Periodic Stationary Patterns Governed by a Convective Cahn--Hilliard Equation

Zaks, Michael A.; Podolny, A.; Nepomnyashchy, Alexander; Golovin, A. A.

doi:10.1137/040615766

Cited by 34 publications

(45 citation statements)

References 46 publications

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“…Indeed, reactive plasma techniques stand out as efficient tools for nanostructuring [21]. This system admits an accurate description [22] by a convective Cahn-Hilliard (CCH) equation [23], which is a paradigmatic model for nonvariational coarsening systems, from step instabilities in epitaxy to dewetting of a thin film flowing down an inclined plane, see, e.g., [24] and references therein. We analyze the topological properties of the cellular pattern, and obtain as a consequence the coarsening law for the domain size.…”

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confidence: 99%

Pattern-Wavelength Coarsening from Topological Dynamics in Silicon Nanofoams

et al. 2014

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We report the experimental observation of a submicron cellular structure on the surface of silicon targets eroded by an ion plasma. Analysis by atomic force microscopy allows us to assess the time evolution and show that the system can be described quantitatively by the convective Cahn-Hilliard equation, found in the study of domain coarsening for a large class of driven systems. The space-filling trait of the ensuing pattern relates it to evolving foams. Through this connection, we are actually able to derive the coarsening law for the pattern wavelength from the nontrivial topological dynamics of the cellular structure. Thus, the study of the topological properties of patterns in nonvariational spatially extended systems emerges as complementary to morphological approaches to their challenging coarsening properties.

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confidence: 99%

Pattern-Wavelength Coarsening from Topological Dynamics in Silicon Nanofoams

et al. 2014

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“…(see (38)) on the pattern wavenumber K = 2π/l [16], [73], [64]. If dA/dK < 0 for any K, which takes place for [74].…”

Section: Oscillatory Tails Of Domain Walls and Stability Of Stationarmentioning

confidence: 99%

“…If dA/dK < 0 for any K, which takes place for [74]. The region of oscillatory response can contain a subinterval of stable patterns (where σ 2 (K) < 0) [60], [64]. That is possible because of the alternating sign of the interaction between domain walls.…”

Section: Oscillatory Tails Of Domain Walls and Stability Of Stationarmentioning

confidence: 99%

Coarsening versus pattern formation

Nepomnyashchy

2015

Comptes Rendus. Physique

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It is known that similar physical systems can reveal two quite different ways of behavior, either coarsening, which creates a uniform state or a large-scale structure, or formation of ordered or disordered patterns, which are never homogenized. We present a description of coarsening using simple basic models, the Allen-Cahn equation and the Cahn-Hilliard equation, and discuss the factors that may slow down and arrest the process of coarsening. Among them are pinning of domain walls on inhomogeneities, oscillatory tails of domain walls, nonlocal interactions, and others. Coarsening of pattern domains is also discussed.

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“…In their paper, the dynamics of domain walls (kinks) governed by the convective Cahn-Hilliard equation was studied by means of asymptotic and numerical methods. M. A. Zaks et al [19] investigate bifurcations of stations periodic solutions of a convective Cahn-Hilliard equation, they described phase separation in driven systems, and studied the stability of the main family of these solutions. Eden and Kalantarov [3,4] considered the convective Cahn-Hilliard equation as [12] with periodic boundary conditions in one space dimension and three space dimension.…”

Section: Introductionmentioning

confidence: 99%