Scale invariance of continuum size distribution upon irreversible growth of surface islands

The continuum kinetic equation for irreversible heterogeneous growth of a surface island is ana-lyzed given a special form of the dependence of capture coefficient $\sigma$ on size $s$ and coverage of the surface $\Theta$. It is shown that, if $\sigma(s,\Theta)=\alpha(\Theta)(a+s)^\beta$, the function $\alpha(\Theta)$ is arbitrary, and 0 $\le\beta\le$ 1, then the solutions of the continuum equation of the first order satisfy the hypothesis about the scale invariance of the size distribu-tion (scaling) in a single exceptional case – at $\beta$ = 1. The obtained results testify about the presence of a fundamental relation of the scaling and linearity of the dependence $\sigma(s)$. Problems about associations of distri-bution functions in continuum and discrete growth models and about application of the obtained solutions for modeling and interpretation of experimental data in different systems are discussed.