We study the island-size distributions in homogeneous nucleation and growth during submonolayer deposition depending on the form of capture numbers $\sigma \left(s\right)$ in the rate equations. For the power-law size dependences of the capture numbers $\sigma \left(s\right)\sim s^{\alpha }$, our solutions to the continuum rate equation are reduced to the known Family-Vicsek scaling functions of the scaled size $x=s/\overline{s}$ for $0\le \alpha <1/2$. These solutions contain singularities at large $x$. The singularities increase with the growth index $\alpha$ and become not normalizable at $1/2<\alpha <1$, showing the absence of any scaling solutions in this case. The only analytic scaling function is obtained at $\alpha =1$. Using these results, we revisit the Bartelt-Evans theory for the scaled capture numbers $\sigma \left(s\right)=\overline{\sigma }C\left(x\right)$. We show that the source of singularities in the scaling functions is the zero growth rate $dx/dln\tau$ 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 $C\left(x\right)$, which tends to a constant at $x\to 0$ 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.

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• Nucleation