Double-exponential scaling function for island-size distributions

We study the island-size distributions in homogeneous nucleation and growth during submonolayer deposition depending on the form of capture numbers σ(s) in the rate equations. For the power-law size dependences of the capture numbers σ(s)∼s^{α}, our solutions to the continuum rate equation are reduced to the known Family-Vicsek scaling functions of the scaled size x=s/s[over ¯] for 0≤α<1/2. These solutions contain singularities at large x. The singularities increase with the growth index α and become not normalizable at 1/2<α<1, showing the absence of any scaling solutions in this case. The only analytic scaling function is obtained at α=1. Using these results, we revisit the Bartelt-Evans theory for the scaled capture numbers σ(s)=σ[over ¯]C(x). We show that the source of singularities in the scaling functions is the zero growth rate dx/dlnτ at a maximum size that collects islands of any initial size in the single point of attraction. To circumvent the singularity, we propose an analytic form of C(x), which tends to a constant at x→0 and becomes linear at large x. The resulting analytic scaling function has the double-exponential shape, satisfies the sum rules for the island density, size, and scaled capture number, and is very close to the functions previously obtained by kinetic Monte Carlo simulations.