A Model for Ion-Sputtering: from Pattern Formation to Rough Surfaces

1995 ◽  
Vol 407 ◽  
Author(s):  
R. Cuerno ◽  
H. A. Makse ◽  
S. Tomassone ◽  
S. T. Harrington ◽  
H. E. Tanley
Keyword(s):  
Pattern Formation ◽  
Periodic Pattern ◽  
Affine Scaling ◽  
Ion Sputtering ◽  
Unified Framework ◽  
Surface Evolution ◽  
Continuum Equation ◽  
Time Regime ◽  
Sivashinsky Equation ◽  
The Continuum

ABSTRACTMany surfaces eroded by ion-sputtering have been observed to develop morphologies which are either periodic, or rough and non-periodic. We have introduced a discrete stochastic model that allows to interpret these experimental observations within a unified framework. A simple periodic pattern characterizes the initial stages of the surface evolution, whereas the later time regime is consistent with self-affine scaling. The continuum equation describing the surface height is a noisy version of the Kuramoto-Sivashinsky equation.

THE MORPHOLOGY AND EVOLUTION OF THE SURFACE IN EPITAXIAL AND THIN FILM GROWTH: A CONTINUUM MODEL WITH SURFACE DIFFUSION

Fractals ◽  
1993 ◽  
Vol 01 (04) ◽  
pp. 753-766 ◽  
Author(s):  
FEREYDOON FAMILY ◽  
JACQUES G. AMAR
Keyword(s):  
Thin Film ◽  
Surface Diffusion ◽  
Film Growth ◽  
Vertical Jump ◽  
Thin Film Growth ◽  
Discrete Models ◽  
Dynamic Scaling ◽  
Diffusion Operator ◽  
Continuum Equation ◽  
The Continuum

A number of discrete models as well as continuum equations have been proposed for describing epitaxial and thin film growth. We have shown that there exists a macroscopic groove instability in many of these models. This unphysical feature in the continuum equations arises from the truncation or linearization of the diffusion operator along the surface. A similar artifact occurs in the discrete models, because in these models adatoms only diffuse horizontally and must take an unphysical vertical jump at step edges. We have proposed and studied a continuum equation for epitaxial and thin-film growth in which the full diffusion along the surface is taken into account. The results of the solutions of this continuum equation, for the growth and the morphology of the surface, are in excellent agreement with recent low temperature molecular-beam epitaxy and ion-sputtering experiments. In particular, we find that at late times dynamic scaling breaks down and the surface is no longer a self-affine fractal. The surface develops a characteristic morphology whose dependence on deposition rate and surface diffusion is similar to that found in experiments.

CONTINUUM COUPLED-MAP APPROACH TO PATTERN FORMATION IN OSCILLATING GRANULAR LAYERS: ROBUSTNESS AND LIMITATION

2008 ◽  
Vol 18 (06) ◽  
pp. 1627-1643 ◽  
Author(s):  
MARY ANN F. HARRISON ◽  
YING-CHENG LAI
Keyword(s):  
Pattern Formation ◽  
Period Doubling ◽  
Spatial Inhomogeneity ◽  
Spatial Coupling ◽  
Quantitative Level ◽  
Qualitative Level ◽  
Coupled Maps ◽  
Granular Layers ◽  
Map Approach ◽  
The Continuum

Continuum coupled maps have been proposed as a generic and universal class of models to understand pattern formation in oscillating granular layers. Such models usually involve two features: Temporal period doubling in local maps and spatial coupling. The models can generate various patterns that bear striking similarities to those observed in real experiments. Here we ask two questions: (1) How robust are patterns generated by continuum coupled maps? and (2) Are there limitations, at a quantitative level, to the continuum coupled-map approach? We address the first question by investigating the effects of noise and spatial inhomogeneity on patterns generated. We also propose a measure to characterize the sharpness of the patterns. This allows us to demonstrate that patterns generated by the model are robust to random perturbations in both space and time. For the second question, we investigate the temporal scaling behavior of the disorder function, which has been proposed to characterize experimental patterns in granular layers. We find that patterns generated by continuum coupled maps do not exhibit scaling behaviors observed in experiments, suggesting that the coupled map approach, while insightful at a qualitative level, may not yield behaviors that are of importance to pattern characterization at a more quantitative level.

Periodic pattern formation in solid state reactions related to the Kirkendall effect

1998 ◽  
Vol 46 (18) ◽  
pp. 6521-6528 ◽  
Author(s):  
A.A. Kodentsov ◽  
M.R. Rijnders ◽  
F.J.J. van Loo
Keyword(s):  
Solid State ◽  
Pattern Formation ◽  
Kirkendall Effect ◽  
Solid State Reactions ◽  
Periodic Pattern

Laser Ablation Modeling of Periodic Pattern Formation on Polymer Substrates

2008 ◽  
Vol 10 (5) ◽  
pp. 488-493 ◽  
Author(s):  
A. Lasagni ◽  
M. Cornejo ◽  
F. Lasagni ◽  
F. Muecklich
Keyword(s):  
Laser Ablation ◽  
Pattern Formation ◽  
Periodic Pattern ◽  
Polymer Substrates ◽  
Ablation Modeling

Nano-patterning of surfaces by ion sputtering: numerical study of the damping effect on the anisotropic Kuramoto-Sivashinsky equation

2015 ◽  
Author(s):  
Eduardo Vitral ◽  
Daniel Walgraef ◽  
José Pontes ◽  
Gustavo Rabello dos Anjos ◽  
Norberto Mangiavacchi
Keyword(s):  
Numerical Study ◽  
Ion Sputtering ◽  
Damping Effect ◽  
Sivashinsky Equation ◽  
Nano Patterning

Bifurcations and Pattern Formation in a Predator–Prey Model

2018 ◽  
Vol 28 (11) ◽  
pp. 1850140 ◽  
Author(s):  
Yongli Cai ◽  
Zhanji Gui ◽  
Xuebing Zhang ◽  
Hongbo Shi ◽  
Weiming Wang
Keyword(s):  
Steady State ◽  
Hopf Bifurcation ◽  
Pattern Formation ◽  
Global Bifurcation ◽  
Periodic Pattern ◽  
Spatiotemporal Pattern ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey Model ◽  
Steady State Bifurcation

In this paper, we investigate the spatiotemporal dynamics of a Leslie–Gower predator–prey model incorporating a prey refuge subject to the Neumann boundary conditions. We mainly consider Hopf bifurcation and steady-state bifurcation which bifurcate from the constant positive steady-state of the model. In the case of Hopf bifurcation, by the center manifold theory and the normal form method, we establish the bifurcation direction and stability of bifurcating periodic solutions; in the case of steady-state bifurcation, by the local and global bifurcation theories, we prove the existence of the steady-state bifurcation, and find that there are two typical bifurcations, Turing bifurcation and Turing–Hopf bifurcation. Via numerical simulations, we find that the model exhibits not only stationary Turing pattern induced by diffusion which is dependent on space and independent of time, but also temporal periodic pattern induced by Hopf bifurcation which is dependent on time and independent of space, and spatiotemporal pattern induced by Turing–Hopf bifurcation which is dependent on both time and space. These results may enrich the pattern formation in the predator–prey model.

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