A stationary core-shell assembly in a ternary inhibitory system

Ren, Xiaofeng; Wu, Chong

doi:10.3934/dcds.2017041

Cited by 14 publications

(10 citation statements)

References 31 publications

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“…The mathematical study of the triblock copolmyer problem is still in the early stage. There are existence theorems about stationary assemblies of core-shells [13], double bubbles [18], and discs [14], with the last work being the most relevant to this paper. Here we treat two of the three monomer types of a triblock copolymer as species and view the third type as the surrounding environment, dependent on the two species.…”

Section: Introductionmentioning

confidence: 99%

Nonhexagonal Lattices From a Two Species Interacting System

Luo¹,

Ren²,

Wei³

2020

SIAM J. Math. Anal.

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A two species interacting system motivated by the density functional theory for triblock copolymers contains long range interaction that affects the two species differently. In a two species periodic assembly of discs, the two species appear alternately on a lattice. A minimal two species periodic assembly is one with the least energy per lattice cell area. There is a parameter b in [0, 1] and the type of the lattice associated with a minimal assembly varies depending on b. There are several thresholds defined by a number B = 0.1867... If b ∈ [0, B), a minimal assembly is associated with a rectangular lattice whose ratio of the longer side and the shorter side is in [, a minimal assembly is associated with a square lattice; if b ∈ (1 − B, 1], a minimal assembly is associated with a rhombic lattice with an acute angle in [ π 3 , π 2 ). Only when b = 1, this rhombic lattice is a hexagonal lattice. None of the other values of b yields a hexagonal lattice, a sharp contrast to the situation for one species interacting systems, where hexagonal lattices are ubiquitously observed.

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Section: Introductionmentioning

confidence: 99%

Nonhexagonal Lattices From a Two Species Interacting System

Luo¹,

Ren²,

Wei³

2020

SIAM J. Math. Anal.

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“…Thus, global minimization should indeed produce a crystalline lattice of double and/or single bubbles, as in the stationary assemblies constructed in [53,46,45]. Theorem 3.1 provides a more precise statement which gives a fine detailed structure of minimizers v η of E η .…”

Section: Introductionmentioning

confidence: 94%

“…One-dimensional stationary points to the Euler-Lagrange equations of (1.1) were found in [48,14]. Two and three dimensional stationary configurations were studied recently in [54,53,45,46,22].…”

Section: Introductionmentioning

confidence: 99%

“…A recent series of papers by Ren & Wei [53] and Ren & Wang [46,45] consider the triblock energy in a parameter regime where two of the components are very dilute with respect to the third. They use perturbative arguments to generate stationary configurations consisting of assemblies of double bubbles, core shells, or single bubbles of both species, in an array.…”

Section: Introductionmentioning

confidence: 99%

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Periodic Minimizers of a Ternary Non-Local Isoperimetric Problem

Alama¹,

Bronsard²,

Lu³

et al. 2019

Preprint

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We study a two-dimensional ternary inhibitory system derived as a sharp-interface limit of the Nakazawa-Ohta density functional theory of triblock copolymers. This free energy functional combines an interface energy favoring micro-domain growth with a Coulomb-type long range interaction energy which prevents micro-domains from unlimited spreading. Here we consider a limit in which two species are vanishingly small, but interactions are correspondingly large to maintain a nontrivial limit. In this limit two energy levels are distinguished: the highest order limit encodes information on the geometry of local structures as a two-component isoperimetric problem, while the second level describes the spatial distribution of components in global minimizers. We provide a sharp rigorous derivation of the asymptotic limit, both for minimizers and in the context of Gamma-convergence. Geometrical descriptions of limit configurations are derived; among other results, we will show that, quite unexpectedly, coexistence of single and double bubbles can arise. The main difficulties are hidden in the optimal solution of two-component isoperimetric problem: compared to binary systems, not only it lacks an explicit formula, but, more crucially, it can be neither concave nor convex on parts of its domain.

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“…Numerical work using Cahn--Hilliard-type systems has also revealed a wide range of equilibrium morphologies [9,26]. Stationary assemblies of multidomain structures have also been investigated in a related three-phase system [43].…”

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confidence: 99%

Evolution and Competition of Block Copolymer Nanoparticles

Glasner¹

2019

SIAM J. Appl. Math.

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Nanoparticle structures formed in a mixture of diblock copolymer and solvent are investigated using a three-phase density functional model and its sharp interface approximation. A wide variety of equilibria described by localized domain patterns are quantified both numerically and analytically. Competition among multiple particles is shown to occur through mass diffusion driven by differences in chemical potential, which may or may not lead to Ostwald ripening behavior. Late stage rigid body dynamics is shown to result from interaction through dipolar fields, leading to orientational alignment and long-range attraction.

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A stationary core-shell assembly in a ternary inhibitory system

Cited by 14 publications

References 31 publications

Nonhexagonal Lattices From a Two Species Interacting System

Nonhexagonal Lattices From a Two Species Interacting System

Periodic Minimizers of a Ternary Non-Local Isoperimetric Problem

Evolution and Competition of Block Copolymer Nanoparticles

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