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Invalidity of Near-Field Heat Transfer
Maxwell’s equations in near-field heat transfer for the tunneling of evanescent waves through nanoscale gaps are invalid as temperature fluctuations of surface atoms required by the fluctuation-dissipation theorem are negated by quantum mechanics
By: Thomas Prevenslik
Historically, Planck’s theory  of BB radiation giving the dispersion of thermal EM radiation with wavelength (or frequency) as a function of temperature provided the basis for QM. BB stands for blackbody, EM for electromagnetic, and QM for quantum mechanics. Planck’s theory has served well in far-field heat transfer where the separation between the surfaces of bodies is longer than the wavelength of the emitted thermal radiation. In contrast, near-field heat transfer is directed to bodies separated by nanoscale gaps shorter than the wavelength of the thermal radiation.
Today, near-field heat transfer by evanescent waves is claimed [2,3] to produce heat fluxes 1000 times greater than given by Planck’s theory of BB radiation. Even though Planck never claimed his theory was applicable to gaps shorter than the emitted thermal radiation, claims  have been made that near-field heat transfer exceeds the Planck limit. The notion of near-field heat transfer is based on tunneling of evanescent waves through nanoscale gaps. In the absence of other bodies, evanescent waves travelling along surfaces of a body decay exponentially normal to the surface at a distance less than a wavelength. Bringing the bodies close to each other so the surfaces are within their evanescent fields is thought to excite dipoles in the surfaces, the motions of which are dissipated by Joule heating, thereby inducing net radiative transfer by tunneling that exceeds the values predicted by Planck’s theory. Confirmation of the physics of near-field heat transfer is therefore very important in energy harvesting, e.g., the development of high-energy density thermo-photovoltaic devices. See http://www.sciencedaily.com/
Near-field heat transfer is supported  by experimental data for gaps between flat glass plates that show enhancement of heat transfer above that given by Planck theory. However, difficulty in plate flatness limits supporting data to micron gaps. The claim  of enhancement of near-field heat transfer between microspheres and flat surfaces is encouraging. Regardless, near-field enhancement relies almost entirely on classical EM analysis of evanescent waves by the Maxwell equations.
Maxwell’s equations provide solutions of EM fields, but in thermal heat transfer a relation between the fields and temperature is required. Traditionally, the FDT satisfies this requirement by relating the oscillations of dipoles in the BB surfaces to thermal fluctuations, the frequencies of which are given in the QM of Planck’s theory by the Einstein-Hopf relation. FDT stands for fluctuation-
The validity of Maxwell’s equations in supporting the enhancement of near-field heat transfer across nanoscale gaps by evanescent waves is negated because oscillations of the dipoles in the surfaces of nanoscale gaps cannot be related to temperature fluctuations as required by the FDT. Nanoscale gaps place the atoms in the BB surfaces under high EM confinement thereby precluding the atoms from having the heat capacity necessary to respond to absorption of heat by fluctuations in temperature. Lacking temperature fluctuations, the FDT is not applicable to nanoscale gaps. However, the EM confinement of atoms in macroscale gaps allows valid Maxwell solutions with temperature fluctuations consistent with the FDT, but the near-field enhancement is minimal and near-field heat transfer approaches that in the far-field given by Planck’s BB theory.
QM by the Einstein-Hopf relation for the harmonic oscillator shows the Planck energy of the atom depends on the temperature and wavelength (or frequency) of EM confinement. At 300K, the Planck energy as the measure of the capacity of the atom to absorb heat is shown in the thumbnail. Clearly, atoms in gaps are under EM confinement at wavelengths W = 2d, where d is the gap dimension. Unlike classical physics, QM limits the heat capacity of the atoms below the thermal wavelength Wt = hc/kT, where h is Planck’s constant, c the speed of light, k Boltzmann’s constant and T absolute temperature. At ambient temperature, Wt ~ 50 microns, and therefore surface atoms in gaps d > 25 microns have the capacity to absorb heat and produce temperature fluctuations in the near-field as required by the FDT. At W < Wt, the heat capacity of the atoms is diminished, e.g., for W < 6 microns, d < 3 microns and atoms have little heat capacity. In nanoscale gaps, the atom no longer has any capacity to produce temperature fluctuations as required by the FDT.
What this means is QM refutes the validity of solutions of Maxwell’s equations in near-field heat transfer by evanescent waves that tunnel through nanoscale gaps, i.e., temperature fluctuations do not occur because the EM confinement of atoms precludes the heat capacity necessary to conserve heat by an increase in temperature. Alternatively, the notion of temperature for surface atoms is refuted by QM. Hence, claims  of near-field enhancement between a microsphere and flat surfaces that assume atoms in gap surfaces have the same temperatures measured away from the gap may be safely dismissed.
Instead of evanescent waves, the heat flowing by conduction to the hot surface or from the cold surface is conserved by QED inducing the surface atoms to create standing wave photons having Planck energy E = hc/2d. QED stands for quantum electrodynamics. A single QED photon therefore transfers EM energy across the gap d at the rate q = E*c/2d, the total number N of QED photons determined by conserving the Stefan-Boltzmann heat Q given by Planck theory, i.e., N = Q/q. Hence, near-field heat transfer is implicitly bounded by Planck theory. See QED induced heat transfer given in “Invalidity of Near-Field Heat Transfer,” at http://www.nanoqed.org, 2012.
1. Near-field heat transfer by tunneling of evanescent waves across gaps between flat surfaces is not supported by solutions of Maxwell’s equations because EM confinement of surface atoms precludes the heat capacity necessary for temperature fluctuations required by the FDT. Similarly, near-field heat transfer enhancement between microspheres and flat surfaces is not supported.
2. In macroscale gaps, the EM confinement of surface atoms allows the FDT to be satisfied, but the near-field enhancement is not significant and heat transfer approaches that of far-field heat transfer given by that of Planck theory.
3. Planck theory is a reasonable upper bound to near-field heat transfer provided the gaps are macroscale.
 M. Planck, The Theory of Heat Radiation (Dover, New York, 1991)
 A. Narayanaswamy et al., “Breakdown of the Planck blackbody radiation law at nanoscale gaps,” Appl Phys A, 96: 357–362 (2009)
 R. S. Ottens, et al., “Near-Field Radiative Heat Transfer between Macroscopic Planar Surfaces,” PRL 107, 014301 (2011)
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Classically, thermal EM radiation conserves heat by an increase in temperature. But at the nanoscale, temperature increases are forbidden by quantum mechanics. QED radiation explains how heat is conserved by the emission of nonthermal EM radiation.