We study theoretically the size distributions of nanoparticles (islands, droplets, nanowires) whose time evolution obeys the kinetic rate equations with size-dependent condensation and evaporation rates. Different effects are studied which contribute to the size distribution broadening, including kinetic fluctuations, evaporation, nucleation delay, and size-dependent growth rates. Under rather general assumptions, an analytic form of the size distribution is obtained in terms of the natural variable $s$ which equals the number of monomers in the nanoparticle. Green's function of the continuum rate equation is shown to be Gaussian, with the size-dependent variance. We consider particular examples of the size distributions in either linear growth systems (at a constant supersaturation) or classical nucleation theory with pumping (at a time-dependent supersaturation) and compare the spectrum broadening in terms of $s$ versus the invariant variable $\rho$ for which the regular growth rate is size independent. For the growth rate scaling with $s$ as $s^{\alpha }$ (with the growth index $\alpha$ between 0 and 1), the size distribution broadens for larger $\alpha$ in terms of $s$, while it narrows with $\alpha$ if presented in terms of $\rho$. We establish the conditions for obtaining a time-invariant size distribution over a given variable for different growth laws. This result applies for a wide range of systems and shows how the growth method can be optimized to narrow the size distribution over a required variable, for example, the volume, surface area, radius or length of a nanoparticle. An analysis of some concrete growth systems is presented from the viewpoint of the obtained results.

• Received 1 September 2020
• Revised 28 November 2020
• Accepted 23 December 2020

DOI:https://doi.org/10.1103/PhysRevE.103.012112