# Non-hexagonal lattices from a two species interacting system

Date: 2021

Location: UBC, Online

Conference: PIMS Workshop on New Trends in Localized Patterns in PDES

Subject: Mathematics

Class: Scientific

### 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 is in [0, B), a minimal assembly is associated with a rectangular lattice; if b is in [B, 1-B], a minimal assembly is associated with a square lattice; if b is in (1-B, 1], a minimal assembly is associated with a rhombic lattice. 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.