Density-functional theory (DFT) calculations for idealized nucleation processes of (5,5) and (10,0) single-walled carbon nanotubes (SWCNTs) on a 55 atom nickel cluster (Ni_(55)) showed that it requires a larger chemical potential to grow a carbon island (which is the simplest structure that can lead to formation of the SWCNTs) on the cluster than to extend the island into a SWCNT or to have the carbon atoms dispersed on the cluster surface. Hence, in the thermodynamic limit the island will only form once the (surface of the) cluster is saturated with carbon, and the island will spontaneously form a SWCNT at the chemical potentials required to create the island. The DFT (zero Kelvin) and tight binding Monte Carlo (1000 K) also show that there is a minimum cluster size required to support SWCNT growth, and that this cluster size can be used to control the diameter, but probably not the chirality, of the SWCNT at temperatures relevant to carbon nanotube growth. It also imposes a minimum size of clusters that are used for SWCNT regrowth.
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