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Exact island-size distribution and Family-Vicsek scaling function in homogeneous growth with size-linear capture numbers
Phys. Rev. E 114, 054117 – Published 9 November, 2026
DOI: https://doi.org/10.1103/lvk8-jmkk
Abstract
Family-Vicsek scaling of island-size distribution (ISD) is well known for homogeneous epitaxial islands with infinitely large ratios of surface diffusion coefficient over deposition rate, where the capture numbers are asymptotically linear in size. Linear rows of surface adatoms, droplets in the cavitation regime, and vertical nanowires present the examples of growth systems with size-linear capture rates for any size. Theory of Bartelt and Evans does not provide any solution in this case, because the ISD is established in the short-range nucleation step rather than in a much longer asymptotic growth stage. Here, we present the exact solution of discrete rate equations for the monomer concentration and ISD in homogeneous irreversible growth with size-linear capture rates for any size, and obtain the corresponding scaling function. The monomer concentration is a unimodal function of time, whereas the ISD is monotonically decreasing with size at any time. The Family-Viscek scaling function is also monotonically decreasing and close to exponential decay. These parameter-free solutions of the well-defined problem should be useful for general understanding of the ISD scaling properties and the analysis of concrete systems.
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