  Phonons (collective atomic vibrations in solids) are more effective in
transporting heat than photons. This is the reason why the conduction mode of
heat transport in nonmetals (mediated by phonons) is dominant compared to the
radiation mode of heat transport (mediated by photons). However, since phonons
are unable to traverse a vacuum gap (unlike photons) it is commonly believed
that two bodies separated by a gap cannot exchange heat via phonons. Recently,
a mechanism was proposed by which phonons can transport heat across a vacuum
gap - through Van der Waals interaction between two bodies with gap less than
wavelength of light. Such heat transfer mechanisms are highly relevant for
heating (and cooling) of nanostructures; the heating of the flying heads in
magnetic storage disks is a case in point. Here, the theoretical derivation for
modeling phonon transmission is revisited and extended to the case of two
bodies made of different materials separated by a vacuum gap. Magnitudes of
phonon transmission, and hence the heat transfer, for commonly used materials
in the micro and nano-electromechanical industry are calculated and compared
with the calculation of conduction heat transfer through air for small gaps.
