Surface pattern formation and scaling described by conserved lattice gases

TL;DR

This paper models surface pattern formation using a conserved lattice gas approach, revealing how competing diffusion processes lead to diverse surface patterns and confirming the stability of KPZ scaling in two dimensions.

Contribution

The authors extend a discrete growth model to include conserved local exchange dynamics, enabling large-scale simulations and analysis of surface pattern formation and scaling behaviors.

Findings

Surface diffusion can be described by attracting or repelling dimers.

KPZ scaling remains stable despite surface diffusion.

Different patterns like dots or ripples emerge from competing diffusion processes.

Abstract

We extend our 2+1 dimensional discrete growth model (PRE 79, 021125 (2009)) with conserved, local exchange dynamics of octahedra, describing surface diffusion. A roughening process was realized by uphill diffusion and curvature dependence. By mapping the slopes onto particles two-dimensional, nonequilibrium binary lattice model emerge, in which the (smoothing/roughening) surface diffusion can be described by attracting or repelling motion of oriented dimers. The binary representation allows simulations on very large size and time scales. We provide numerical evidence for Mullins-Herring or molecular beam epitaxy class scaling of the surface width. The competition of inverse Mullins-Herring diffusion with a smoothing deposition, which corresponds to a Kardar-Parisi-Zhang (KPZ) process generates different patterns: dots or ripples. We analyze numerically the scaling and wavelength growth behavior in these models. In particular we confirm by large size simulations that the KPZ type of scaling is stable against the addition of this surface diffusion, hence this is the asymptotic behavior of the Kuramoto-Sivashinsky equation as conjectured by field theory in two dimensions, but has been debated numerically. If very strong, normal surface diffusion is added to a KPZ process we observe smooth surfaces with logarithmic growth, which can describe the mean-field behavior of the strong-coupling KPZ class. We show that ripple coarsening occurs if parallel surface currents are present, otherwise logarithmic behavior emerges.