Logarithmic Clustering in Submonolayer Epitaxial Growth

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We investigate submonolayer epitaxial growth with a fixed monomer flux and irreversible aggregation of adatom islands due to their effective diffusion. When the diffusivity Dₖ of an island of mass k is proportional to k⁻μ, a Smoluchowski rate equation approach predicts steady behavior for 0<μ<1, with the concentration cₖ of islands of mass k varying as k⁻⁽³⁻μ⁾/². For μ>1, continuous evolution occurs in which cₖ(t) (t)⁻⁽²ᵏ⁻¹⁾μ/², while the total island density increases as N(t) (t)μ/². Monte Carlo simulations support these predictions.