# Nonhexagonal Lattices From a Two Species Interacting System

@article{Luo2020NonhexagonalLF,
title={Nonhexagonal Lattices From a Two Species Interacting System},
author={Senping Luo and Xiaofeng Ren and Juncheng Wei},
journal={SIAM J. Math. Anal.},
year={2020},
volume={52},
pages={1903-1942}
}
• Published 25 February 2019
• Materials Science
• SIAM J. Math. Anal.
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…

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## References

SHOWING 1-10 OF 22 REFERENCES
A stationary core-shell assembly in a ternary inhibitory system
• Chemistry
• 2016
A ternary inhibitory system motivated by the triblock copolymer theoryis studied as a nonlocal geometric variational problem. The free energyof the system is the sum of two terms: the total size of
A Proof of Crystallization in Two Dimensions
This work shows rigorously that under suitable assumptions on the potential V which are compatible with the growth behavior of the Lennard-Jones potential the ground state energy per particle converges to an explicit constant E*: where E* ∈ ℝ is the minimum of a simple function on [0,∞).
A Double Bubble Assembly as a New Phase of a Ternary Inhibitory System
• Mathematics
• 2015
A ternary inhibitory system is a three component system characterized by two properties: growth and inhibition. A deviation from homogeneity has a strong positive feedback on its further increase. In
Uniform energy distribution for an isoperimetric problem with long-range interactions
• Mathematics
• 2008
We study minimizers of a nonlocal variational problem. The problem is a mathematical paradigm for the ubiquitous phenomenon of energy-driven pattern formation induced by competing short- and
From the Ginzburg-Landau Model to Vortex Lattice Problems
• Mathematics, Physics
• 2010
It is shown that the vortices of minimizer of Ginzburg-Landau, blown-up at a suitable scale, converge to minimizers of W, thus providing a first rigorous hint at the Abrikosov lattices, which is a next order effect compared to the mean-field type results.
Small Volume-Fraction Limit of the Diblock Copolymer Problem: II. Diffuse-Interface Functional
• Mathematics
SIAM J. Math. Anal.
• 2011
This work addresses the limit in which e and the volume fraction tend to zero but the number of minority phases (called particles) remains O(1), and derives first- and second-order effective energies, whose energy landscapes are simpler and more transparent.
Small Volume Fraction Limit of the Diblock Copolymer Problem: I. Sharp-Interface Functional
• Mathematics
SIAM J. Math. Anal.
• 2010
This article addresses the limit in which epsilon and the volume fraction tend to zero but the number of minority phases (called particles) remains O(1), and focuses on two levels of Gamma-convergence, which derive first- and second-order effective energies, whose energy landscapes are simpler and more transparent.
The Γ-Limit of the Two-Dimensional Ohta–Kawasaki Energy. I. Droplet Density
• Mathematics
• 2013
This is the first in a series of two papers in which we derive a Γ-expansion for a two-dimensional non-local Ginzburg–Landau energy with Coulomb repulsion, also known as the Ohta–Kawasaki model, in
Microphase separation of ABC-type triblock copolymers
• Materials Science
• 1993
Microphase separation of ABC-type triblock copolymers in the strong segregation limit was investigated by generalizing our previous theory for diblock copolymers. The free energy functional in terms
MANY DROPLET PATTERN IN THE CYLINDRICAL PHASE OF DIBLOCK COPOLYMER MORPHOLOGY
• Physics
• 2007
The Ohta–Kawasaki density functional theory of diblock copolymers gives rise to a nonlocal free boundary problem. In a proper range of the block composition parameter and the nonlocal interaction