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 kinetic critical radius rather than the 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 postcritical 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.

Figure 1

A model of a core-shell-type composite nucleus in a three-phase system. The phase transition occurs from metastable solution ${\mathrm{L}}_2$ to stable crystal C via the intermediate metastable solution ${\mathrm{L}}_1$ (${\mathrm{L}}_2\to {\mathrm{L}}_1\to \mathrm{C}$) according to the Ostwald's step rule (left). The spherical crystal nucleus C is grown from the solution ${\mathrm{L}}_1$, which is phase separated from the original mother solution ${\mathrm{L}}_2$ to form a spherical shell around the crystal C. Therefore, the critical nucleus is the composite nucleus of core-shell structure (right).

Figure 2

A diffusion flux into the core-shell composite nucleus. A diffusion flux in the solution ${\mathrm{L}}_2$ is absorbed into the shell of the solution ${\mathrm{L}}_1$ and continues to diffuse in the solution ${\mathrm{L}}_1$ into the stable crystal C. Two fluxes in solution ${\mathrm{L}}_2$ and ${\mathrm{L}}_1$ must connect continuously at the boundary of radius $R_{\mathrm{L}}$.

Figure 3

Concentrations $c_1\left(r\right)$ and $c_2\left(r\right)$ of monomers around the growing nucleus as a function of distance $r$ from the center of the nucleus. The critical nucleus having the radii $R_{\mathrm{C}}^{*}$ and $R_{\mathrm{L}}^{*}$ is surrounded by a wetting layer of uniform concentration $c_1\left(r\right)=c_1\left(R_{\mathrm{C}}^{*}\right)=c_1\left(R_{\mathrm{L}}^{*}\right)=c_{1,\infty }$, which is surrounded further by a uniform solution with the concentrations $c_2\left(r\right)=c_{2,\infty }$ of the metastable mother solution to form a core-shell critical nucleus. As the postcritical nucleus begins to grow ($R_{\mathrm{C}}^{*}\to R_{\mathrm{C}}, R_{\mathrm{L}}^{*}\to R_{\mathrm{L}}$), the concentrations $c_{1,\mathrm{eq}}\left(R_{\mathrm{C}}\right)$ and $c_{2,\mathrm{eq}}\left(R_{\mathrm{L}};R_{\mathrm{C}}\right)$ at the surface of both the core nucleus at $R_{\mathrm{C}}$ and outer boundary at $R_{\mathrm{L}}$ given by the Ostwald-Freundlich boundary condition in Eqs. (30) and (32) start to decrease. Then, the concentrations $c_1\left(r\right)$ of the solution ${\mathrm{L}}_1$ and $c_2\left(r\right)$ of the solution ${\mathrm{L}}_2$ are no longer uniform and concentration gradients appear around the nucleus, according to Eqs. (6) and (7), and diffusional flux ensures that the growing nucleus is fed.