4.7 Article

Kinetic broadening of size distribution in terms of natural versus invariant variables

Journal

PHYSICAL REVIEW E
Volume 103, Issue 1, Pages -

Publisher

AMER PHYSICAL SOC
DOI: 10.1103/PhysRevE.103.012112

Keywords

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Funding

  1. Russian Science Foundation [19-72-30004]
  2. Russian Science Foundation [19-72-30004] Funding Source: Russian Science Foundation

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This study theoretically explores the size distributions of nanoparticles, demonstrating a correlation between size distribution and the natural variable s. The Green's function is shown to be Gaussian, with size-dependent variance. Different growth systems exhibit varying size distributions under different levels of supersaturation. The study also establishes conditions for obtaining a time-invariant distribution under different growth laws.
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. for which the regular growth rate is size independent. For the growth rate scaling with s as sa (with the growth index a between 0 and 1), the size distribution broadens for larger a in terms of s, while it narrows with a if presented in terms of.. 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.

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