Continuation for Thin Film Hydrodynamics and Related Scalar Problems

Engelnkemper, Sebastian; Gurevich, Svetlana V.; Uecker, Hannes; Wetzel, Daniel; Thiele, Uwe

doi:10.1007/978-3-319-91494-7_13

Cited by 37 publications

(66 citation statements)

References 137 publications

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“…Since the Cahn-Hilliard and PFC models are both continuity equations and we only consider steady states with periodic or no-flux boundary conditions, we can use the integrated equations (5) and (8), respectively. There are then two possibilities for studying the dependence of the various steady states on the mean concentration f 0 [26]. Direct continuation in the control parameter f 0 is incorporated through an integral condition.…”

Section: Numerical Approachmentioning

confidence: 99%

“…These states exhibit the phenomenon of slanted snaking [13,61,62] but take the form of standard homoclinic snaking when the order parameter f 0 is plotted as a function of the chemical potential, a property that is expected to carry over to other models with a conserved order parameter. See the conclusion of [20,26] for further discussion. Note that figure 9 highlights the states representing global energy minima using thick line segments, an approach also taken for other systems showing multiple steady states, as done, e.g.…”

Section: Phase Behaviormentioning

confidence: 99%

“…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. However, this changes the properties of the associated bifurcation diagram [26]. Strictly speaking, μ is actually a scaled chemical potential difference, since the local concentration f is a scaled difference in the local densities, although below we refer to it simply as the chemical potential.…”

Section: Introductionmentioning

confidence: 99%

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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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Section: Numerical Approachmentioning

confidence: 99%

Section: Phase Behaviormentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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First order phase transitions and the thermodynamic limit

et al. 2019

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“…The introduction of Ref. [42] gives an overview of common models that can be brought in the form (2).…”

Section: General Formmentioning

confidence: 99%

“…As we are interested in the bifurcation structure of transversally inhomogeneous deposition in coating problems we entirely focus on results obtained with numerical path continuation methods [38,41,42] that have to our knowledge not yet been applied to such a problem.…”

Section: Numerical Path Continuationmentioning

confidence: 99%

Two-dimensional patterns in dip coating - first steps on the continuation path

Mitas

Thiele

et al. 2020

Physica D: Nonlinear Phenomena

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We present a brief comparative investigation of the bifurcation structure related to the formation of two-dimensional deposition patterns as described by continuum models of Cahn-Hilliard type. These are, on the one hand a driven Cahn-Hilliard model for Langmuir-Blodgett transfer of a surfactant layer from the surface of a bath onto a moving plate and on the other hand a driven thin-film equation modelling the surface acoustic wave-driven coating of a plate by a simple liquid. In both cases, we present selected two-dimensional steady states corresponding to deposition patterns and discuss the main structure of the corresponding bifurcation diagrams.

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Pattern formation on regular polygons and circles

Woolley

2024

J Nonlinear Sci

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We investigate the formation of Turing patterns on regular polygonal domains, as the number of edges grow, leading to the limiting case of the circle. Using linear and weakly nonlinear analysis, and evidence by simulations, we demonstrate how the domain shape can fundamentally change the expected bifurcation structure. Specifically, on the square domain we are able to derive pitchfork bifurcations for stripe and spot solutions, as well as show that both branches cannot bifurcate to produce stable patterns. This compares with the case of the equilateral triangle domain that causes the Turing bifurcation to be generically transcritical and, in some cases, none of the bifurcating branches are stable. Moreover, we find a monotonically increasing, but nonlinear relationship, between the minimal bifurcation area and the number of edges. Thus, patterns can occur on triangles with much smaller areas than circles. Overall, this work raises questions for researchers who are simulating applications on domains with simple shapes. Specifically, even small changes to domain geometry can have large impacts on the produced patterns; thus, domain perturbations should be considered in any sensitivity analyses.

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Continuation for Thin Film Hydrodynamics and Related Scalar Problems

Cited by 37 publications

References 137 publications

First order phase transitions and the thermodynamic limit

First order phase transitions and the thermodynamic limit

Two-dimensional patterns in dip coating - first steps on the continuation path

Pattern formation on regular polygons and circles

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