Self-assembly of quasicrystals under cyclic shear

1. Problem Statement

The self-assembly of complex structures, particularly quasicrystals, remains a significant challenge in soft matter physics. Unlike simple periodic crystals, quasicrystals possess long-range orientational order but lack translational periodicity, often exhibiting "forbidden" rotational symmetries (e.g., 12-fold).

• Thermal Limitations: In thermal equilibrium, quasicrystals are typically stable only in narrow regions of the phase diagram, often bordered by competing periodic crystals (square or hexagonal). Their formation is frequently hindered by kinetic trapping in metastable amorphous states due to rugged energy landscapes and the slow dynamics of colloidal systems.
• The Gap: While mechanical driving (cyclic shear) has been used to "train" amorphous glasses and induce crystallization in simple systems, it is unclear if cyclic shear alone can drive a system into a complex quasicrystalline state, especially in regimes where thermal fluctuations fail due to kinetic barriers.

2. Methodology

The authors employed numerical simulations to investigate a two-dimensional system of $N$ particles interacting via a smoothed square-shoulder potential.

• Interaction Potential: The potential $U(r)$ features a repulsive core and a smoothed shoulder at $r = \delta = 1.35\sigma$. This dual-length-scale interaction promotes competition between hexagonal packing (high density) and square-based tilings (lower density), a known mechanism for 12-fold quasicrystal formation.
• Simulation Protocols:
  1. Athermal Quasi-Static (AQS) Shear: The system is subjected to cyclic shear cycles ($0 \to \gamma_{max} \to 0 \to -\gamma_{max} \to 0$) with energy minimization (Conjugate Gradient) after each strain increment. This effectively replaces thermal fluctuations with plastic rearrangements.
  2. Finite Shear Rate: Overdamped dynamics were used to explore regimes where strain localization (shear bands) might be suppressed.
  3. Monotonic Shear: Used as a control to understand the mechanical response of pre-formed quasicrystals.
• Analysis Tools:
  • Bond Orientational Order Parameters (BOOP): Local ($q_{12}$) and global ($Q_{12}$) order parameters were calculated to distinguish between local structural motifs and global orientational coherence.
  • Tiling Reconstruction: The dodecagonal phase was decomposed into square and triangular tiles to visualize structural order.
  • Phase Diagram Mapping: The steady-state structures were mapped as a function of density ($\rho$) and shear amplitude ($\gamma_{max}$).

3. Key Contributions and Results

A. Shear-Driven Self-Assembly

The study demonstrates that cyclic shear alone can drive initially random, amorphous configurations into ordered 12-fold quasicrystalline states.

• Phase Diagram: The non-equilibrium phase diagram qualitatively mirrors the thermal equilibrium diagram, exhibiting square, hexagonal, and quasicrystalline phases, as well as phase coexistence regions.
• Stabilization Mechanism: Remarkably, shear-stabilized quasicrystals appear in density regions where the zero-temperature ($T=0$) equilibrium ground state favors square-hexagonal coexistence. This suggests that mechanical driving can stabilize quasicrystalline order in a manner analogous to entropic effects in thermal systems, effectively overcoming the energetic preference for mixed phases.

B. Optimization near Yielding

The structural quality of the self-assembled quasicrystals is maximized near the yielding transition ($\gamma_{max} \approx \gamma_{yield}$).

• Quality vs. Speed: While dynamics are slowest near yielding, the resulting quasicrystals exhibit the highest global orientational order ($Q_{12}$).
• Efficiency: Despite the slow dynamics near yielding, the system forms monodomain quasicrystals rapidly (within $\sim 100$ cycles) without complex annealing protocols. This is significantly faster than thermal molecular dynamics at low temperatures, which often get kinetically trapped.
• Mechanism: The authors propose that the fast assembly is driven by system-spanning plastic rearrangements (avalanches) triggered near yielding. These non-local events act similarly to "cluster moves" in Monte Carlo simulations, allowing the system to explore the energy landscape more effectively than local thermal diffusion.

C. Finite-Size Scaling and Order

• Quasi-Long-Range Order: Finite-size scaling analysis reveals that the global orientational order parameter $\langle Q_{12} \rangle$ decays slowly with system size ($N$). This indicates quasi-long-range order, comparable to equilibrium hexatic phases, rather than true long-range order.
• Shear Bands: In large systems under AQS, strain localizes into shear bands. These bands fragment the system, preventing global equilibration of domains outside the band and reducing $\langle Q_{12} \rangle$.
• Finite Rate Shear: Switching to finite shear rates suppresses the formation of a single dominant shear band, leading to more homogeneous deformation. However, even in this regime, the order remains quasi-long-range, suggesting that the cyclic driving itself induces fluctuations large enough to disrupt true long-range order (analogous to dislocation unbinding in 2D melting).

D. Structural Characteristics

• Residual Stress: Unlike equilibrium systems, shear-driven quasicrystals retain residual stress ($\sigma_{xy} \neq 0$) at zero strain, indicating they occupy distinct regions of the configurational landscape.
• Coexistence Regions: Phase coexistence (e.g., square + quasicrystal) is not a simple lever-rule mixture. Instead, it features stripe-like motifs (e.g., hexagonal bands within square domains) that form to relieve shear stress via domain sliding.

4. Significance

• New Assembly Pathway: The work establishes cyclic shear as a robust, non-thermal pathway for the self-assembly of complex quasicrystalline structures, bypassing the kinetic traps that plague thermal equilibration.
• Mechanical vs. Thermal: It highlights a fundamental analogy between mechanical driving and thermal entropy, showing that mechanical work can stabilize phases that are entropically favored but energetically disfavored at $T=0$.
• Material Design: The ability to generate high-quality, oriented quasicrystalline domains rapidly without complex thermal annealing protocols offers practical implications for designing materials with exotic photonic and mechanical properties (e.g., low friction).
• Non-Equilibrium Physics: The findings contribute to the understanding of non-equilibrium steady states, revealing that cyclic driving can induce quasi-long-range order and specific structural motifs distinct from equilibrium counterparts.

In conclusion, the paper provides a comprehensive numerical demonstration that mechanical cycling is not just a tool for disordering or simple crystallization, but a potent mechanism for guiding soft matter into complex, aperiodic, yet highly ordered quasicrystalline states.