A temperature gradient (TG) is often included in the Vapor-Liquid-Solid model to describe catalyzed nucleation and growth of carbon nanotubes (CNTs). Although the TG may be important to describe the growth of large CNTs, it has been suggested that a carbon concentration gradient (CG) is more important for the growth of single-walled carbon nanotubes (SWNTs) on small catalyst particles. The molecular dynamics study performed here shows that the CG is more important than the TG for the growth of SWNTs on small metal particles and that SWNTs growth can, in fact, even occur in the presence of an opposing TG (i.e., the SWNTs grow from the hot region of the catalyst particle). This shows that a TG is not required for SWNT growth on small catalyst particles.