TY - JOUR

T1 - Kinetic broadening of size distribution in terms of natural versus invariant variables

AU - Dubrovskii, Vladimir G.

AU - Sibirev, Nickolay V.

AU - Sokolovskii, Andrei S.

N1 - Publisher Copyright: © 2021 American Physical Society.

PY - 2021/1/13

Y1 - 2021/1/13

N2 - 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 ρ for which the regular growth rate is size independent. For the growth rate scaling with s as sα (with the growth index α between 0 and 1), the size distribution broadens for larger α in terms of s, while it narrows with α if presented in terms of ρ. 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.

AB - 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 ρ for which the regular growth rate is size independent. For the growth rate scaling with s as sα (with the growth index α between 0 and 1), the size distribution broadens for larger α in terms of s, while it narrows with α if presented in terms of ρ. 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.

KW - LENGTH DISTRIBUTIONS

KW - GROWTH

KW - NUCLEATION

KW - RELAXATION

KW - CONDENSATION

KW - EQUATIONS

KW - DYNAMICS

KW - SURFACES

KW - SYSTEM

KW - VAPOR

UR - http://www.scopus.com/inward/record.url?scp=85099627758&partnerID=8YFLogxK

UR - https://www.mendeley.com/catalogue/1f8e8c03-ac84-38cb-b8b7-490459c7430f/

U2 - 10.1103/physreve.103.012112

DO - 10.1103/physreve.103.012112

M3 - Article

C2 - 33601594

AN - SCOPUS:85099627758

VL - 103

JO - Physical Review E

JF - Physical Review E

SN - 1539-3755

IS - 1

M1 - 012112

ER -