🔬 condensed matter

A mechanical approach to facilitate the formation of dodecagonal quasicrystals and their approximants

This study demonstrates that mechanical perturbations applied to unstable square packings of identical hard disks can spontaneously generate dodecagonal quasicrystals and their approximants using a single length scale, thereby challenging existing theories and offering a highly efficient pathway for synthesizing quasicrystalline materials.

Original authors: Zhehua Jiang, Jianhua Zhang, Mengyuan Zhan, Jiaqi Si, Junchao Huang, Hua Tong, Ning Xu

Published 2026-04-30

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CC BY 4.0

Original authors: Zhehua Jiang, Jianhua Zhang, Mengyuan Zhan, Jiaqi Si, Junchao Huang, Hua Tong, Ning Xu

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you have a box full of identical, perfectly round marbles. If you shake them up and let them settle, they naturally want to arrange themselves into a honeycomb pattern (hexagonal), because that's the most efficient way to pack them. This is the "standard" behavior for these marbles.

Now, imagine you want to build something much more complex: a dodecagonal quasicrystal. Think of this as a magical, intricate mosaic that has 12-fold symmetry (like a snowflake with 12 points) but never repeats its pattern exactly. Usually, to build this, scientists need a very complicated recipe: different-sized marbles, special magnetic fields, or particles that repel each other in weird, multi-layered ways. The rule of thumb has always been: "You need multiple different sizes or forces to make this complex shape."

The Big Surprise
The researchers in this paper found a way to break that rule. They asked: Can we make this complex 12-point mosaic using only identical marbles and simple mechanical nudges?

They discovered that the answer is yes, but with a twist.

The "Unstable Square" Trick
Instead of starting with the natural honeycomb, they forced the marbles into a square grid (like a checkerboard).

The Problem: A square grid of marbles is inherently wobbly. It's like balancing a house of cards; it's unstable and wants to collapse into a honeycomb.
The Solution: The researchers realized this instability was actually a superpower. By giving the square grid a tiny, random mechanical "nudge" (or removing a few marbles to create empty spots), they triggered a chain reaction.
The Result: The marbles didn't just collapse randomly. They rearranged themselves into a beautiful, complex pattern made of squares and triangles. This pattern is a "sibling" to the 12-point quasicrystal, called an approximant. It's a crystalline version of the quasicrystal, sharing the same local building blocks but arranged in a repeating pattern.

The "Vacancy" Recipe
To make the actual 12-point motifs (the specific 19-marble clusters that make up the quasicrystal), they used a specific mechanical trick:

Start with the wobbly square grid.
Remove two specific marbles (creating a "vacancy pair") in a precise spot.
Let the system settle.
The surrounding marbles, trying to fill the gap, naturally rotate and shift into the exact shape needed for the quasicrystal. It's like knocking out two bricks in a wall, causing the surrounding bricks to slide and lock into a new, intricate design without anyone pushing them individually.

Why This Matters (According to the Paper)

It Challenges Old Theories: Scientists thought you needed different-sized particles or complex forces to make these shapes. This paper shows that with just one size of particle and a simple mechanical push, you can get there.
It's a Fast Track: They tested this on a system known to make quasicrystals (using particles with a "soft" repulsion that mimics two different sizes).
- Normal Way: If you just let these particles cool down from a liquid, it takes a long time to form the quasicrystal, and at very low temperatures, they get stuck and can't move fast enough to find the right shape.
- Their Way: By starting with the square grid and adding the "nudge" (vacancies), they jump-started the process. The quasicrystal formed almost instantly.
Thermal "Polishing": While the mechanical nudge created the structure quickly, it wasn't perfect everywhere. They then applied a tiny bit of heat (thermal treatment). This acted like a gentle massage, allowing the particles to wiggle just enough to fix the imperfect spots, resulting in a much higher quality quasicrystal than if they had just waited for it to form naturally.

The Bottom Line
The authors found a "mechanical shortcut." By exploiting the fact that a square packing of identical marbles is unstable, they can force the system to spontaneously rearrange into complex, quasicrystal-like patterns. This method is faster and works even in conditions where the particles are usually too sluggish to organize themselves.

What the Paper Does NOT Claim

It does not claim this will immediately lead to new commercial materials or medical devices.
It does not claim this works for all types of quasicrystals (specifically those that don't use squares as building blocks).
It focuses entirely on the physics of how these structures form, not on industrial applications.

1. Problem Statement

The formation of quasicrystals (QCs) and their approximants typically requires stringent conditions, most notably the presence of multiple characteristic length scales to stabilize the complex, non-periodic order. Conventional theories and experimental approaches rely on complex interparticle interactions, polydisperse systems, or anisotropic particles to achieve this.

The Challenge: It is generally accepted that the simplest system of monodisperse identical hard disks (possessing only a single length scale, the disk diameter) cannot spontaneously form quasicrystals or complex crystalline structures. The thermodynamic equilibrium state for such a system is a hexagonal packing. While metastable states exist, no spontaneous formation of dodecagonal quasicrystals (DDQCs) or their approximants has been reported in this simplest system, even under non-equilibrium conditions.
The Question: Can the simplest system of monodisperse hard disks spontaneously form DDQCs or their approximants, and can a purely mechanical approach facilitate this process in systems known to form QCs?

2. Methodology

The authors employed computational simulations using two distinct interaction models:

Hard Disk Approximation (Harmonic Repulsion):
- Particles interact via a harmonic repulsion potential. At low pressures ( $p = 10^{-4}$ ), this mimics hard disks.
- Protocol: The system starts in a square packing configuration. Due to the presence of "soft modes" (zero-frequency normal modes), this square lattice is mechanically unstable.
- Perturbation: The authors introduced mechanical perturbations by:
  - Randomly displacing disks slightly from equilibrium.
  - Vacancy Pair Removal: Systematically removing pairs of disks (vacancies) from the square lattice along specific symmetry axes ( $45^\circ$ and $-45^\circ$ ).
- Relaxation: The system underwent enthalpy minimization at constant pressure to reach a mechanically stable state.
Two-Length-Scale (TLS) System:
- A system known to self-assemble into DDQCs, utilizing a potential with a soft core and a shoulder (inverse-power-law + shoulder potential).
- Protocol: The same mechanical approach (square lattice initialization + vacancy pair introduction) was applied to this TLS system.
- Thermal Treatment: To optimize the resulting structures, the mechanically transformed states were subjected to molecular dynamics simulations at low temperatures ( $T=0.12$ and $T=0.06$ ) to allow for structural relaxation.

3. Key Contributions

Discovery in Hard Disks: The authors demonstrated that the unstable feature of square packing in monodisperse hard disks can be exploited to spontaneously form a DDQC approximant and local DDQC motifs (specifically the $(3^2.4.3.4)$ Archimedean tiling) without any multiple length scales.
Mechanical Facilitation: They developed a purely mechanical protocol (introducing vacancy pairs into a square lattice) that acts as a catalyst for DDQC formation.
Thermal Enhancement: They showed that combining this mechanical approach with mild thermal treatment yields significantly higher quality quasicrystalline order compared to direct self-assembly from a liquid state within the same timeframe.
Low-Temperature Viability: The approach remains effective even at temperatures where thermal self-assembly from a liquid is kinetically hindered (sluggish dynamics), offering a non-diffusive pathway for QC growth.

4. Key Results

A. Hard Disk System (Single Length Scale)

Approximant Formation: Perturbing a square packing of hard disks led to the spontaneous formation of a 4-fold symmetric DDQC approximant. The structure consists of a mix of squares and equilateral triangles, with a triangle-to-square ratio of 2 (distinct from the perfect DDQC ratio of $4/\sqrt{3}$ ).
Motif Generation: By removing specific pairs of disks (vacancies) along symmetry axes, the system spontaneously rearranged into DDQC motifs (19-disk clusters with concentric hexagon and dodecagon shells).
Limitation: While local motifs formed, global long-range DDQC order could not be sustained in pure hard disks. Increasing the number of vacancy pairs destabilized the complex motifs, driving the system back to the thermodynamically stable hexagonal phase. This suggests that while the mechanism exists, hard disks lack the energetic depth to stabilize global QC order against the hexagonal phase.

B. TLS System (Two-Length-Scale)

Rapid Order Induction: Applying the vacancy-pair protocol to the TLS system successfully induced long-range DDQC order.
- With $N=900$ disks and 4 vacancy pairs, four distinct motifs formed.
- With 54 vacancy pairs, the system exhibited strong 12-fold symmetry in diffraction patterns, with bond angle distributions matching the six characteristic angles of DDQCs.
Order Parameter Evolution: The 12-fold bond orientational order parameter ( $\chi_{12}$ ) saturated at a high value ( $\approx 0.8$ ), while the 4-fold symmetry ( $\chi_4$ ) dropped significantly, indicating a successful transition from square to dodecagonal order.
Thermal Optimization:
- Mechanical vs. Thermal: The mechanical approach instantly created a "seed" of order. Subsequent thermal treatment ( $T=0.12$ ) reduced local deviations ( $\Delta\theta$ ) and sharpened diffraction spots, optimizing the global order.
- Efficiency: Compared to direct quenching from a liquid state, the mechanical approach achieved a much higher correlation function ( $G_{12}(r)$ ) in the same amount of time.
- Low-Temperature Performance: At very low temperatures ( $T=0.06$ ), liquid self-assembly was effectively frozen (no growth). However, the mechanically pre-structured system continued to relax and grow DDQC order, proving the method's efficacy in non-diffusive regimes.

C. Predictive Capability

The authors demonstrated that the mechanical approach can serve as a diagnostic tool. In a system where DDQCs are thermodynamically unstable (high density), the mechanical approach failed to produce QC structures, resulting instead in polycrystalline hexagonal phases. Thus, the emergence of QC structures via this method indicates the system's inherent capability to form QCs.

5. Significance

Theoretical Challenge: The finding challenges the prevailing hypothesis that multiple length scales are strictly necessary for QC formation. It reveals that the instability of square packing in single-scale systems is a sufficient condition to generate complex motifs, expanding the theoretical understanding of quasicrystalline assembly.
Synthesis Strategy: The study proposes a highly efficient, mechanical pathway for synthesizing quasicrystals. By bypassing the slow nucleation and growth barriers of liquid self-assembly, this method accelerates the formation of high-quality DDQCs.
Experimental Relevance: The approach is experimentally feasible using colloidal suspensions manipulated by optical tweezers to construct square packings and introduce controlled defects.
Material Design: This method offers a new route for the design and synthesis of quasicrystalline materials with applications in thermal insulation, surface coatings, and advanced engineering, particularly in regimes where thermal self-assembly is inefficient.