Nucleation and growth of a core-shell composite nucleus by diffusion

The critical radius of a core-shell-type nucleus grown by diffusion in a phase-separated solution is studied. A {\it kinetic} critical radius rather than the {\it thermodynamic} critical radius of standard classical nucleation theory can be defined from the diffusional growth equations. It is shown that there exist two kinetic critical radii for the core-shell-type nucleus, for which both the inner core radius and the outer shell radius will be stationary. Therefore, these two critical radii correspond to a single critical point of the nucleation path with a single energy barrier even though the nucleation looks like a two-step process. The two radii are given by formulas similar to that of classical nucleation theory if the Ostwald-Freundlich boundary condition is imposed at the surface of the inner nucleus and that of the outer shell. The subsequent growth of a core-shell-type post-critical nucleus follows the classical picture of Ostwald's step rule. Our result is consistent with some of the experimental and numerical results which suggest the core-shell-type critical nucleus.