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We present a theoretical analysis of rate equations for heterogeneous nucleation of
nanomaterials with linear size dependences of the aggregation and fragmentation rate
constants. Two scenarios are considered, one relating to stable growth and the other
describing unstable situation with a time-dependent critical size. An interesting analytical
solution is obtained which is exact in the stable case and only asymptotic in the unstable
growth. This solution is expressed through the Polya distribution. Its continuum form features
scaling properties for all but very small sizes, which is an intrinsic property of the model. Our
scaled size distribution is capable of reproducing both monomodal and monotonically
decreasing shapes depending on the value of the dimerization constant. The obtained solution
is shown to reproduce fairly well some experimental size spectra of linear chains of metal
adatoms on Si(100) surfaces.
Keywords: nanomaterials; heterogeneous nucleation; rate equations |
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