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#### Analysis of PDEsMathematical Physicsmathscidoc:2006.03001

SIAM Journal on Mathematical Analysis, 52, (2), 1903-1942, 2020
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. The minimal two species periodic assembly is the 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 the minimal assembly varies depending on $b$. There are several threshold defined by a number $B=0.1867...$ If $b \in [0, B)$, the minimal assembly is associated with a rectangular lattice whose ratio of the longer side and the shorter side is in $[\sqrt{3}, 1)$; if $b \in [B, 1-B]$, the minimal assembly is associated with a square lattice; if $b \in (1-B, 1]$, the 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 the hexagonal lattice. None of the other values of $b$ yields the hexagonal lattice, a sharp contrast to the situation for one species interacting systems, where the hexagonal lattice is ubiquitously observed.
Two species interacting system, triblock copolymer, two species periodic assembly of discs, rectangular lattice, square lattice, rhombic lattice, hexagonal lattice, duality property.
• This is the first discovery and proof of the phase transitions of a triangular-rhombic-square-rectangular phenomenon.
@inproceedings{senping2020nonhexagonal,
title={Nonhexagonal lattices from a two species interacting system},
author={Senping Luo, Xiaofeng Ren, and Juncheng Wei},
url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20200606061247544313696},
booktitle={SIAM Journal on Mathematical Analysis},
volume={52},
number={2},
pages={1903-1942},
year={2020},
}

Senping Luo, Xiaofeng Ren, and Juncheng Wei. Nonhexagonal lattices from a two species interacting system. 2020. Vol. 52. In SIAM Journal on Mathematical Analysis. pp.1903-1942. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20200606061247544313696.