Optimized growth of single wall carbon nanotubes requires full knowledge of the actual state of the catalyst nanoparticle and its interface with the tube. Using tight binding based atomistic computer simulations, we calculate carbon adsorption isotherms on nanoparticles of nickel, a typical catalyst, and show that carbon solubility increases for smaller nanoparticles that are either molten or surface molten under experimental conditions. Increasing carbon content favors the dewetting of Ni nanoparticles with respect to sp(2) carbon walls, a necessary property to limit catalyst encapsulation and deactivation. Grand canonical Monte Carlo simulations of the growth of tube embryos show that wetting properties of the nanoparticles, controlled by carbon solubility, are of fundamental importance to enable the growth, shedding new light on the growth mechanisms.