Bubble assemblies in ternary systems with long range interaction

Wang, Chong; Ren, Xiaofeng; Zhao, Yanxiang

doi:10.4310/cms.2019.v17.n8.a10

Cited by 16 publications

(8 citation statements)

References 46 publications

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“…However Ren and Wang have shown the existence of statoinary points that are unions of perturbed discs in a bounded domain with the Neumann bounary condition [14]. Numerical evidence strongly suggests the existence of stationary points similar to two species assemblies [20].…”

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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“…, by using the estimates (35) and (36), respectively. Using the conditions (31) and (32), respectively, leads to the desired inequality for the energy stability.…”

Section: 2mentioning

confidence: 99%

“…The other one is to start with a small M so that the initial stiffness is easily controlled, then gradually but slowly increase the value of M . This method has been adopted in some of our earlier work [41,32].…”

Section: 4mentioning

confidence: 99%

Second-order stabilized semi-implicit energy stable schemes for bubble assemblies in binary and ternary systems

Choi

Zhao

2022

DCDS-B

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<p style='text-indent:20px;'>In this paper, we propose some second-order stabilized semi-implicit methods for solving the Allen-Cahn-Ohta-Kawasaki and the Allen-Cahn-Ohta-Nakazawa equations. In the numerical methods, some nonlocal linear stabilizing terms are introduced and treated implicitly with other linear terms, while other nonlinear and nonlocal terms are treated explicitly. We consider two different forms of such stabilizers and compare the difference regarding the energy stability. The spatial discretization is performed by the Fourier collocation method with FFT-based fast implementations. Numerically, we verify the second order temporal convergence rate of the proposed schemes. In both binary and ternary systems, the coarsening dynamics is visualized as bubble assemblies in hexagonal or square patterns.</p>

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“…In higher space dimensions, one only has upper bounds. Another related three-components model was considered in [18,19], where the existence and stability of equilibria, which are minimizers of the energy, were studied (see also [36] for a similar model with nonlocal interactions). Efficient numerical simulations for the coupled Cahn-Hilliard model were performed in [23], based on an uncoupled and second-order unconditionally energy stable scheme.…”

Section: Introductionmentioning

confidence: 99%