Scaling and Exponent Equalities in Island Nucleation: Novel Results and Application to Organic Films

It is known in thin-film deposition that the density of nucleated clusters N varies with the deposition rate F as a power law, N ∼ F^α. The exponent α is a function of the critical nucleus size i in a way that changes with the aggregation limiting process. We extend here the derivation of the analytical capture-zone distribution function P_β(s) = a ·s ·exp(-bs²) of Pimpinelli and Einstein to generic aggregation-limiting processes. We show that the parameter β is generally related to the critical nucleus size i and to the exponent α by the equality α·β = i, in the case of compact islands. This remarkable result 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 for pentacene deposition on mica. This system shows a crossover from diffusion-limited to attachment-limited aggregation with increasing deposition rates.