Hopf Bifurcation from Fronts in the Cahn–Hilliard Equation

Goh, Ryan; Scheel, Arnd

doi:10.1007/s00205-015-0853-2

Cited by 18 publications

(16 citation statements)

References 54 publications

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“…First, one could work in exponentially weighted spaces, resorting to stronger assumptions on the inhomogeneity. Fredholm properties of differential operators on the real line in exponentially weighted spaces are well known [20,25] and have been used in the context of perturbation and bifurcation theory in the presence of essential spectrum [25,8].…”

Section: Discussionmentioning

confidence: 99%

The effect of impurities on striped phases

Jaramillo

Scheel

2018

Proceedings of the Royal Society of Edinburgh: Section A Mathem

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We study the effect of algebraically localized impurities on striped phases in one space-dimension. We therefore develop a functional-analytic framework which allows us to cast the perturbation problem as a regular Fredholm problem despite the presence of essential spectrum, caused by the soft translational mode. Our results establish the selection of jumps in wavenumber and phase, depending on the location of the impurity and the average wavenumber in the system. We also show that, for select locations, the jump in the wavenumber vanishes.

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Section: Discussionmentioning

confidence: 99%

The effect of impurities on striped phases

Jaramillo

Scheel

2018

Proceedings of the Royal Society of Edinburgh: Section A Mathem

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“…See, e.g. [34][35][36] for situations where this condition is not fulfilled. We note in passing that in situations where the total concentration is not controlled, the parameter μ becomes a relevant control parameter representing an external field or imposed chemical potential.…”

Section: Introductionmentioning

confidence: 99%

First order phase transitions and the thermodynamic limit

et al. 2019

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We consider simple mean field continuum models for first order liquid-liquid demixing and solidliquid phase transitions and show how the Maxwell construction at phase coexistence emerges on going from finite-size closed systems to the thermodynamic limit. The theories considered are the Cahn-Hilliard model of phase separation, which is also a model for the liquid-gas transition, and the phase field crystal model of the solid-liquid transition. Our results show that states comprising the Maxwell line depend strongly on the mean density with spatially localized structures playing a key role in the approach to the thermodynamic limit.However, in a system of finite size or when a finite time horizon is considered, metastable states often play an important role and even unstable states may be crucial, as transient states, for extended time periods. The full set of states and their dependence on the various control parameters is conveniently presented in the form of bifurcation diagrams, well known in the context of dynamical systems and pattern formation theory [3][4][5]. The place of thermodynamic phase diagrams is taken by 'morphological phase diagrams' or state diagrams and stability diagrams [3,4,6]. The notion of a Maxwell point is often used in the context of pattern formation in nonconserved systems [7][8][9][10][11] to indicate equal energy states because of its dynamical significance [12,13]. In this context this notion applies equally to finite and infinite systems [7,8] although for finite systems it lacks the thermodynamic relevance as the condition for phase coexistence. In the context of buckling the corresponding concept is the Maxwell load [14].In this paper we show and discuss how the discontinuities in the TL represented by the Maxwell construction arise from the bifurcation diagrams relating stable, metastable and unstable steady states in finite-size systems. We focus on two systems: (i) phase decomposition of a binary liquid mixture and (ii) the liquid to crystalline solid phase transition. We investigate the transitions that occur in the context of the most basic mean-field continuum models for these two different phase transitions, namely, the Cahn-Hilliard equation [15][16][17] and the phase field crystal (PFC) model (or conserved Swift-Hohenberg equation) [18][19][20].Some aspects related to this question have been considered previously, in particular in relation to the nature of some of the states that can arise in finite-size systems in the two-phase region. References [21-24] describe theory and computer simulation results for atomistic models exhibiting gas-liquid, liquid-hexatic and hexaticsolid phase transitions that indicate how the Maxwell construction develops as the system size increases or the temperature decreases. For example, figures 3-5 of [23] compare Monte-Carlo computer simulation results in a finite three-dimensional domain (see also figures 2 and 3 of [22]) with the results from a capillary drop type model, also in a finite domain, with a mean-field expression for the chemi...

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“…This externally imposed front velocity is a control parameter of the system. Examples for investigations of such triggered pattern formation range from the experimentally and theoretically investigated structure formation in Langmuir-Blodgett films [29][30][31][32][33][34], over the study of Cahn-Hilliard-type model equations in one (1D) and two (2D) dimensions for externally quenched phase separation (e.g., by a moving temperature jump for films of polymer blends or binary mixtures) [35][36][37] to the rigorous mathematical analysis of trigger fronts in a complex Ginzburg-Landau equation as well as in a Cahn-Hilliard and an Allen-Cahn equation [38][39][40]. In the aforementioned systems, a switch from a linearly stable to an unstable state takes place at a certain position within the considered domain.…”

Section: Introductionmentioning

confidence: 99%

Self-organized dip-coating patterns of simple, partially wetting, nonvolatile liquids

et al. 2019

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When a solid substrate is withdrawn from a bath of simple, partially wetting, nonvolatile liquid, one typically distinguishes two regimes, namely, after withdrawal the substrate is macroscopically dry or homogeneously coated by a liquid film. In the latter case, the coating is called a Landau-Levich film. Its thickness depends on the angle and velocity of substrate withdrawal. We predict by means of a numerical and analytical investigation of a hydrodynamic thin-film model the existence of a third regime. It consists of the deposition of a regular pattern of liquid ridges oriented parallel to the meniscus. We establish that the mechanism of the underlying meniscus instability originates from competing film dewetting and Landau-Levich film deposition. Our analysis combines a marginal stability analysis, numerical time simulations and a numerical bifurcation study via path-continuation.

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Hopf Bifurcation from Fronts in the Cahn–Hilliard Equation

Cited by 18 publications

References 54 publications

The effect of impurities on striped phases

The effect of impurities on striped phases

First order phase transitions and the thermodynamic limit

Self-organized dip-coating patterns of simple, partially wetting, nonvolatile liquids

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