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}
}
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… 

Figures from this paper

Crystallization to the Square Lattice for a Two-Body Potential
We consider two-dimensional zero-temperature systems of $N$ particles to which we associate an energy of the form $$ \mathcal{E}[V](X):=\sum_{1\le i<j\le N}V(|X(i)-X(j)|), $$ where $X(j)\in\mathbb
On the optimality of the rock-salt structure among lattices with charge distributions
The goal of this work is to investigate the optimality of the $d$-dimensional rock-salt structure, i.e., the cubic lattice $V^{1/d}\mathbb{Z}^d$ of volume $V$ with an alternation of charges $\pm 1$
On energy ground states among crystal lattice structures with prescribed bonds
  • Laurent Bétermin
  • Computer Science
    Journal of Physics A: Mathematical and Theoretical
  • 2021
TLDR
The universal minimality—i.e. the optimality for all completely monotone interaction potentials— of strongly eutactic lattices among these structures gives new optimality results for the square, triangular, simple cubic (sc), face-centred-cubic (fcc) and body-centre-cUBic (bcc) lattices in dimensions 2 and 3 when points are interacting through completely Monotone potentials.
On universally optimal lattice phase transitions and energy minimizers of completely monotone potentials
. We consider the minimizing problem for energy functionals with two types of competing particles and completely monotone potential on a lattice. We prove that the minima of sum of two completely
Theta functions and optimal lattices for a grid cells model
TLDR
The question of the existence of an optimal grid is transformed into a maximization problem among all possible unit density lattices for a Fisher Information which measures the accuracy of grid-cells representations in $\mathbb{R}^d$.
Minimal Soft Lattice Theta Functions
We study the minimality properties of a new type of “soft” theta functions. For a lattice $$L\subset {\mathbb {R}}^d$$ L ⊂ R d , an L -periodic distribution of mass $$\mu _L$$ μ L , and another mass
Vortex patterns and sheets in segregated two component Bose–Einstein condensates
We study minimizers of a Gross-Pitaevskii energy describing a two-component Bose-Einstein condensate set into rotation. We consider the case of segregation of the components in the Thomas-Fermi
A note on the maximum of a lattice generalization of the logarithm and a deformation of the Dedekind eta function
We consider a deformation $E_{L,\Lambda}^{(m)}(it)$ of the Dedekind eta function depending on two $d$-dimensional simple lattices $(L,\Lambda)$ and two parameters $(m,t)\in (0,\infty)$, initially
On minima of difference of theta functions and application to hexagonal crystallization
. Let z = x + iy ∈ H := { z = x + iy ∈ C : y > 0 } and θ ( α ; z ) = (cid:80) ( m,n ) ∈ Z 2 e − α πy | mz + n | 2 be the theta function associated with the lattice L = Z ⊕ z Z . In this paper we
ON A LATTICE GENERALISATION OF THE LOGARITHM AND A DEFORMATION OF THE DEDEKIND ETA FUNCTION
We consider a deformation $E_{L,\unicode[STIX]{x1D6EC}}^{(m)}(it)$ of the Dedekind eta function depending on two $d$ -dimensional simple lattices $(L,\unicode[STIX]{x1D6EC})$ and two parameters
...
...

References

SHOWING 1-10 OF 22 REFERENCES
A stationary core-shell assembly in a ternary inhibitory system
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
TLDR
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
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
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
TLDR
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
TLDR
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
TLDR
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
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
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
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
...
...