Non-hexagonal lattices from a two species interacting system

TL;DR

This paper investigates how a two-species interacting system forms various lattice structures, revealing that unlike single-species systems, hexagonal lattices are only optimal at a specific parameter value, with other parameters favoring different lattice types.

Contribution

It characterizes the minimal energy lattice configurations for a two-species system, showing the dependence on a parameter and highlighting the absence of hexagonal lattices except at a single point.

Findings

Minimal assemblies vary with parameter b, forming rectangular, square, or rhombic lattices.

Hexagonal lattices only occur at b=1, unlike in single-species systems.

Different lattice types are optimal depending on the value of b.

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.