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Exponent equality for capture-zone scaling in island nucleation: Theory and application to organic films

It is known in thin-film deposition that the density of nucleated clusters $N$ varies with the deposition rate $R$ as a power law, $N \sim R^α$. The exponent $α$ is a function of the critical nucleus size $i$ in a way that changes with the aggregation-limiting process active in a given system. We extend here to generic aggregation-limiting processes the derivation of the analytical capture-zone distribution function $P_β(s) = a_βs^β\exp(-b_βs^2)$ of Pimpinelli and Einstein [Phys. Rev. Lett. 99, 226102 (2007)]. We show that the exponent $β$ is generally related to the critical nucleus size $i$ and to the exponent $α$ by the equality $α(2β+ d_f - 2) = 2i$ where $d_f$ is the fractal dimensionality of the clusters. This remarkable results allows one to measure $i$ with no a priori knowledge of the actual aggregation mechanism. We apply this equality to measuring the critical nucleus size in pentacene deposition on mica.