Scenarios of domain pattern formation in a reaction-diffusion system

Muratov, Cyrill B.; Осипов, В. В.

doi:10.1103/physreve.54.4860

Cited by 52 publications

(74 citation statements)

References 35 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…For pure diblock copolymers, on the other hand, the region between micelletype structures must be filled with a homogeneous mixture, which may or may not be unstable. This paper's results have some qualitative similarity to a variety of systems that arise in the reaction-diffusion literature [13,21,27,45]. Pattern formation in variational models similar to ours has been studied by Muratov [26].…”

Section: Tionsmentioning

confidence: 51%

Spatially Localized Structures in Diblock Copolymer Mixtures

Glasner¹

2010

SIAM J. Appl. Math.

View full text Add to dashboard Cite

Abstract. Above the critical temperature for the order-disorder transition, diblock copolymer melts have been observed to exhibit localized structures that exist within the homogeneous mixture. This paper uses an Ohta-Kawasaki-type density functional to explore this regime. Spatially localized peak-shaped equilibria are studied in one, two, and three dimensions, corresponding to amphiphilic bilayers, cylindrical micelles, and spherical micelles, respectively. A combination of rigorous estimates, asymptotic analysis, and numerical computation is used to characterize solutions and the regime where they exist. The interaction of superpositions of these solutions is studied by a perturbation analysis and shows how steady multipeak configurations can be achieved. Evidence is found for a secondary bifurcation slightly below the spinodal instability threshold, beyond which self-replication phenomena are observed. Dynamics in two dimensions are also illustrated, suggesting other mechanisms for instability and growth.Key words. diblock copolymer, micelle, density functional theory, self-replication AMS subject classifications. 82D60, 35K55DOI. 10.1137/080743913 1. Introduction. Diblock copolymers are linear chain molecules composed of two distinct subchains. At low temperatures, repulsion between subchains can result in phase segregation, but macroscopic segregation cannot occur because of covalent bonding between subchains. This incomplete segregation can give rise to a variety of stable microstructures (see, e.g., [1,11,14,24]), which are typically characterized as spatially periodic patterns whose domain boundaries closely resemble minimal surfaces, such as spheres, cylinders, and gyroids [39].Outside of the regime where mixtures are unstable to spinodal decomposition, the situation for copolymers is more complicated than simple binary systems, such as those described by the Cahn-Hilliard theory [5]. In addition to a stable homogeneous state, there are other observed behaviors. Physical experiments have identified inhomogeneous states above the critical temperature, such as sparse arrangements of micellar structures [35,36,43] and anisotropic lamellar grains [15]. This paper seeks to investigate this regime and its dynamics using density functional theory and the corresponding gradient flow dynamics.Theoretical work on the disordered regime has focused on spherical micelles. Dormidontova and Lodge [9] use Semenov's strong segregation theory [37] to predict the concentration of micelle assemblies. Their theory predicts that the number of micelles increases rapidly below a "critical micelle temperature" (CMT), which is greater than the temperature at which a homogeneous mixture is spinodally unstable. Wang, Wang, and Yang [42] use self-consistent field theory calculations to find the density profile of individual spherical micelles and evaluate their excess free energy. They find that micelles exist (within the mean field approximation) for volume frac-

show abstract

Section: Tionsmentioning

confidence: 51%

Spatially Localized Structures in Diblock Copolymer Mixtures

Glasner¹

2010

SIAM J. Appl. Math.

View full text Add to dashboard Cite

show abstract

“…II B). Thus, at the threshold of the instability the domain pattern with characteristic size ∼ λ will start to form [2,57]. These domains will still be smaller than the equilibrium size R ∼ ǫ −2/3 , so the Let us emphasize that the patterns that form at the end of the simulations of Fig.…”

Section: Coarsening and Disordermentioning

confidence: 93%

“…We will analyze the spectrum of L for simple geometries below (see also [28,45,48,50,51,57] [48,50,57] based on the stability analysis of the localized and periodic patterns). Indeed, suppose a pattern is made of a collection of droplets of size and distance between each other of order R…”

Section: Properties Of the Domain Patterns A Equations For Statmentioning

confidence: 99%

Theory of domain patterns in systems with long-range interactions of Coulomb type

2002

Self Cite

View full text Add to dashboard Cite

We develop a theory of the domain patterns in systems with competing short-range attractive interactions and long range repulsive Coulomb interactions. We take an energetic approach, in which patterns are considered as critical points of a mean-field free energy functional. Close to the microphase separation transition, this functional takes on a universal form, allowing to treat a number of diverse physical situations within a unified framework. We use asymptotic analysis to study domain patterns with sharp interfaces. We derived an interfacial representation of the pattern's free energy which remains valid in the fluctuating system, with a suitable renormalization of the Coulomb interaction's coupling constant. We also derived integrodifferential equations describing the stationary domain patterns of arbitrary shapes and their thermodynamic stability, coming from the first and second variation of the interfacial free energy. We showed that the length scale of a stable domain pattern must obey a certain scaling law with the strength of the Coulomb interaction. We analyzed existence and stability of localized (spots, stripes, annuli) and periodic (lamellar, hexagonal) patterns in two dimensions. We showed that these patterns are metastable in certain ranges of the parameters and that they can undergo morphological instabilities leading to the formation of more complex patterns. We discuss nucleation of the domain patterns by thermal fluctuations and pattern formation scenarios for various thermal quenches. We argue that self-induced disorder is an intrinsic property of the domain patterns in the systems under consideration.

show abstract

“…The examples we present in figures 2-8 and the nonlinear simulations used a form of the FitzHugh-Nagumo (FN) model [30], for which the local dynamics is given…”

Section: Linear Analysis Of Rda Equationsmentioning

confidence: 99%

Pattern formation by boundary forcing in convectively unstable, oscillatory media with and without differential transport

McGraw

Menzinger

2005

Phys. Rev. E

View full text Add to dashboard Cite

Motivated by recent experiments and models of biological segmentation, we analyze the excitation of pattern-forming instabilities in convectively unstable reaction-diffusion-advection systems, occuring by constant or periodic forcing at the upstream boundary. Such boundary-controlled pattern selection is a generalization of the flow-distributed-oscillation (FDO) mechanism that can be modified to include differential diffusion (Turing) and differential flow (DIFI) modes. Our goal is to clarify the relationships among these mechanisms in the general case where there is differential flow as well as differential diffusion. We do so by analyzing the dispersion relation for linear perturbations and showing how its solutions are affected by differential transport. We find a close relationship between DIFI and FDO modes, while the Turing mechanism gives rise to a distinct set of unstable modes. Finally, we illustrate the relevance of the dispersion relations using nonlinear simulations and we discuss experimental implications of our results.

show abstract

Scenarios of domain pattern formation in a reaction-diffusion system

Cited by 52 publications

References 35 publications

Spatially Localized Structures in Diblock Copolymer Mixtures

Spatially Localized Structures in Diblock Copolymer Mixtures

Theory of domain patterns in systems with long-range interactions of Coulomb type

Pattern formation by boundary forcing in convectively unstable, oscillatory media with and without differential transport

Contact Info

Product

Resources

About