  We consider the adsorption properties of superfluid $^4$He films on carbon
nanotubes. One major factor in the adsorption is the surface tension force
arising from the very small diameter of the nanotubes. Calculations show that
surface tension keeps the film thickness on the tubes very thin even when the
helium vapor is increased to the saturated pressure. The weakened Van der Waals
force due to the cylindrical geometry also contributes to this. Both of these
effects act to lower the predicted velocity of third sound propagation along
the tubes. It does not appear that superfluidity will be possible on
single-walled nanotubes of diameter about one nm, since the film thickness is
less than 3 atomic layers even at saturation. Superfluidity is possible on
larger-diameter nanotube bundles and multi-walled nanotubes, however. We have
observed third sound signals on nanotube bundles of average diameter 5 nm which
are sprayed onto a Plexiglass surface, forming a network of tubes.
