• DOI:10.1137/19m1245980
• Corpus ID: 119159377

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}
}

• S. Luo, X. Ren, Juncheng Wei
• 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 several thresholds defined by a number $B=0.1867...$ If $b \in [0, B)$, a minimal assembly is associated with a rectangular lattice whose ratio of the longer side and the shorter side is in $[\sqrt{3}, 1)$; if $b \in [B, 1-B]$, a minimal assembly is associated with a square lattice; if $b \in (1-B, 1]$, a minimal assembly is associated with a rhombic lattice with an acute angle in $[\frac{\pi}{3}, \frac{\pi}{2})$. 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.