Surface structures of equilibrium restricted curvature model on two fractal substrates

Abstract

With the aim to probe the effects of the microscopic details of fractal substrates on the scaling of discrete growth models, the surface structures of the equilibrium restricted curvature (ERC) model on Sierpinski arrowhead and crab substrates are analyzed by means of Monte Carlo simulations. These two fractal substrates have the same fractal dimension df, but possess different dynamic exponents of random walk zrw. The results show that the surface structure of the ERC model on fractal substrates are related to not only the fractal dimension df, but also to the microscopic structures of the substrates expressed by the dynamic exponent of random walk zrw. The ERC model growing on the two substrates follows the well-known Family—Vicsek scaling law and satisfies the scaling relations 2α + df ≍ z ≍ 2zrw. In addition, the values of the scaling exponents are in good agreement with the analytical prediction of the fractional Mullins—Herring equation.

A fractal langevin equation describing the kinetic roughening growth on fractal lattices

“…After that, the validity of the fractal MH equation and the scaling relation were confirmed by the model of ERC on other too fractal substrates, i.e. the Sierpinski arrowhead and the crab fractal substrate [25].…”

Dynamic scaling behaviors of linear fractal Langevin-type equation driven by nonconserved and conserved noise

“…After that, the validity of the fractional MH equation and the scaling relation were confirmed by numerical work [16]. In this paper, to probe the effects of the microscopic details of fractal substrates on the scaling behavior of the growth model deeply, the generalized linear fractal Langevin-type equations, driven by both nonconserved and conserved noise, are proposed and investigated theoretically based on scaling analysis.…”

“…(ii) The scaling relation 2α + d f = z is satisfied, independent of linear diffusion type (this can be seen from different values of factor m) on the surfaces and interfaces. (iii) These results of the general linear fractal Langevin-type equation can be tested by numerical works[13,16,19].…”

“…Among previous theoretical investigations of continuum equations, as well as numerical simulations of discrete atomistic models, much more were performed on regular or Euclidean substrates with integer dimension, however, less were devoted to fractal substrates. As a result, there is no simple and clear understanding about the interplay between the dynamical growth rules of the system and the self-similarity of fractal structures until the recent works [12][13][14][15][16][17][18][19][20].…”