A restricted solid-on-solid model for growth on fractal substrates

Abstract

The restricted solid-on-solid (RSOS) model for growth on twodifferent fractal substrates having the same fractal dimensiondf = 2 is studied. The Sierpinski gasket and the checkerboard fractal embedded in threedimensions are considered as the substrates. It is found that the interface widthW growsas tβ withβ≈0.26 for growth on a Sierpinskigasket and with β≈0.30 for growth on a checkerboard fractal. For the saturated regime,W followsW∼Lα,L being the size of thesystem, with α≈0.50 for aSierpinski gasket and α≈0.56 for a checkerboard fractal, implying that the growing surfaces of fractal substratesare rougher than that of a regular substrate. The growth exponent is notfully determined by the fractal dimension only, and the dynamic exponentz, obtained fromthe relation z = α/β, for both fractal lattices does not satisfy the scaling relationα+z = 2 due to the intrinsic fractal nature of the substrate. The RSOS model for growth on aregular lattice is generally believed to be described by the Kardar–Parisi–Zhang (KPZ)equation. However, the RSOS model for fractal substrates does not appear to follow theKPZ type universality. Generalization to the equilibrium RSOS model for growth on thefractal substrates is also investigated.