  Heat transfer between two surfaces separated by a nanometre gap is important
for a number of applications ranging from spaced head disk systems, scanning
thermal microscopy and thermal transport in aerogels. At these separation
distances, near field radiative heat transfer competes with heat transfer
mediated by phonons.
  Here we quantify the contribution of phonon assisted heat transfer between
apolar solids using lattice dynamics combined with ab-initio calculations. We
clearly demonstrate that phonons dominate heat transfer for subnanometre gaps.
Strikingly, we conclude that even in the situation where the gap is filled with
air molecules, phonons provide the dominant energy channel between the two
solids nearly in contact. Our results predict orders of magnitude enhanced
phonon heat transfer compared to previous works and bring forward a methodology
to analyse phonon transmission across nanoscale vacuum gaps between apolar
materials.
