Self-assembly of quasicrystals under cyclic shear

TL;DR

This work tackles the challenge of self-assembling complex quasicrystalline order by using cyclic shear as a non-thermal driving mechanism. It employs a two-dimensional model with a smoothed square-shoulder potential and analyzes both athermal quasi-static cycles and finite-rate driving, quantifying structure with bond-orientational order parameters $q_\ell$ and $Q_\ell$ and by tiling to identify dodecagonal order. The main findings show that a shear-stabilised dodecagonal quasicrystal can emerge even when the zero-temperature equilibrium favours square–hexagonal coexistence, with the strongest order near the yielding amplitude $\gamma_{\text{yield}}$ and rapid, annealing-free self-assembly; however, the global orientational order is only quasi-long-range and limited by system-size shear bands, even under finite-rate driving. Overall, cyclic shear is demonstrated as an effective route to complex non-trivial structures, with implications for material design and non-equilibrium statistical physics, while prompting further study of defect topology, phason strain, and extension to three dimensions.

Abstract

We investigate the self-assembly of two-dimensional dodecagonal quasicrystals driven by cyclic shear, effectively replacing thermal fluctuations with plastic rearrangements. Using particles interacting via a smoothed square-shoulder potential, we demonstrate that cyclic shearing drives initially random configurations into ordered quasicrystalline states. The resulting non-equilibrium phase diagram qualitatively mirrors that of thermal equilibrium, exhibiting square, quasicrystalline, and hexagonal phases, as well as phase coexistence. Remarkably, the shear-stabilised quasicrystal appears even where the zero-temperature equilibrium ground state favours square-hexagonal coexistence, suggesting that mechanical driving can stabilise quasicrystalline order in a way analogous to entropic effects in thermal systems. The structural quality of the self-assembled state is maximised near the yielding transition, even though the dynamics are slowest there. Yet, the system still quickly forms monodomain quasicrystals without any complex annealing protocols, unlike at equilibrium, where thermal annealing would be required. Finite-size scaling analysis reveals that global orientational order decays slowly with system size, indicative of quasi-long-range order comparable to equilibrium hexatic phases. Overall, our results establish cyclic shear as an efficient pathway for the self-assembly of complex structures.

Figures (8)

• Figure 1: Thermally self-assembled quasicrystal. (a) Part of a typical snapshot with particles coloured by $|q_{12}^{(j)}|$ and the reconstructed tiling. (b) and (c) Orientation distribution of the tiles for the corresponding snapshot. $N = 5000$, $\rho\sigma^2=0.933$ and $k_BT/\varepsilon=0.13$.
• Figure 2: Typical behaviour of the system under monotonic AQS. (a) Stress as a function of strain for various configurations equilibrated at different temperatures. The inset represents the average $Q_{12}$ of the unstrained state for different equilibration temperatures. Each curve corresponds to an average over 20 realisations of systems with $N = 10^4$ particles and $\rho\sigma^2 = 0.937$. (b) and (c) show the reconstructed tiling for a system with $N=5\times 10^3$ at $k_BT/\varepsilon=0.2$ for $\gamma = 0$ and $\gamma =0.2$, respectively. (d, e) and (f, g) show the observed probability distributions of tiles in (b) and (c), respectively. Yellow tiles are tiles considered misaligned with the global orientation of the system, based on an arbitrary angle deviation threshold.
• Figure 3: Self-assembly of a crystalline and quasicrystalline structure via cyclic shear. (a-c) Evolution of the system after 0, 50, and 100 cyclic shear cycles at $\gamma_{\max} = 0.1$ and $\rho\sigma^2 = 0.94$ with $\gamma_{\rm acc}=4\gamma_{\max}n_c$, the accumulated strain, where $n_c$ is the number of cycles. The first configuration is an energy-minimized configuration equilibrated at $k_BT/\varepsilon=1$. (d-i) Global and local BOOP (Eq. \ref{['eq:boop']}) as a function of $\rho$ for various $\gamma_{\max}$ in the steady state. The initial configuration is completely random; each point contains an average over 3 different initial configurations and 25 uncorrelated snapshots each. (j) Observed phase diagram. The top part corresponds to the thermal one, reproduced from Refs. kryuchkov2018complex and padilla2020phase. The middle part corresponds to the theoretical phase diagram at $T=0$, obtained in the Supplementary Information†. The bottom part corresponds to the observed phase in our cyclically sheared system at $\gamma_{\max}=0.1$. The coexistence regions are generally not simply a mixture of the two states at their boundaries, as explained in the main text. (k-m) Typical snapshot at various densities for $\gamma_{\max}=0.1$. All simulations are performed at $N=4000$.
• Figure 4: Dependence of the quality of the self-assembled quasicrystal on $\gamma_{\max}$. All simulations were performed at $N=4000$ and $\rho\sigma^2=0.966$. (a–b) Local and global bond-orientational order parameters (Eq. \ref{['eq:boop']}) as a function of $\gamma_{\max}$. Each data point is an average over 10 independent initial conditions and 50 uncorrelated snapshots. (c–d) Typical snapshots at $\gamma_{\max}=0.07$ and $\gamma_{\max}=0.20$, respectively, with particles coloured by their local 12-fold orientation. (e) Typical number of cycles required to reach a steady state from a fully random configuration as a function of $\gamma_{\max}$. This self-assembly time is determined by averaging $\langle Q_{12}\rangle(\text{cycles})$ over 10 independent initial conditions and finding the first cycle at which $\langle Q_{12}\rangle(n_c)$ enters and remains within 10% of its long-time average. The inset shows the probability distribution of the 12-fold orientation angle. (f) Characteristic angle decorrelation timescale (in cycles), obtained from an exponential fit of the 12-fold orientation autocorrelation function $C_{12}^{\text{angle}}(n_c)\sim e^{-n_c/\tau}$ (Eq. \ref{['eq:C(t)']}). Each autocorrelation curve is averaged over at least 5000 configurations sampled every 10 cycles.
• Figure 5: Quasicrystal quality dependence on the system size. (a) and (b) are snapshots of a large system ($N = 25000$) at $\rho\sigma^2 = 0.966$ and $\gamma_{\max}=0.07$. (a) displacement field (in arbitrary units of length) long after the band formation, between cycles 850 and 1000. (b) snapshot with particles coloured according to their 12-fold local orientation. (c) Evolution of the local and global BOOPs (Eq. \ref{['eq:boop']}) as $N$ is increased for $\gamma_{\max}=0.085$. Averages are performed over 3 realisations for 25 uncorrelated cycles after 1000 cycles for each system size.