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

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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 Rα. 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 Pβ(s) = aβsβ(-bβs²) 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+ df - 2) = 2i where df 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.