Phason-Driven Diversity of Nucleation Pathways in Icosahedral Quasicrystals

Abstract

The nucleation of quasicrystals remains a fundamental puzzle, primarily due to the absence of a periodic translational template. Here, we demonstrate that phasons - hidden degrees of freedom unique to quasiperiodic order - drive diverse nucleation pathways in icosahedral quasicrystals (IQCs). Combining a Landau free-energy model with the spring pair method, we compute distinct critical nuclei and their corresponding minimum energy paths. At low temperatures, a direct, symmetry-preserving pathway dominates. In contrast, higher temperatures promote a "symmetry detour" that reduces the nucleation barrier via a lower-symmetry critical nucleus. Remarkably, while the resulting bulk IQCs exhibit distinct real-space symmetries, they remain thermodynamically degenerate with identical diffraction patterns. We resolve this paradox within the high-dimensional projection framework, showing that phason shifts modulate real-space symmetry without altering bulk thermodynamics. Our findings establish phasons as the structural origin of pathway diversity, offering a new physical picture for the emergence of quasiperiodic order.

Figures (8)

• Figure 1: Schematic of the nucleation selection dilemma. The liquid (center) is surrounded by a thermodynamically degenerate manifold generated by phason shifts. While these bulk states share the same free energy and diffraction intensities, they can differ in real-space symmetry, illustrated by the ideal id-IQC ($\mathcal{I}_h$) and representative nid-IQCs ($C_5$, $C_3$, $C_2$). Dashed arrows indicate competing kinetic routes from the liquid to different symmetry endpoints. The question mark highlights that bulk degeneracy does not determine which symmetry is selected during nucleation.
• Figure 2: Pair interaction potentials and the ordered structures they stabilize in the present study. (a) Real-space pair interaction potential $G(r)$ that is purely repulsive. (b) Fourier transform $\hat{G}(k)$ exhibiting a single dominant characteristic length scale. (c) The body-centered cubic (BCC) crystal stabilized by this interaction. (d) Real-space pair interaction potential $G(r)$ with alternating attractive and repulsive components (sign-changing). (e) Fourier transform $\hat{G}(k)$ exhibiting two characteristic length scales whose ratio equals the golden ratio. (f) The icosahedral quasicrystal (IQC) stabilized by this interaction.
• Figure 3: Nucleation of the BCC crystal as a reference case. (a) Evolution of BCC CNs at increasing temperatures ($\varepsilon = 0.0015$, $0.0020$, and $0.0025$; $\alpha = 0.2$). Upper panels show density distributions, and lower panels show the corresponding atomic arrangements with BCC connectivity. The CNs preserve BCC symmetry while increasing in size. (b) Nucleation energy barrier as a function of temperature. (c) Free energy of the bulk BCC phase grown from the CNs, indicating a weakening thermodynamic driving force with increasing temperature. (d) MEP for BCC nucleation and growth at $\varepsilon = 0.0025, \alpha = 0.2$.
• Figure 4: Temperature-dependent CNs for id-IQC and nid-IQC and the crossover in nucleation barriers. (a1--a3) Density distributions (upper) and corresponding atomic arrangements (lower) of id-IQC CNs at $\varepsilon=0.0861$, $0.0865$, and $0.0070$ ($\alpha=0.3$), showing robust $\mathcal{I}_h$ symmetry. (b) Atomic arrangement of (a3) viewed along 2-fold, 3-fold, and 5-fold axes, confirming full icosahedral symmetry. (c1--c3) CNs along the nid-IQC pathway at the same temperatures, exhibiting reduced symmetry with approximate 6-fold motifs. (d) Atomic arrangement of (c3) viewed along a 2-fold axis and an approximate 6-fold axis, highlighting symmetry reduction. (e) Nucleation barriers for id-IQC and nid-IQC versus $\varepsilon$, showing a crossover near $\varepsilon\approx0.0865$. (f) Energies of the fully developed id-IQC and nid-IQC states versus $\varepsilon$, indicating degenerate thermodynamic stability.
• Figure 5: Energy penalty for density fluctuations ($E_f = \int_{\Omega} \frac{\varepsilon}{2}\varphi^2 dr$) in CNs as a function of temperature parameter $\varepsilon$ for id-IQC (solid line with squares) and nid-IQC (dashed line with circles). At lower temperatures, both types of CNs exhibit similar energy penalties. However, as temperature increases, the energy penalty for id-IQC CNs rises more rapidly than for nid-IQC CNs, creating a significant energy advantage for the lower-symmetry nid-IQC nucleation pathway at higher temperatures. This demonstrates why systems preferentially form CNs with lower symmetry at elevated temperatures.