[Paper Review] Self-assembling two-dimensional quasicrystals in simple systems of monodisperse soft-core disks

Mengjie Zu, Peng Tan+1 more|arXiv (Cornell University)|Mar 26, 2017

Quasicrystal Structures and Properties20 citations

TL;DR

This study demonstrates the unexpected self-assembly of two-dimensional quasicrystals—both octagonal and dodecagonal—using monodisperse, isotropic soft-core disks interacting via a simple pairwise repulsion potential with no explicit multiple length scales. The formation is driven by pentagon-rich local order, revealing a new pathway to quasicrystalline order in simple systems, challenging existing theoretical frameworks.

ABSTRACT

In previous approaches to form quasicrystals, multiple competing length scales involved in particle size, shape or interaction potential are believed to be necessary. It is unexpected that quasicrystals can be self-assembled by monodisperse, isotropic particles interacting via a simple potential without multiple length scales. Here we report the surprising finding of the self-assembly of such quasicrystals in two dimensional systems of soft-core disks interacting via repulsions. We find not only dodecagonal but also octagonal quasicrystals, which have not been found yet in soft quasicrystals. In the self-assembly of such unexpected quasicrystals, particles tend to form pentagons, which are essential elements to form the quasicrystalline order. Our findings pave an unexpected and simple way to form quasicrystals and pose a new challenge for theoretical understanding of quasicrystals.

Motivation & Objective

• To investigate whether quasicrystals can form in simple systems of monodisperse, isotropic particles without multiple competing length scales.
• To determine the role of particle interactions and soft-core potentials in enabling spontaneous quasicrystal formation.
• To identify structural motifs—particularly pentagons—that may underlie the emergence of quasicrystalline order in such systems.
• To challenge existing theoretical models by demonstrating quasicrystal formation in systems previously considered insufficiently complex.

Proposed method

• Molecular dynamics simulations in NVT and NPT ensembles using a soft-core repulsive potential $ U(r) = \frac{\epsilon}{\alpha}\left(1 - r/\sigma\right)^\alpha \Theta(1 - r/\sigma) $ with variable softness $ \alpha $.
• Systematic quenching from high-temperature liquids to explore phase behavior across varying number density $ \rho $ and $ \alpha $, ensuring equilibrium via long relaxation times.
• Use of static structure factor $ S(\vec{k}) $ and radial distribution function $ g(\vec{r}) $ to compute diffraction patterns and density profiles.
• Employment of polygonal order parameter $ \delta = \max\left\{ \left| e_i / \bar{e} - 1 \right| \right\} $ to identify pentagons formed by five neighboring disks.
• Analysis of van Hove autocorrelation function $ G_a(\vec{r}, t) $ to assess particle dynamics and mobility in quasicrystalline phases.
• Contour plots of $ S(k) $ to detect low-$ k $ peaks associated with intermediate-range order in liquids preceding quasicrystal formation.

Experimental results

Research questions

• RQ1 Can quasicrystals form in two-dimensional systems of monodisperse, isotropic soft-core disks without explicit multiple length scales in particle size, shape, or interaction potential?
• RQ2 What structural motifs—particularly local arrangements like pentagons—mediate the emergence of quasicrystalline order in such simple systems?
• RQ3 How does the softness of the inter-particle potential, controlled by exponent $ \alpha $, influence the stability and formation of octagonal and dodecagonal quasicrystals?
• RQ4 What is the role of intermediate-range order in the liquid phase in pre-organizing particles toward quasicrystalline structures?
• RQ5 Why do quasicrystals form only in specific regimes of density and softness, and what underlying mechanisms govern this selectivity?

Key findings

• Octagonal quasicrystals (OQCs) are observed for the first time in soft, monodisperse systems, with no prior experimental or simulation confirmation in such contexts.
• Dodecagonal quasicrystals (DDQCs) form in two distinct density regimes, indicating a complex phase behavior beyond simple ordering.
• Pentagons—defined via a polygonal order parameter—emerge as dominant structural units, with their fraction peaking in QC-forming liquids, indicating their role as building blocks.
• The static structure factor $ S(k) $ reveals two pronounced low-$ k $ peaks in liquids preceding QC formation, signaling intermediate-range order critical for quasicrystal nucleation.
• Particle trajectories and van Hove functions show reduced mobility in QC phases, confirming long-range, non-periodic order with localized dynamics.
• The formation of QCs occurs in the absence of explicit multiple length scales, challenging existing theories that require competing length scales for quasicrystal stability.