Survey of morphologies formed in the wake of an enslaved phase-separation front in two dimensions

Foard, Eric; Wagner, A. James

doi:10.1103/physreve.85.011501

Cited by 37 publications

(52 citation statements)

References 19 publications

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“…Together with the construction of stripe‐formation parallel to the interface in a quenched Cahn–Hilliard equation based on matched asymptotics , one would hope to be able to adapt our strategy to establish the existence of such fronts forming oblique stripes. Again, the prediction of parallel only stripes at larger speeds that we predict to be quite universally valid is in agreement with the observations, here .…”

Section: Applications and Discussionsupporting

confidence: 92%

“…More precisely, we are interested in systems that exhibit stable or metastable ordered states, such as stripes, or spots arranged in hexagonal lattices. Examples of such systems arise for example in di‐block copolymers , phase‐field models , and other phase separative systems , as well as in phyllotaxis , and reaction–diffusion systems . Throughout, we will focus on a paradigmatic model, the Swift–Hohenberg equation

u_{t} = - {(1 + Δ)}^{2} u + μ u - u^{3},

where

u = u (t, x, y) \in R

false(x, y false) \in {double-struckR}^{2}

t \in R

, subscripts denote partial derivatives, and

Δ u = u_{x x} + u_{y y}

.…”

Section: Introductionmentioning

confidence: 99%

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Pattern‐forming fronts in a Swift–Hohenberg equation with directional quenching — parallel and oblique stripes

Goh

Scheel

2018

J. London Math. Soc.

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We study the effect of domain growth on the orientation of striped phases in a Swift–Hohenberg equation. Domain growth is encoded in a step‐like parameter dependence that allows stripe formation in a half plane, and suppresses patterns in the complement, while the boundary of the pattern‐forming region is propagating with fixed normal velocity. We construct front solutions that leave behind stripes in the pattern‐forming region that are parallel to or at a small oblique angle to the boundary. Technically, the construction of stripe formation parallel to the boundary relies on ill‐posed, infinite‐dimensional spatial dynamics. Stripes forming at a small oblique angle are constructed using a functional‐analytic, perturbative approach. Here, the main difficulties are the presence of continuous spectrum and the fact that small oblique angles appear as a singular perturbation in a traveling‐wave problem. We resolve the former difficulty using a farfield‐core decomposition and Fredholm theory in weighted spaces. The singular perturbation problem is resolved using preconditioners and boot‐strapping.

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Section: Applications and Discussionsupporting

confidence: 92%

u_{t} = - {(1 + Δ)}^{2} u + μ u - u^{3},

where

u = u (t, x, y) \in R

false(x, y false) \in {double-struckR}^{2}

t \in R

, subscripts denote partial derivatives, and

Δ u = u_{x x} + u_{y y}

.…”

Section: Introductionmentioning

confidence: 99%

Pattern‐forming fronts in a Swift–Hohenberg equation with directional quenching — parallel and oblique stripes

Goh

Scheel

2018

J. London Math. Soc.

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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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“…Furthermore, the noise widens the available regions of the meta-and unstable states (see earlier morphological phase diagrams of Liesegang patterns [27]). Noiseless Cahn-Hilliard type dynamics where the front moves with fixed velocity have been much studied [28,29]. In these cases, however, noise was present in the initial state, and complex morphologies resulted from complex initial conditions or from complex motion of the reaction front.…”

mentioning

confidence: 99%

Probability of the Emergence of Helical Precipitation Patterns in the Wake of Reaction-Diffusion Fronts

et al. 2013

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Helical and helicoidal precipitation patterns emerging in the wake of reaction-diffusion fronts are studied. In our experiments, these chiral structures arise with well defined probabilities PH controlled by conditions such as e.g., the initial concentration of the reagents. We develop a model which describes the observed experimental trends. The results suggest that PH is determined by a delicate interplay among the time-and length-scales related to the front and to the unstable precipitation modes and, furthermore, the noise amplitude also plays a quantifiable role.PACS numbers: 68.35.Ct Helices and helicoids are present from nano-to macroscale (ZnO nanohelices [1], macromolecules and inorganic crystals with a helical structure [2, 3], precipitation helices [4][5][6], fiber geometry of heart walls [7]). Formation of these fascinating structures generally follows two routes. First, templates with chiral symmetry (e.g., oragogel fibers) may exist in the system, and the symmetry is just transcribed to a structure (e.g., inorganic materials [8]) at a larger scale. Second, spontaneous symmetry breaking may occur through the self-assembly of achiral building blocks into a helical/helicoidal structure, as e.g. in case of crystals with chiral morphology [2,9].Theoretically, the symmetry-breaking route is more interesting. Universal aspects may emerge and the robust features of this self-organization process may be important for applications as well. Indeed, control over creating helical structures would make engineering (in particular, the bottom-up design of micro-patterns [10]) more flexible since chiral morphology of materials are known to affect their physical (electronic) properties [6,11].In order to develop insight into the genesis of helices/helicoids, we investigate an emblematic example of pattern formation, namely the formation of precipitation patterns in the wake of reaction-diffusion fronts [12,13]. The motivation for this choice comes from the observation that helicoidal structures have an axis, and the correlations are simple in the plane perpendicular to the axis. This suggests that building the perpendicular correlations in the wake of an advancing planar front may be a simple and natural mechanism of creating helices/helicoids. Additional motivation comes from the existence of a large body of knowledge in the related Liesegang phenomena [12,13]. It allows the use of well-established experimental and theoretical approaches, thus making it easier to develop a novel view on the formation of helical structures.Our main results concern the probabilistic aspects of the symmetry-breaking route. We determine the probability P H of the emergence of single helices/helicoids in Liesegang-type experiments as the conditions such as the initial concentration of inner or outer electrolytes, or the temperature are changed. P H is found to be well reproducible and large (P H > 0.5 for some parameter range). The results are understood by expanding and simulating a model of formation of precipitation patterns [14]...

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Survey of morphologies formed in the wake of an enslaved phase-separation front in two dimensions

Cited by 37 publications

References 19 publications

Pattern‐forming fronts in a Swift–Hohenberg equation with directional quenching — parallel and oblique stripes

Pattern‐forming fronts in a Swift–Hohenberg equation with directional quenching — parallel and oblique stripes

First order phase transitions and the thermodynamic limit

Probability of the Emergence of Helical Precipitation Patterns in the Wake of Reaction-Diffusion Fronts

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