  Various diameters of alumina-supported Fe catalysts are used to grow
single-walled carbon nanotubes (SWCNTs) with chemical vapor decomposition. We
find that the reduction of the catalyst size requires an increase of the
minimum temperature necessary for the growth. We address this phenomenon in
terms of solubility of C in Fe nanoclusters and, by using first principles
calculations, we devise a simple model to predict the behavior of the phases
competing for stability in Fe-C nano-clusters at low temperature. We show that,
as a function particles size, there are three scenarios compatible with steady
state-, limited- and no-growth of SWCNTs, corresponding to unaffected, reduced
and no solubility of C in the particles. The result raises previously unknown
concerns about the growth feasibility of small and very-long SWCNTs within the
current Fe CVD technology, and suggests new strategies in the search of better
catalysts.
