Non-hexagonal lattices from a two species interacting system

Senping Luo; Xiaofeng Ren; Juncheng Wei

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arXiv · 1902.09611

Non-hexagonal lattices from a two species interacting system

Senping Luo·Xiaofeng Ren·Juncheng Wei

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Abstract

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.

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Senping Luo, Xiaofeng Ren, Juncheng Wei. 2019-02-25. Non-hexagonal lattices from a two species interacting system. https://arxiv.org/abs/1902.09611

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