Non-hexagonal lattices from a two species interacting system

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 $B=\mathrm{0.1867...}$ 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.