Size distributions and scaling in heterogeneous nucleation with size-linear rate constants

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.