Surface-controlled mononuclear growth surface-controlled polynuclear growth or diffusion-controlled growth Mathematically explain why?

Answer 1

Surface-controlled mononuclear growth occurs when the rate of nucleation is limited by the surface reaction, leading to a growth rate proportional to the surface area of the growing phase. In contrast, surface-controlled polynuclear growth involves multiple nuclei forming and growing on the surface, with growth rates influenced by the availability of surface sites. Diffusion-controlled growth, however, is limited by the rate of mass transport to the growth interface, resulting in a growth rate dependent on the diffusion coefficient and concentration gradient. Mathematically, these growth mechanisms can be described using different rate equations, where surface-controlled processes often exhibit linear relationships with respect to time, while diffusion-controlled growth typically follows a parabolic relationship due to the square root dependence on time.