A restricted curvature model on a Sierpinski gasket substrate

Abstract

The surface structure of an equilibrium restricted curvature (RC) model on a Sierpinski gasket substrate is studied. The surface width W increases as tβ at early time t and becomes saturated at Lα for , where L is the system size. The growth exponent β≈0.323, the roughness exponent α≈1.54 and the dynamic exponent z≈4.78 are obtained numerically. They satisfy the scaling relations 2α + df = z and z = 2zrw very well, where zrw is the random walk exponent of the Sierpinski gasket. We introduce a fractional Langevin equation to describe the model.

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

“…The scaling exponents derived satisfied the relations 2α + d f ≈ z and z ≈ 2z rw very well [15]. To describe this discrete model on fractal substrates, they introduced the fractional Langevin equation [15], which is called the fractal Mullins-Herring (MH) equation, written as…”

“…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].…”

“…In one of their recent works, Kim and co-workers [15] performed numerical simulations on the surface structures of the equilibrium restricted curvature (ERC) model on a Sierpinski gasket substrate and obtained the results that α ≈ 1.54, β ≈ 0.323 and z = α/β ≈ 4.78. The scaling exponents derived satisfied the relations 2α + d f ≈ z and z ≈ 2z rw very well [15].…”

Surface structures of equilibrium restricted curvature model on two fractal substrates

“…The growth process of the etching model on the fractal substrate, however, can no longer be described by the original KPZ equation. [15] In one of their recent works, Kim et al [20] performed numerical simulations on the surface structures of the ERC model on Sierpinski gasket substrate and obtained the results α ≈ 1.54, β ≈ 0.323, and z=α/β ≈ 4.78. The scaling exponents derived satisfied the scaling relations 2α + d f ≈ z and z ≈ 2z rw very well.…”

“…The scaling exponents derived satisfied the scaling relations 2α + d f ≈ z and z ≈ 2z rw very well. [20] To describe this discrete model on fractal substrates, they introduced the fractional Langevin equation, [20] written as…”