA.[3-4] The enhancement of heat transfer at the nanoscale has been known for decades, but experiments have been difficult to do, and the exact mechanism was not understood.[4] In the mid-20th century, Russian physicist, Sergei Rytov, proposed a theory called "fluctuational electrodynamics" to describe heat transfer at small distances, but his theory had never been tested at the nanoscale.[4]

For this study, samples at 305 °F were probed with the tip of a customized scanning thermal microscope coated with the same materials and held at 98 °F lower temperature. The tip was brought from 50 nanometers distance to touching while the temperature of the tip was measured.[3-4] Measurements were made at gaps as small as two nanometers.[3] The materials tested were silica, silicon nitride and gold, and all these showed gap-size-dependent enhancements of the radiative heat transfer.[3]

The cause of the rapid heat transfer appears to be an overlap of the surface and evanescent waves of each member of the couple. Both of these types of waves carry heat.[4] Says Bai Song, a graduate student in mechanical engineering at the University of Michigan and an author of the study,
"These waves reach only a small distance into the gap between materials... and their intensity at the extreme near-field is enormous compared to the electromagnetic waves at larger distances. When these waves from two different devices overlap, that's when they allow tremendous heat flux."[4]

The data obtained were consistent with fluctuational electrodynamics. Pramod Reddy, a professor of mechanical engineering at the University of Michigan and co-principal investigator of this study, remarks
"These results disprove current dogma in nanoscale heat transfer, which holds that radiative heat transfer in single digit nanometer-sized gaps cannot be explained by existing theory."[4]

There are potential applications for such enhanced nanoscale radiative heat transfer. Since the heating process is rapid, it might be used for heat-assisted magnetic recording.[4] The process might be utilized to make more efficient thermoelectric devices, which convert a temperature gradient to electricity.[4] This research was funded by the U.S. Department of Energy, the Army Research Office, the National Science Foundation, the Spanish Ministry of Economy and Competitiveness, and other organizations.[4]

### References:

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