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

Abstract

The restricted solid-on-solid (RSOS) model for growth on two different fractal substrates having the same fractal dimension d(f) = 2 is studied. The Sierpinski gasket and the checkerboard fractal embedded in three dimensions are considered as the substrates. It is found that the interface width W grows as t(beta) with beta approximate to 0.26 for growth on a Sierpinski gasket and with beta approximate to 0.30 for growth on a checkerboard fractal. For the saturated regime, W follows W similar to L-alpha, L being the size of the system, with alpha approximate to 0.50 for a Sierpinski gasket and alpha approximate to 0.56 for a checkerboard fractal, implying that the growing surfaces of fractal substrates are rougher than that of a regular substrate. The growth exponent is not fully determined by the fractal dimension only, and the dynamic exponent z, obtained from the relation z = alpha/beta, for both fractal lattices does not satisfy the scaling relation alpha + z = 2 due to the intrinsic fractal nature of the substrate. The RSOS model for growth on a regular 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 the KPZ type universality. Generalization to the equilibrium RSOS model for growth on the fractal substrates is also investigated.