Phase Field Crystal model for particles with n-fold rotational symmetry in two dimensions

TL;DR

This paper develops a Phase Field Crystal model for particles with n-fold rotational symmetry in two dimensions, capturing various phases including crystals, quasicrystals, and liquid crystal states, based on a free energy functional.

Contribution

It introduces a novel PFC model for n-fold symmetric particles that generalizes liquid crystal models and captures diverse ordered phases.

Findings

Model recovers known liquid crystal behavior for n=2.

Identifies stable phases including cluster crystals and quasicrystals.

Describes how patchy colloids with different symmetries can be modeled.

Abstract

We introduce a Phase Field Crystal (PFC) model for particles with n-fold rotational symmetry in two dimensions. Our approach is based on a free energy functional that depends on the reduced one-particle density, the strength of the orientation, and the direction of the orientation, where all these order parameters depend on the position. The functional is constructed such that for particles with axial symmetry (i. e. n = 2) the PFC model for liquid crystals as introduced by H. L\"owen [J. Phys.: Condens. Matter 22, 364105 (2010)] is recovered. We discuss the stability of the functional and explore phases that occur for $1 \leq n \leq 6$. In addition to isotropic, nematic, stripe, and triangular order, we also observe cluster crystals with square, rhombic, honeycomb, and even quasicrystalline symmetry. The n-fold symmetry of the particles corresponds to the one that can be realized for colloids with symmetrically arranged patches. We explain how both, repulsive as well as attractive patches, are described in our model.