Family-Vicsek scaling in classical nucleation theory

Using the framework of classical nucleation theory in open systems with pumping and power-law size dependence of the attachment rates for large enough islands, the analytic island-size distribution (ISD) was previously obtained in the double-exponential form. The double-exponential shape of the distribution over invariant size (for which the regular growth rate of islands is independent of their size) is maintained in time. This result applies to large supercritical islands in systems with attachment-detachment kinetics. On the other hand, irreversible growth of surface islands with zero detachment rates or a small prescribed critical size is often described by the Family-Vicsek scaling functions versus the scaled natural size x=s/〈s〉, with 〈s〉 as the average size. Here, we combine the two approaches by showing that the island-size distributions in classical nucleation theory, converted to the scaled natural size, often satisfy the Family-Vicsek scaling hypothesis in the large time limit. For size-linear capture coefficients σ(s)=s, the analytic scaling functions are monomodal and tend to zero at x→0. For power-law capture coefficients σ(s)=s^{α} with α<1, the ISD transforms to the delta function δ(x-1) for large enough islands. These results reveal a nontrivial relationship between the ISDs in different variables, confirm the Family-Vicsek scaling in classical nucleation theory, and should be useful for understanding and modeling the growth kinetics and statistical properties within the ensembles of different nano-objects.